Research ArticleImplicit Contractive Mappings in Modular Metric andFuzzy Metric Spaces

N Hussain1 and P Salimi2

1 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2 Young Researchers and Elite Club Rasht Branch Islamic Azad University Rasht Iran

Correspondence should be addressed to N Hussain nhusainkauedusa

Received 21 April 2014 Accepted 26 May 2014 Published 5 June 2014

Academic Editor Abdullah Alotaibi

Copyright copy 2014 N Hussain and P Salimi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The notion of modular metric spaces being a natural generalization of classical modulars over linear spaces like Lebesgue OrliczMusielak-Orlicz Lorentz Orlicz-Lorentz and Calderon-Lozanovskii spaces was recently introduced In this paper we investigatethe existence of fixed points of generalized 120572-admissible modular contractive mappings in modular metric spaces As applicationswe derive some new fixed point theorems in partially ordered modular metric spaces Suzuki type fixed point theorems in modularmetric spaces and new fixed point theorems for integral contractions In last section we develop an important relation betweenfuzzy metric and modular metric and deduce certain new fixed point results in triangular fuzzy metric spaces Moreover someexamples are provided here to illustrate the usability of the obtained results

1 Introduction and Basic Definitions

Chistyakov introduced the notion of modular metric spacesin [1 2]Themain idea behind this new concept is the physicalinterpretation of the modular Informally speaking whereasa metric on a set represents nonnegative finite distancesbetween any two points of the set a modular on a setattributes a nonnegative (possibly infinite valued) ldquofield of(generalized) velocitiesrdquo to each ldquotimerdquo 120582 gt 0 the absolutevalue of an average velocity 120596120582(119909 119910) is associated in such away that in order to cover the ldquodistancerdquo between points119909 119910 isin119883 it takes time 120582 to move from 119909 to 119910 with velocity 120596120582(119909 119910)But the way we approached the concept of modular metricspaces is different Indeed we look at these spaces as thenonlinear version of the classical modular spaces introducedby Nakano [3] on vector spaces and modular function spacesintroduced by Musielak [4] and Orlicz [5 6]

For the study of electrorheological fluids (for instancelithiumpolymethacrylate) modelingwith sufficient accuracyusing classical Lebesgue and Sobolev spaces 119871119901 and 119882

1119901where 119901 is a fixed constant is not adequate but rather theexponent 119901 should be able to vary [7] One of the mostinteresting problems in this setting is the famous Dirichlet

energy problem [8 9] The classical technique used so far instudying this problem is to convert the energy functional nat-urally defined by a modular to a convoluted and complicatedproblem which involves a norm (the Luxemburg norm) Themodular metric approach is more natural and has not beenused extensively In recent years there was a strong interestto study the fixed point property in modular function spacesafter the first paper [10] was published in 1990 For more onmetric fixed point theory the reader may consult the book[11] and for modular function spaces [12 13]

Let 119883 be a nonempty set Throughout this paper for afunction 120596 (0infin) times 119883 times 119883 rarr [0infin] we will write

120596120582 (119909 119910) = 120596 (120582 119909 119910) (1)

for all 120582 gt 0 and 119909 119910 isin 119883

Definition 1 (see [1 2]) A function 120596 (0infin) times 119883 times 119883 rarr

[0infin] is said to be modular metric on 119883 if it satisfies thefollowing axioms

(i) 119909 = 119910 if and only if 120596120582(119909 119910) = 0 for all 120582 gt 0(ii) 120596120582(119909 119910) = 120596120582(119910 119909) for all 120582 gt 0 and 119909 119910 isin 119883

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 981578 12 pageshttpdxdoiorg1011552014981578

2 The Scientific World Journal

(iii) 120596120582+120583(119909 119910) le 120596120582(119909 119911) + 120596120583(119911 119910) for all 120582 120583 gt 0 and119909 119910 119911 isin 119883

If instead of (i) we have only the condition (i1015840)

120596120582 (119909 119909) = 0 forall120582 gt 0 119909 isin 119883 (2)

then 120596 is said to be a pseudomodular (metric) on 119883 Amodular metric 120596 on 119883 is said to be regular if the followingweaker version of (i) is satisfied

119909 = 119910 iff 120596120582 (119909 119910) = 0 for some 120582 gt 0 (3)

Finally 120596 is said to be convex if for 120582 120583 gt 0 and 119909 119910 119911 isin 119883it satisfies the inequality

120596120582+120583 (119909 119910) le

120582

120582 + 120583

120596120582 (119909 119911) +

120583

120582 + 120583

120596120583 (119911 119910) (4)

Note that for a metric pseudomodular 120596 on a set 119883 and any119909 119910 isin 119883 the function 120582 rarr 120596120582(119909 119910) is nonincreasing on(0infin) Indeed if 0 lt 120583 lt 120582 then

120596120582 (119909 119910) le 120596120582minus120583 (119909 119909) + 120596120583 (119909 119910) = 120596120583 (119909 119910) (5)

Following example presented by Abdou and Khamsi [14]is an important motivation of the concept of modular metricspaces

Example 2 Let 119883 be a nonempty set and Σ a nontrivial 120590-algebra of subsets of 119883 Let P be a 120575-ring of subsets of 119883such that 119864 cap 119860 isin P for any 119864 isin P and 119860 isin Σ Letus assume that there exists an increasing sequence of sets119870119899 isin P such that119883 = ⋃119870119899 ByEwe denote the linear spaceof all simple functions with supports fromP ByMinfin we willdenote the space of all extended measurable functions thatis all functions 119891 119883 rarr [minusinfininfin] such that there exists asequence 119892119899 sub E |119892119899| le |119891| and 119892119899(119909) rarr 119891(119909) for all119909 isin 119883 By 1119860 we denote the characteristic function of the set119860 Let 120588 Minfin rarr [0infin] be a nontrivial convex and evenfunction We say that 120588 is a regular convex function pseudo-modular if

(i) 120588(0) = 0(ii) 120588 is monotone that is |119891(119909)| le |119892(119909)| for all 119909 isin 119883

implies 120588(119891) le 120588(119892) where 119891 119892 isin Minfin(iii) 120588 is orthogonally subadditive that is 120588(1198911119860cup119861) le

120588(1198911119860) + 120588(1198911119861) for any 119860 119861 isin Σ such that 119860 cap

119861 = 0 119891 isin M(iv) 120588 has the Fatou property that is |119891119899(119909)| uarr |119891(119909)| for

all 119909 isin 119883 implies 120588(119891119899) uarr 120588(119891) where 119891 isin Minfin(v) 120588 is order continuous in E that is 119892119899 isin E and

|119892119899(119909)| darr 0 implies 120588(119892119899) darr 0

Similarly as in the case of measure spaces we say that a set119860 isin Σ is 120588-null if 120588(1198921119860) = 0 for every 119892 isin E We saythat a property holds 120588-almost everywhere if the exceptionalset is 120588-null As usual we identify any pair of measurable setswhose symmetric difference is 120588-null as well as any pair of

measurable functions differing only on a 120588-null set With thisin mind we define

M (119883 ΣP 120588) = 119891 isin Minfin1003816100381610038161003816119891 (119909)

1003816100381610038161003816lt infin120588-ae (6)

where each119891 isin M(119883 ΣP 120588) is actually an equivalence classof functions equal to120588-ae rather than an individual functionWhere no confusion exists we will write M instead ofM(119883 ΣP 120588) Let 120588 be a regular function pseudomodular

(a) We say that 120588 is a regular function semimodular if120588(120572119891) = 0 for every 120572 gt 0 implies 119891 = 0 120588-ae

(b) We say that120588 is a regular functionmodular if120588(119891) = 0implies 119891 = 0 120588-ae

The class of all nonzero regular convex function modularsdefined on 119883 will be denoted by R Let us denote 120588(119891 119864) =120588(1198911119864) for 119891 isin M 119864 isin Σ It is easy to prove that 120588(119891 119864) isa function pseudomodular in the sense of Definition 211 in[13] (see also [15 16]) Let 120588 be a convex function modular

(a) The associated modular function space is the vectorspace 119871120588(119883 Σ) or briefly 119871120588 defined by

119871120588 = 119891 isin M 120588 (120582119891) 997888rarr 0 as 120582 997888rarr 0 (7)

(b) The following formula defines a norm in 119871120588 (fre-quently called Luxemburg norm)

10038171003817100381710038171198911003817100381710038171003817120588= inf 120572 gt 0 120588 (

119891

120572

) le 1 (8)

A modular function space furnishes a wonderful example ofa modular metric space Indeed let 119871120588 be amodular functionspace Define the function modular 120596 by

120596120582 (119891 119892) = 120588(

119891 minus 119892

120582

) (9)

for all 120582 gt 0 and 119891 119892 isin 119871120588 Then 120596 is a modular metric on119871120588 Note that 120596 is convex if and only if 120588 is convex Moreoverwe have

1003817100381710038171003817119891 minus 119892

1003817100381710038171003817120588= 119889lowast

120596(119891 119892) (10)

for any 119891 119892 isin 119871120588

Other easy examples may be found in [1 2]

Definition 3 Let119883120596 be a modular metric space

(1) The sequence (119909119899)119899isinN in119883120596 is said to be120596-convergentto119909 isin 119883120596 if and only if for each120582 gt 0120596120582(119909119899 119909) rarr 0as 119899 rarr infin 119909 will be called the 120596-limit of (119909119899)

(2) The sequence (119909119899)119899isinN in 119883120596 is said to be 120596-Cauchy iffor each 120582 gt 0 120596120582(119909119898 119909119899) rarr 0 as119898 119899 rarr infin

(3) A subset119872 of119883120596 is said to be 120596-closed if the 120596-limitof a 120596-convergent sequence of 119872 always belongs to119872

The Scientific World Journal 3

(4) A subset119872 of 119883120596 is said to be 120596-complete if any 120596-Cauchy sequence in 119872 is a 120596-convergent sequenceand its 120596-limit is in119872

(5) A subset119872 of119883120596 is said to be 120596-bounded if we have

120575120596 (119872) = sup 120596120582 (119909 119910) 119909 119910 isin 119872 lt infin (11)

In 2012 Samet et al [17] introduced the concepts of 120572-120595-contractive and 120572-admissible mappings and establishedvarious fixed point theorems for such mappings defined oncomplete metric spaces Afterwards Salimi et al [18] andHussain et al [19ndash21]modified the notions of120572-120595-contractiveand 120572-admissible mappings and established certain fixedpoint theorems

Definition 4 (see [17]) Let 119879 be self-mapping on 119883 and 120572

119883times119883 rarr [0 +infin) a function One says that 119879 is an 120572-admis-sible mapping if

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119879119909 119879119910) ge 1 (12)

Definition 5 (see [18]) Let 119879 be self-mapping on119883 and 120572 120578 119883 times 119883 rarr [0 +infin) two functions One says that 119879 is an 120572-admissible mapping with respect to 120578 if

119909 119910 isin 119883 120572 (119909 119910) ge 120578 (119909 119910) 997904rArr 120572 (119879119909 119879119910)

ge 120578 (119879119909 119879119910)

(13)

Note that if we take 120578(119909 119910) = 1 then this definition reducesto Definition 4 Also if we take 120572(119909 119910) = 1 then we say that119879 is an 120578-subadmissible mapping

Definition 6 (see [20]) Let (119883 119889) be a metric space Let 120572 120578 119883 times 119883 rarr [0infin) and 119879 119883 rarr 119883 be functions One says 119879is an 120572-120578-continuous mapping on (119883 119889) if for given 119909 isin 119883

and sequence 119909119899 with

119909119899 997888rarr 119909 as 119899 997888rarr infin 120572 (119909119899 119909119899+1) ge 120578 (119909119899 119909119899+1)

forall119899 isin N 997904rArr 119879119909119899 997888rarr 119879119909

(14)

Example 7 (see [20]) Let 119883 = [0infin) and 119889(119909 119910) = |119909 minus 119910|

be a metric on 119883 Assume 119879 119883 rarr 119883 and 120572 120578 119883 times 119883 rarr

[0 +infin) are defined by

119879119909 =

1199095 if 119909 isin [0 1]

sin120587119909 + 2 if (1infin)

120572 (119909 119910) =

1199092+ 1199102+ 1 if 119909 119910 isin [0 1]

0 otherwise

(15)

and 120578(119909 119910) = 1199092 Clearly 119879 is not continuous but 119879 is 120572-120578-

continuous on (119883 119889)

A mapping 119879 119883 rarr 119883 is called orbitally continuous at119901 isin 119883 if lim119899rarrinfin119879

119899119909 = 119901 implies that lim119899rarrinfin119879119879

119899119909 = 119879119901

The mapping 119879 is orbitally continuous on 119883 if 119879 is orbitallycontinuous for all 119901 isin 119883

Remark 8 (see [20]) Let 119879 119883 rarr 119883 be self-mapping on anorbitally 119879-complete metric space 119883 Define 120572 120578 119883 times 119883 rarr

[0 +infin) by

120572 (119909 119910) =

3 if 119909 119910 isin 119874 (119908)0 otherwise

120578 (119909 119910) = 1 (16)

where 119874(119908) is an orbit of a point 119908 isin 119883 If 119879 119883 rarr 119883 is anorbitally continuous map on (119883 119889) then 119879 is 120572-120578-continuouson (119883 119889)

In this paper we investigate existence and uniqueness offixed points of generalized 120572-admissible modular contractivemappings in modular metric spaces As applications wederive some new fixed point theorems in partially orderedmodular metric spaces Suzuki type fixed point theorems inmodular metric spaces and new fixed point theorems forintegral contractions At the end we develop an importantrelation between fuzzy metric and modular metric anddeduce certain new fixed point results in triangular fuzzymetric spaces Moreover some examples are provided hereto illustrate the usability of the obtained results

2 Fixed Point Results forImplicit Contractions

Let us first start this section with a definition of a family offunctions

Definition 9 Assume that ΔH denotes the collection of allcontinuous functionsH R+

6

rarr R satisfying the following

(H1) H is increasing in its 1th variable and nonincreasingin its 5th variable

(H2) if 119906 V isin R+ with 119906 V gt 0 andH(119906 V V 119906 V+119906 0) le 0then there exists 120595 isin Ψ such that

119906 le 120595 (V) (17)

Notice that here we denote with Ψ the family of non-decreasing functions 120595 [0 +infin) rarr [0 +infin) such thatsuminfin

119899=1120595119899(119905) lt +infin for all 119905 gt 0 where 120595119899 is the 119899th iterate

of 120595

Example 10 Let H(1199051 1199052 1199053 1199054 1199055 1199056) = 1199051 minus 120595(max1199052 (1199053 +1199054)2 (1199055 + 1199056)2) where 120595 isin Ψ thenH isin ΔH

Example 11 Let H(1199051 1199052 1199053 1199054 1199055 1199056) = 1199051 minus 120595(max1199052 (1 +1199053)1199054(1 + 1199052)) where 120595 isin Ψ thenH isin ΔH

Theorem 12 Let 119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

4 The Scientific World Journal

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(18)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Let 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090) For1199090 isin 119883 we define the sequence 119909119899 by 119909119899 = 119879

1198991199090 = 119879119909119899

Now since 119879 is an 120572-admissible mapping with respect to 120578then 120572(1199090 1199091) = 120572(1199090 1198791199090) ge 120578(1199090 1198791199090) = 120578(1199090 1199091) Bycontinuing this process we have

120578 (119909119899minus1 119879119909119899minus1) = 120578 (119909119899minus1 119909119899) le 120572 (119909119899minus1 119909119899) (19)

for all 119899 isin N Also let there exists 1198990 isin 119883 such that 1199091198990 =1199091198990+1

Then 1199091198990is fixed point of 119879 and we have nothing to

prove Hence we assume 119909119899 = 119909119899+1 for all 119899 isin N cup 0 Nowby taking 119909 = 119909119899minus1 and 119910 = 119909119899 in (iv) we get

H (120596120582119888 (119879119909119899minus1 119879119909119899) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119879119909119899minus1)

120596120582119897 (119909119899 119879119909119899) 1205962120582119897 (119909119899minus1 119879119909119899) 120596120582119897 (119909119899 119879119909119899minus1)) le 0

(20)

which implies

H (120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119909119899)

120596120582119897 (119909119899 119909119899+1) 1205962120582119897 (119909119899minus1 119909119899+1) 0) le 0

(21)

On the other hand

1205962120582119897 (119909119899minus1 119909119899+1) le 120596120582119897 (119909119899minus1 119909119899) + 120596120582119897 (119909119899 119909119899+1)

le 120596120582119897 (119909119899minus1 119909119899) + 120596120582119888 (119909119899 119909119899+1)

120596120582119897 (119909119899 119909119899+1) le 120596120582119888 (119909119899 119909119899+1)

(22)

Now since H is nonincreasing in its 5th variable so by(21) and (22) we obtain

H (120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119909119899)

120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899)

+120596120582119888 (119909119899 119909119899+1) 0) le 0

(23)

From (H1) we deduce that

120596120582119888 (119909119899 119909119899+1) le 120595 (120596120582119897 (119909119899minus1 119909119899)) le 120595 (120596120582119888 (119909119899minus1 119909119899))

(24)

Inductively we obtain

120596120582119888 (119909119899 119909119899+1) le 120595119899(120596120582119888 (1199090 1199091)) (25)

Taking limit as 119899 rarr infin in the above inequality we get

lim119899rarrinfin

120596120582119888 (119909119899 119909119899+1) = 0 (26)

Suppose119898 119899 isin N with119898 gt 119899 and 120598 gt 0 be given Then thereexists 119899120582(119898minus119899) isin N such that

120596120582119888(119898minus119899) (119909119899 119909119899+1) lt

120598

119888 (119898 minus 119899)(27)

for all 119899 ge 119899120582(119898minus119899) Therefore we get

120596120582119888 (119909119899 119909119898) le 120596120582119888(119898minus119899) (119909119899 119909119899+1)

+ 120596120582119888(119898minus119899) (119909119899+1 119909119899+2) + sdot sdot sdot

+ 120596120582119888(119898minus119899) (119909119898minus1 119909119898)

lt

120598

119888 (119898 minus 119899)

+

120598

119888 (119898 minus 119899)

+

120598

119888 (119898 minus 119899)

+ sdot sdot sdot

+

120598

119888 (119898 minus 119899)

=

120598

119888

(28)

for all 119898 119899 ge 119899120582(119898minus119899) This shows that 119909119899 is a Cauchysequence Since119883120596 is complete so there exists 119909lowast isin 119883120596 suchthat lim119899rarrinfin119909119899 = 119909

lowast Now since 119879 is an 120572-120578-continuousmapping so 119879119909119899 rarr 119879119909

lowast as 119899 rarr infin Therefore

119879119909lowast= lim119899rarrinfin

119879119909119899 = lim119899rarrinfin

119909119899+1 = 119909lowast (29)

Thus 119879 has a fixed point Let all 119909 119910 isin Fix(119879) we have120578(119909 119909) le 120572(119909 119910) and H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0Then by (iv)

H (120596120582119897 (119909 119910) 120596120582119897 (119909 119910) 0 0 120596120582119897 (119909 119910) 120596120582119897 (119910 119909))

le H (120596120582119888 (119909 119910) 120596120582119897 (119909 119910) 0 0

1205962120582119897 (119909 119910) 120596120582119897 (119910 119909))

= H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909)

120596120582119897 (119910 119879119910) 1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(30)

Now if 120596120582119897(119909 119910) gt 0 then

0 lt H (120596120582119897 (119909 119910) 120596120582119897 (119909 119910) 0 0

120596120582119897 (119909 119910) 120596120582119897 (119910 119909)) le 0

(31)

which is a contradictionHence120596120582119897(119909 119910) = 0That is 119909 = 119910Thus 119879 has a unique fixed point

The Scientific World Journal 5

Corollary 13 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 a self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 1(iii) 119879 is an 120572-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 120572(119909 119910) ge 1 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(32)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 1 le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then119879 has a unique fixed point

Theorem 14 Let 119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(33)

holds for all 119899 isin N(iv) condition (iv) of Theorem 12 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof As in proof of Theorem 12 we can deduce a sequence119909119899 such that 119909119899+1 = 119879119909119899 with 120572(119909119899 119909119899+1) ge 120578(119909119899 119909119899+1) andthere exists 119909lowast isin 119883120596 such that 119909119899 rarr 119909

lowast as 119899 rarr infin From(iii) either

120578 (119909119899 119879119909119899) le 120572 (119909119899 119909lowast) or

120578 (119909119899+1 119879119909119899+1) le 120572 (119909119899+1 119909lowast)

(34)

holds for all 119899 isin N Let 120578(119909119899 119879119909119899) le 120572(119909119899 119909lowast)Then by taking

119909 = 119909119899 and 119910 = 119909lowast in (iv) we have

H (120596120582119888 (119879119909119899 119879119909lowast) 120596120582119897 (119909119899 119909

lowast) 120596120582119897 (119909119899 119879119909119899)

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909119899 119879119909

lowast) 120596120582119897 (119909

lowast 119879119909119899)) le 0

(35)which implies

H (120596120582119888 (119909119899+1 119879119909lowast) 120596120582119897 (119909119899 119909

lowast)

120596120582119897 (119909119899 119909119899+1) 120596120582119897 (119909lowast 119879119909lowast)

1205962120582119897 (119909119899 119879119909lowast) 120596120582119897 (119909

lowast 119909119899+1)) le 0

(36)

Taking limit as 119899 rarr infin in the above inequality we obtain

H (120596120582119888 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909

lowast 119879119909lowast) 0) le 0

(37)

Now since 1205962120582119897(119909lowast 119879119909lowast) lt 120596120582119897(119909

lowast 119879119909lowast) 120596120582119897(119909

lowast 119879119909lowast) lt

120596120582119888(119909lowast 119879119909lowast) H is increasing in its 1th variable and nonin-

creasing in its 5th variable so we obtain

H (120596120582119897 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 120596120582119897 (119909

lowast 119879119909lowast) 0) le 0

(38)

which is a contradiction Now by taking 119906 = 120596120582119897(119909lowast 119879119909lowast)

and V = 0 from (H2) we have

120596120582119897 (119909lowast 119879119909lowast) le 120595 (0) = 0 (39)

Hence 120596120582119897(119909lowast 119879119909lowast) = 0 that is 119909lowast = 119879119909lowast Similarly we can

deduce that 119879119909lowast = 119909lowast when 120578120582(119909119899+1 119879119909119899+1) le 120572120582(119909119899+1 119909

lowast)

By using Example 10 and Theorem 14 we can obtain thefollowing corollary

Corollary 15 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(40)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 120595(max120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(41)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

6 The Scientific World Journal

Corollary 16 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(42)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(43)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Corollary 17 Let119883120596 be a complete modular metric space Let119879 119883120596 rarr 119883120596 be self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(44)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(45)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Example 18 Let 119883120596 = [0 +infin) and 120596120582(119909 119910) = (1120582)|119909 minus 119910|Define 119879 119883120596 rarr 119883120596 120572 120578 119883120596 times 119883120596 rarr [0infin) and 120595

[0infin) rarr [0infin) by

119879119909 =

1

16

1199092 if 119909 isin [0 1]

sin2119909 + cos119909 + 1radic1199092+ 1

if 119909 isin (1 2]

7119909 + ln119909 if 119909 isin (2infin)

120572 (119909 119910) =

1

2

if 119909 119910 isin [0 1]

1

8

otherwise

120578 (119909 119910) =

1

4

120595 (119905) =

1

2

119905

(46)

Let 120572(119909 119910) ge 120578(119909 119910) then 119909 119910 isin [0 1] On the otherhand 119879119908 isin [0 1] for all 119908 isin [0 1] Then 120572(119879119909 119879119910) ge

120578(119879119909 119879119910) That is 119879 is an 120572-admissible mapping with respectto 120578 If 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then 119879119909119899 1198792119909119899 1198793119909119899 isin

[0 1] for all 119899 isin N That is

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909)

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(47)

hold for all 119899 isin N Clearly 120572(0 1198790) ge 120578(0 1198790) Let 120572(119909 119910) ge120578(119909 119879119909) Now if 119909 notin [0 1] or 119910 notin [0 1] then 18 ge 14which is a contradiction So 119909 119910 isin [0 1] Therefore

1205961205822 (119879119909 119879119910) =

2

120582

1

16

100381610038161003816100381610038161199092minus 119910210038161003816100381610038161003816

=

1

8120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 + 119910

1003816100381610038161003816le

1

4120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

1

120582 (12)

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

120596120582(12) (119909 119910) = 120595 (120596120582(12) (119909 119910))

le 120595(max120596120582119897 (119909 119910)

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(48)

Hence all conditions of Corollary 16 hold and 119879 has a uniquefixed point

Let (119883120596 ⪯) be a partially ordered modular metric spaceRecall that 119879 119883120596 rarr 119883120596 is nondecreasing if for all 119909119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) Fixed point theoremsfor monotone operators in ordered metric spaces are widely

The Scientific World Journal 7

investigated and have found various applications in differ-ential and integral equations (see [20 22ndash25] and referencestherein) From results proved above we derive the followingnew results in partially ordered modular metric spaces

Theorem 19 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfy-ing the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(49)

where 0 lt l lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 20 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(50)

holds for all 119899 isin N(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(51)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Corollary 21 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(52)

holds for all 119899 isin N

(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 wehave

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(53)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 22 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(54)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(55)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 23 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(56)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 119886120596120582119897 (119909 119910) + 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(57)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

8 The Scientific World Journal

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

3 Suzuki Type Fixed Point Results inModular Metric Spaces

In 2008 Suzuki proved a remarkable fixedpoint theorem thatis a generalization of the Banach contraction principle andcharacterizes themetric completeness Consequently a num-ber of extensions and generalizations of this result appearedin the literature (see [26ndash30] and references therein) As anapplication of our results proved abovewededuce Suzuki typefixed point theorems in the setting of modular metric spaces

Theorem 24 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

120596120582119897 (119909 119879119909) le 120596120582119897 (119909 119910)

997904rArr H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(58)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has afixed point Moreover if H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Define 120572 120578 (0infin) times 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 120596120582119897 (119909 119910) 120578 (119909 119910) = 120596120582119897 (119909 119910) (59)

Clearly 120578(119909 119910) le 120572(119909 119910) for all 119909 119910 isin 119883120596 and 120582 gt 0Since 119879 is continuous 119879 is 120572-120578-continuous Thus conditions(i)ndash(iii) of Theorem 12 hold Let 120578(119909 119879119909) le 120572(119909 119910) Then120596120582119897(119909 119879119909) le 120596120582119897(119909 119910) So from (58) we obtain

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(60)

Therefore all conditions of Theorem 12 hold and 119879 has aunique fixed point

Theorem 25 Let119883120596 be a complete modular metric space and119879 self-mapping on 119883 Assume that

1 minus 119887

1 minus 119887 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(61)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Proof Define 120572 120578 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 1205961205822119897 (119909 119910) 120578 (119909 119910) = 119896120596120582119897 (119909 119910) (62)

where 119896 = (1 minus 119887)(1 minus 119887 + 119886) Clearly 120578(119909 119910) le 120572(119909 119910) for all119909 119910 isin 119883120596 and 120582 gt 0 Then conditions (i)-(ii) of Corollary 17hold Let 119909119899 be a sequence such that 119909119899 rarr 119909 as 119899 rarr infinSince 119896120596120582119897(119879119909119899 119879

2119909119899) le 1205961205822119897(119879119909119899 119879

2119909119899) for all 119899 isin N from

(61) we obtain

120596120582119897 (1198792119909119899 1198793119909119899)

le 120596120582119888 (1198792119909119899 1198793119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899)

+ 119887

[1 + 120596120582119897 (119879119909119899 1198792119909119899)] 120596120582119897 (119879

2119909119899 1198793119909119899)

1 + 120596120582119897 (119879119909119899 1198792119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899) + 119887120596120582119897 (119879

2119909119899 1198793119909119899)

(63)

which implies

120596120582119897 (1198792119909119899 1198793119909119899) le

119886

1 minus 119887

120596120582119897 (119879119909119899 1198792119909119899) (64)

Suppose there exists 1198990 isin N such that

120578 (1198791199091198990 11987921199091198990

) gt 120572 (1198791199091198990 119909) or

120578 (11987921199091198990

11987931199091198990

) gt 120572 (11987921199091198990

119909)

(65)

then119896120596120582119897 (1198791199091198990

11987921199091198990

) gt 1205961205822119897 (1198791199091198990 119909) or

119896120596120582119897 (11987921199091198990

11987931199091198990

) gt 1205961205822119897 (11987921199091198990

119909)

(66)

Therefore

120596120582119897 (1198791199091198990 11987921199091198990

)

le 1205961205822119897 (1198791199091198990 119909) + 1205961205822119897 (119879

21199091198990

119909)

lt 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896120596120582119897 (11987921199091198990

11987931199091198990

)

le 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896

119886

1 minus 119887

120596120582119897 (1198791199091198990 11987921199091198990

)

le (119896 + 119896

119886

1 minus 119887

)120596120582119897 (1198791199091198990 11987921199091198990

)

= 120596120582119897 (1198791199091198990 11987921199091198990

)

(67)

which is a contradiction Hence (iii) of Corollary 17 holdsThus all conditions of Corollary 17 hold and 119879 has a uniquefixed point

Corollary 26 Let119883120596 be a complete modularmetric space and119879 self-mapping on119883 Assume that

1

1 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

(68)

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

The Scientific World Journal 3

(4) A subset119872 of 119883120596 is said to be 120596-complete if any 120596-Cauchy sequence in 119872 is a 120596-convergent sequenceand its 120596-limit is in119872

(5) A subset119872 of119883120596 is said to be 120596-bounded if we have

120575120596 (119872) = sup 120596120582 (119909 119910) 119909 119910 isin 119872 lt infin (11)

In 2012 Samet et al [17] introduced the concepts of 120572-120595-contractive and 120572-admissible mappings and establishedvarious fixed point theorems for such mappings defined oncomplete metric spaces Afterwards Salimi et al [18] andHussain et al [19ndash21]modified the notions of120572-120595-contractiveand 120572-admissible mappings and established certain fixedpoint theorems

Definition 4 (see [17]) Let 119879 be self-mapping on 119883 and 120572

119883times119883 rarr [0 +infin) a function One says that 119879 is an 120572-admis-sible mapping if

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119879119909 119879119910) ge 1 (12)

Definition 5 (see [18]) Let 119879 be self-mapping on119883 and 120572 120578 119883 times 119883 rarr [0 +infin) two functions One says that 119879 is an 120572-admissible mapping with respect to 120578 if

119909 119910 isin 119883 120572 (119909 119910) ge 120578 (119909 119910) 997904rArr 120572 (119879119909 119879119910)

ge 120578 (119879119909 119879119910)

(13)

Note that if we take 120578(119909 119910) = 1 then this definition reducesto Definition 4 Also if we take 120572(119909 119910) = 1 then we say that119879 is an 120578-subadmissible mapping

Definition 6 (see [20]) Let (119883 119889) be a metric space Let 120572 120578 119883 times 119883 rarr [0infin) and 119879 119883 rarr 119883 be functions One says 119879is an 120572-120578-continuous mapping on (119883 119889) if for given 119909 isin 119883

and sequence 119909119899 with

119909119899 997888rarr 119909 as 119899 997888rarr infin 120572 (119909119899 119909119899+1) ge 120578 (119909119899 119909119899+1)

forall119899 isin N 997904rArr 119879119909119899 997888rarr 119879119909

(14)

Example 7 (see [20]) Let 119883 = [0infin) and 119889(119909 119910) = |119909 minus 119910|

be a metric on 119883 Assume 119879 119883 rarr 119883 and 120572 120578 119883 times 119883 rarr

[0 +infin) are defined by

119879119909 =

1199095 if 119909 isin [0 1]

sin120587119909 + 2 if (1infin)

120572 (119909 119910) =

1199092+ 1199102+ 1 if 119909 119910 isin [0 1]

0 otherwise

(15)

and 120578(119909 119910) = 1199092 Clearly 119879 is not continuous but 119879 is 120572-120578-

continuous on (119883 119889)

A mapping 119879 119883 rarr 119883 is called orbitally continuous at119901 isin 119883 if lim119899rarrinfin119879

119899119909 = 119901 implies that lim119899rarrinfin119879119879

119899119909 = 119879119901

The mapping 119879 is orbitally continuous on 119883 if 119879 is orbitallycontinuous for all 119901 isin 119883

Remark 8 (see [20]) Let 119879 119883 rarr 119883 be self-mapping on anorbitally 119879-complete metric space 119883 Define 120572 120578 119883 times 119883 rarr

[0 +infin) by

120572 (119909 119910) =

3 if 119909 119910 isin 119874 (119908)0 otherwise

120578 (119909 119910) = 1 (16)

where 119874(119908) is an orbit of a point 119908 isin 119883 If 119879 119883 rarr 119883 is anorbitally continuous map on (119883 119889) then 119879 is 120572-120578-continuouson (119883 119889)

In this paper we investigate existence and uniqueness offixed points of generalized 120572-admissible modular contractivemappings in modular metric spaces As applications wederive some new fixed point theorems in partially orderedmodular metric spaces Suzuki type fixed point theorems inmodular metric spaces and new fixed point theorems forintegral contractions At the end we develop an importantrelation between fuzzy metric and modular metric anddeduce certain new fixed point results in triangular fuzzymetric spaces Moreover some examples are provided hereto illustrate the usability of the obtained results

2 Fixed Point Results forImplicit Contractions

Let us first start this section with a definition of a family offunctions

Definition 9 Assume that ΔH denotes the collection of allcontinuous functionsH R+

6

rarr R satisfying the following

(H1) H is increasing in its 1th variable and nonincreasingin its 5th variable

(H2) if 119906 V isin R+ with 119906 V gt 0 andH(119906 V V 119906 V+119906 0) le 0then there exists 120595 isin Ψ such that

119906 le 120595 (V) (17)

Notice that here we denote with Ψ the family of non-decreasing functions 120595 [0 +infin) rarr [0 +infin) such thatsuminfin

119899=1120595119899(119905) lt +infin for all 119905 gt 0 where 120595119899 is the 119899th iterate

of 120595

Example 10 Let H(1199051 1199052 1199053 1199054 1199055 1199056) = 1199051 minus 120595(max1199052 (1199053 +1199054)2 (1199055 + 1199056)2) where 120595 isin Ψ thenH isin ΔH

Example 11 Let H(1199051 1199052 1199053 1199054 1199055 1199056) = 1199051 minus 120595(max1199052 (1 +1199053)1199054(1 + 1199052)) where 120595 isin Ψ thenH isin ΔH

Theorem 12 Let 119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

4 The Scientific World Journal

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(18)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Let 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090) For1199090 isin 119883 we define the sequence 119909119899 by 119909119899 = 119879

1198991199090 = 119879119909119899

Now since 119879 is an 120572-admissible mapping with respect to 120578then 120572(1199090 1199091) = 120572(1199090 1198791199090) ge 120578(1199090 1198791199090) = 120578(1199090 1199091) Bycontinuing this process we have

120578 (119909119899minus1 119879119909119899minus1) = 120578 (119909119899minus1 119909119899) le 120572 (119909119899minus1 119909119899) (19)

for all 119899 isin N Also let there exists 1198990 isin 119883 such that 1199091198990 =1199091198990+1

Then 1199091198990is fixed point of 119879 and we have nothing to

prove Hence we assume 119909119899 = 119909119899+1 for all 119899 isin N cup 0 Nowby taking 119909 = 119909119899minus1 and 119910 = 119909119899 in (iv) we get

H (120596120582119888 (119879119909119899minus1 119879119909119899) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119879119909119899minus1)

120596120582119897 (119909119899 119879119909119899) 1205962120582119897 (119909119899minus1 119879119909119899) 120596120582119897 (119909119899 119879119909119899minus1)) le 0

(20)

which implies

H (120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119909119899)

120596120582119897 (119909119899 119909119899+1) 1205962120582119897 (119909119899minus1 119909119899+1) 0) le 0

(21)

On the other hand

1205962120582119897 (119909119899minus1 119909119899+1) le 120596120582119897 (119909119899minus1 119909119899) + 120596120582119897 (119909119899 119909119899+1)

le 120596120582119897 (119909119899minus1 119909119899) + 120596120582119888 (119909119899 119909119899+1)

120596120582119897 (119909119899 119909119899+1) le 120596120582119888 (119909119899 119909119899+1)

(22)

Now since H is nonincreasing in its 5th variable so by(21) and (22) we obtain

H (120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119909119899)

120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899)

+120596120582119888 (119909119899 119909119899+1) 0) le 0

(23)

From (H1) we deduce that

120596120582119888 (119909119899 119909119899+1) le 120595 (120596120582119897 (119909119899minus1 119909119899)) le 120595 (120596120582119888 (119909119899minus1 119909119899))

(24)

Inductively we obtain

120596120582119888 (119909119899 119909119899+1) le 120595119899(120596120582119888 (1199090 1199091)) (25)

Taking limit as 119899 rarr infin in the above inequality we get

lim119899rarrinfin

120596120582119888 (119909119899 119909119899+1) = 0 (26)

Suppose119898 119899 isin N with119898 gt 119899 and 120598 gt 0 be given Then thereexists 119899120582(119898minus119899) isin N such that

120596120582119888(119898minus119899) (119909119899 119909119899+1) lt

120598

119888 (119898 minus 119899)(27)

for all 119899 ge 119899120582(119898minus119899) Therefore we get

120596120582119888 (119909119899 119909119898) le 120596120582119888(119898minus119899) (119909119899 119909119899+1)

+ 120596120582119888(119898minus119899) (119909119899+1 119909119899+2) + sdot sdot sdot

+ 120596120582119888(119898minus119899) (119909119898minus1 119909119898)

lt

120598

119888 (119898 minus 119899)

+

120598

119888 (119898 minus 119899)

+

120598

119888 (119898 minus 119899)

+ sdot sdot sdot

+

120598

119888 (119898 minus 119899)

=

120598

119888

(28)

for all 119898 119899 ge 119899120582(119898minus119899) This shows that 119909119899 is a Cauchysequence Since119883120596 is complete so there exists 119909lowast isin 119883120596 suchthat lim119899rarrinfin119909119899 = 119909

lowast Now since 119879 is an 120572-120578-continuousmapping so 119879119909119899 rarr 119879119909

lowast as 119899 rarr infin Therefore

119879119909lowast= lim119899rarrinfin

119879119909119899 = lim119899rarrinfin

119909119899+1 = 119909lowast (29)

Thus 119879 has a fixed point Let all 119909 119910 isin Fix(119879) we have120578(119909 119909) le 120572(119909 119910) and H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0Then by (iv)

H (120596120582119897 (119909 119910) 120596120582119897 (119909 119910) 0 0 120596120582119897 (119909 119910) 120596120582119897 (119910 119909))

le H (120596120582119888 (119909 119910) 120596120582119897 (119909 119910) 0 0

1205962120582119897 (119909 119910) 120596120582119897 (119910 119909))

= H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909)

120596120582119897 (119910 119879119910) 1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(30)

Now if 120596120582119897(119909 119910) gt 0 then

0 lt H (120596120582119897 (119909 119910) 120596120582119897 (119909 119910) 0 0

120596120582119897 (119909 119910) 120596120582119897 (119910 119909)) le 0

(31)

which is a contradictionHence120596120582119897(119909 119910) = 0That is 119909 = 119910Thus 119879 has a unique fixed point

The Scientific World Journal 5

Corollary 13 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 a self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 1(iii) 119879 is an 120572-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 120572(119909 119910) ge 1 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(32)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 1 le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then119879 has a unique fixed point

Theorem 14 Let 119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(33)

holds for all 119899 isin N(iv) condition (iv) of Theorem 12 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof As in proof of Theorem 12 we can deduce a sequence119909119899 such that 119909119899+1 = 119879119909119899 with 120572(119909119899 119909119899+1) ge 120578(119909119899 119909119899+1) andthere exists 119909lowast isin 119883120596 such that 119909119899 rarr 119909

lowast as 119899 rarr infin From(iii) either

120578 (119909119899 119879119909119899) le 120572 (119909119899 119909lowast) or

120578 (119909119899+1 119879119909119899+1) le 120572 (119909119899+1 119909lowast)

(34)

holds for all 119899 isin N Let 120578(119909119899 119879119909119899) le 120572(119909119899 119909lowast)Then by taking

119909 = 119909119899 and 119910 = 119909lowast in (iv) we have

H (120596120582119888 (119879119909119899 119879119909lowast) 120596120582119897 (119909119899 119909

lowast) 120596120582119897 (119909119899 119879119909119899)

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909119899 119879119909

lowast) 120596120582119897 (119909

lowast 119879119909119899)) le 0

(35)which implies

H (120596120582119888 (119909119899+1 119879119909lowast) 120596120582119897 (119909119899 119909

lowast)

120596120582119897 (119909119899 119909119899+1) 120596120582119897 (119909lowast 119879119909lowast)

1205962120582119897 (119909119899 119879119909lowast) 120596120582119897 (119909

lowast 119909119899+1)) le 0

(36)

Taking limit as 119899 rarr infin in the above inequality we obtain

H (120596120582119888 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909

lowast 119879119909lowast) 0) le 0

(37)

Now since 1205962120582119897(119909lowast 119879119909lowast) lt 120596120582119897(119909

lowast 119879119909lowast) 120596120582119897(119909

lowast 119879119909lowast) lt

120596120582119888(119909lowast 119879119909lowast) H is increasing in its 1th variable and nonin-

creasing in its 5th variable so we obtain

H (120596120582119897 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 120596120582119897 (119909

lowast 119879119909lowast) 0) le 0

(38)

which is a contradiction Now by taking 119906 = 120596120582119897(119909lowast 119879119909lowast)

and V = 0 from (H2) we have

120596120582119897 (119909lowast 119879119909lowast) le 120595 (0) = 0 (39)

Hence 120596120582119897(119909lowast 119879119909lowast) = 0 that is 119909lowast = 119879119909lowast Similarly we can

deduce that 119879119909lowast = 119909lowast when 120578120582(119909119899+1 119879119909119899+1) le 120572120582(119909119899+1 119909

lowast)

By using Example 10 and Theorem 14 we can obtain thefollowing corollary

Corollary 15 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(40)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 120595(max120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(41)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

6 The Scientific World Journal

Corollary 16 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(42)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(43)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Corollary 17 Let119883120596 be a complete modular metric space Let119879 119883120596 rarr 119883120596 be self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(44)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(45)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Example 18 Let 119883120596 = [0 +infin) and 120596120582(119909 119910) = (1120582)|119909 minus 119910|Define 119879 119883120596 rarr 119883120596 120572 120578 119883120596 times 119883120596 rarr [0infin) and 120595

[0infin) rarr [0infin) by

119879119909 =

1

16

1199092 if 119909 isin [0 1]

sin2119909 + cos119909 + 1radic1199092+ 1

if 119909 isin (1 2]

7119909 + ln119909 if 119909 isin (2infin)

120572 (119909 119910) =

1

2

if 119909 119910 isin [0 1]

1

8

otherwise

120578 (119909 119910) =

1

4

120595 (119905) =

1

2

119905

(46)

Let 120572(119909 119910) ge 120578(119909 119910) then 119909 119910 isin [0 1] On the otherhand 119879119908 isin [0 1] for all 119908 isin [0 1] Then 120572(119879119909 119879119910) ge

120578(119879119909 119879119910) That is 119879 is an 120572-admissible mapping with respectto 120578 If 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then 119879119909119899 1198792119909119899 1198793119909119899 isin

[0 1] for all 119899 isin N That is

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909)

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(47)

hold for all 119899 isin N Clearly 120572(0 1198790) ge 120578(0 1198790) Let 120572(119909 119910) ge120578(119909 119879119909) Now if 119909 notin [0 1] or 119910 notin [0 1] then 18 ge 14which is a contradiction So 119909 119910 isin [0 1] Therefore

1205961205822 (119879119909 119879119910) =

2

120582

1

16

100381610038161003816100381610038161199092minus 119910210038161003816100381610038161003816

=

1

8120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 + 119910

1003816100381610038161003816le

1

4120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

1

120582 (12)

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

120596120582(12) (119909 119910) = 120595 (120596120582(12) (119909 119910))

le 120595(max120596120582119897 (119909 119910)

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(48)

Hence all conditions of Corollary 16 hold and 119879 has a uniquefixed point

Let (119883120596 ⪯) be a partially ordered modular metric spaceRecall that 119879 119883120596 rarr 119883120596 is nondecreasing if for all 119909119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) Fixed point theoremsfor monotone operators in ordered metric spaces are widely

The Scientific World Journal 7

investigated and have found various applications in differ-ential and integral equations (see [20 22ndash25] and referencestherein) From results proved above we derive the followingnew results in partially ordered modular metric spaces

Theorem 19 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfy-ing the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(49)

where 0 lt l lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 20 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(50)

holds for all 119899 isin N(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(51)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Corollary 21 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(52)

holds for all 119899 isin N

(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 wehave

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(53)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 22 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(54)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(55)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 23 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(56)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 119886120596120582119897 (119909 119910) + 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(57)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

8 The Scientific World Journal

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

3 Suzuki Type Fixed Point Results inModular Metric Spaces

In 2008 Suzuki proved a remarkable fixedpoint theorem thatis a generalization of the Banach contraction principle andcharacterizes themetric completeness Consequently a num-ber of extensions and generalizations of this result appearedin the literature (see [26ndash30] and references therein) As anapplication of our results proved abovewededuce Suzuki typefixed point theorems in the setting of modular metric spaces

Theorem 24 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

120596120582119897 (119909 119879119909) le 120596120582119897 (119909 119910)

997904rArr H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(58)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has afixed point Moreover if H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Define 120572 120578 (0infin) times 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 120596120582119897 (119909 119910) 120578 (119909 119910) = 120596120582119897 (119909 119910) (59)

Clearly 120578(119909 119910) le 120572(119909 119910) for all 119909 119910 isin 119883120596 and 120582 gt 0Since 119879 is continuous 119879 is 120572-120578-continuous Thus conditions(i)ndash(iii) of Theorem 12 hold Let 120578(119909 119879119909) le 120572(119909 119910) Then120596120582119897(119909 119879119909) le 120596120582119897(119909 119910) So from (58) we obtain

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(60)

Therefore all conditions of Theorem 12 hold and 119879 has aunique fixed point

Theorem 25 Let119883120596 be a complete modular metric space and119879 self-mapping on 119883 Assume that

1 minus 119887

1 minus 119887 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(61)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Proof Define 120572 120578 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 1205961205822119897 (119909 119910) 120578 (119909 119910) = 119896120596120582119897 (119909 119910) (62)

where 119896 = (1 minus 119887)(1 minus 119887 + 119886) Clearly 120578(119909 119910) le 120572(119909 119910) for all119909 119910 isin 119883120596 and 120582 gt 0 Then conditions (i)-(ii) of Corollary 17hold Let 119909119899 be a sequence such that 119909119899 rarr 119909 as 119899 rarr infinSince 119896120596120582119897(119879119909119899 119879

2119909119899) le 1205961205822119897(119879119909119899 119879

2119909119899) for all 119899 isin N from

(61) we obtain

120596120582119897 (1198792119909119899 1198793119909119899)

le 120596120582119888 (1198792119909119899 1198793119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899)

+ 119887

[1 + 120596120582119897 (119879119909119899 1198792119909119899)] 120596120582119897 (119879

2119909119899 1198793119909119899)

1 + 120596120582119897 (119879119909119899 1198792119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899) + 119887120596120582119897 (119879

2119909119899 1198793119909119899)

(63)

which implies

120596120582119897 (1198792119909119899 1198793119909119899) le

119886

1 minus 119887

120596120582119897 (119879119909119899 1198792119909119899) (64)

Suppose there exists 1198990 isin N such that

120578 (1198791199091198990 11987921199091198990

) gt 120572 (1198791199091198990 119909) or

120578 (11987921199091198990

11987931199091198990

) gt 120572 (11987921199091198990

119909)

(65)

then119896120596120582119897 (1198791199091198990

11987921199091198990

) gt 1205961205822119897 (1198791199091198990 119909) or

119896120596120582119897 (11987921199091198990

11987931199091198990

) gt 1205961205822119897 (11987921199091198990

119909)

(66)

Therefore

120596120582119897 (1198791199091198990 11987921199091198990

)

le 1205961205822119897 (1198791199091198990 119909) + 1205961205822119897 (119879

21199091198990

119909)

lt 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896120596120582119897 (11987921199091198990

11987931199091198990

)

le 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896

119886

1 minus 119887

120596120582119897 (1198791199091198990 11987921199091198990

)

le (119896 + 119896

119886

1 minus 119887

)120596120582119897 (1198791199091198990 11987921199091198990

)

= 120596120582119897 (1198791199091198990 11987921199091198990

)

(67)

which is a contradiction Hence (iii) of Corollary 17 holdsThus all conditions of Corollary 17 hold and 119879 has a uniquefixed point

Corollary 26 Let119883120596 be a complete modularmetric space and119879 self-mapping on119883 Assume that

1

1 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

(68)

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

4 The Scientific World Journal

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(18)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Let 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090) For1199090 isin 119883 we define the sequence 119909119899 by 119909119899 = 119879

1198991199090 = 119879119909119899

Now since 119879 is an 120572-admissible mapping with respect to 120578then 120572(1199090 1199091) = 120572(1199090 1198791199090) ge 120578(1199090 1198791199090) = 120578(1199090 1199091) Bycontinuing this process we have

120578 (119909119899minus1 119879119909119899minus1) = 120578 (119909119899minus1 119909119899) le 120572 (119909119899minus1 119909119899) (19)

for all 119899 isin N Also let there exists 1198990 isin 119883 such that 1199091198990 =1199091198990+1

Then 1199091198990is fixed point of 119879 and we have nothing to

prove Hence we assume 119909119899 = 119909119899+1 for all 119899 isin N cup 0 Nowby taking 119909 = 119909119899minus1 and 119910 = 119909119899 in (iv) we get

H (120596120582119888 (119879119909119899minus1 119879119909119899) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119879119909119899minus1)

120596120582119897 (119909119899 119879119909119899) 1205962120582119897 (119909119899minus1 119879119909119899) 120596120582119897 (119909119899 119879119909119899minus1)) le 0

(20)

which implies

H (120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119909119899)

120596120582119897 (119909119899 119909119899+1) 1205962120582119897 (119909119899minus1 119909119899+1) 0) le 0

(21)

On the other hand

1205962120582119897 (119909119899minus1 119909119899+1) le 120596120582119897 (119909119899minus1 119909119899) + 120596120582119897 (119909119899 119909119899+1)

le 120596120582119897 (119909119899minus1 119909119899) + 120596120582119888 (119909119899 119909119899+1)

120596120582119897 (119909119899 119909119899+1) le 120596120582119888 (119909119899 119909119899+1)

(22)

Now since H is nonincreasing in its 5th variable so by(21) and (22) we obtain

H (120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899) 120596120582119897 (119909119899minus1 119909119899)

120596120582119888 (119909119899 119909119899+1) 120596120582119897 (119909119899minus1 119909119899)

+120596120582119888 (119909119899 119909119899+1) 0) le 0

(23)

From (H1) we deduce that

120596120582119888 (119909119899 119909119899+1) le 120595 (120596120582119897 (119909119899minus1 119909119899)) le 120595 (120596120582119888 (119909119899minus1 119909119899))

(24)

Inductively we obtain

120596120582119888 (119909119899 119909119899+1) le 120595119899(120596120582119888 (1199090 1199091)) (25)

Taking limit as 119899 rarr infin in the above inequality we get

lim119899rarrinfin

120596120582119888 (119909119899 119909119899+1) = 0 (26)

Suppose119898 119899 isin N with119898 gt 119899 and 120598 gt 0 be given Then thereexists 119899120582(119898minus119899) isin N such that

120596120582119888(119898minus119899) (119909119899 119909119899+1) lt

120598

119888 (119898 minus 119899)(27)

for all 119899 ge 119899120582(119898minus119899) Therefore we get

120596120582119888 (119909119899 119909119898) le 120596120582119888(119898minus119899) (119909119899 119909119899+1)

+ 120596120582119888(119898minus119899) (119909119899+1 119909119899+2) + sdot sdot sdot

+ 120596120582119888(119898minus119899) (119909119898minus1 119909119898)

lt

120598

119888 (119898 minus 119899)

+

120598

119888 (119898 minus 119899)

+

120598

119888 (119898 minus 119899)

+ sdot sdot sdot

+

120598

119888 (119898 minus 119899)

=

120598

119888

(28)

for all 119898 119899 ge 119899120582(119898minus119899) This shows that 119909119899 is a Cauchysequence Since119883120596 is complete so there exists 119909lowast isin 119883120596 suchthat lim119899rarrinfin119909119899 = 119909

lowast Now since 119879 is an 120572-120578-continuousmapping so 119879119909119899 rarr 119879119909

lowast as 119899 rarr infin Therefore

119879119909lowast= lim119899rarrinfin

119879119909119899 = lim119899rarrinfin

119909119899+1 = 119909lowast (29)

Thus 119879 has a fixed point Let all 119909 119910 isin Fix(119879) we have120578(119909 119909) le 120572(119909 119910) and H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0Then by (iv)

H (120596120582119897 (119909 119910) 120596120582119897 (119909 119910) 0 0 120596120582119897 (119909 119910) 120596120582119897 (119910 119909))

le H (120596120582119888 (119909 119910) 120596120582119897 (119909 119910) 0 0

1205962120582119897 (119909 119910) 120596120582119897 (119910 119909))

= H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909)

120596120582119897 (119910 119879119910) 1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(30)

Now if 120596120582119897(119909 119910) gt 0 then

0 lt H (120596120582119897 (119909 119910) 120596120582119897 (119909 119910) 0 0

120596120582119897 (119909 119910) 120596120582119897 (119910 119909)) le 0

(31)

which is a contradictionHence120596120582119897(119909 119910) = 0That is 119909 = 119910Thus 119879 has a unique fixed point

The Scientific World Journal 5

Corollary 13 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 a self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 1(iii) 119879 is an 120572-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 120572(119909 119910) ge 1 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(32)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 1 le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then119879 has a unique fixed point

Theorem 14 Let 119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(33)

holds for all 119899 isin N(iv) condition (iv) of Theorem 12 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof As in proof of Theorem 12 we can deduce a sequence119909119899 such that 119909119899+1 = 119879119909119899 with 120572(119909119899 119909119899+1) ge 120578(119909119899 119909119899+1) andthere exists 119909lowast isin 119883120596 such that 119909119899 rarr 119909

lowast as 119899 rarr infin From(iii) either

120578 (119909119899 119879119909119899) le 120572 (119909119899 119909lowast) or

120578 (119909119899+1 119879119909119899+1) le 120572 (119909119899+1 119909lowast)

(34)

holds for all 119899 isin N Let 120578(119909119899 119879119909119899) le 120572(119909119899 119909lowast)Then by taking

119909 = 119909119899 and 119910 = 119909lowast in (iv) we have

H (120596120582119888 (119879119909119899 119879119909lowast) 120596120582119897 (119909119899 119909

lowast) 120596120582119897 (119909119899 119879119909119899)

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909119899 119879119909

lowast) 120596120582119897 (119909

lowast 119879119909119899)) le 0

(35)which implies

H (120596120582119888 (119909119899+1 119879119909lowast) 120596120582119897 (119909119899 119909

lowast)

120596120582119897 (119909119899 119909119899+1) 120596120582119897 (119909lowast 119879119909lowast)

1205962120582119897 (119909119899 119879119909lowast) 120596120582119897 (119909

lowast 119909119899+1)) le 0

(36)

Taking limit as 119899 rarr infin in the above inequality we obtain

H (120596120582119888 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909

lowast 119879119909lowast) 0) le 0

(37)

Now since 1205962120582119897(119909lowast 119879119909lowast) lt 120596120582119897(119909

lowast 119879119909lowast) 120596120582119897(119909

lowast 119879119909lowast) lt

120596120582119888(119909lowast 119879119909lowast) H is increasing in its 1th variable and nonin-

creasing in its 5th variable so we obtain

H (120596120582119897 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 120596120582119897 (119909

lowast 119879119909lowast) 0) le 0

(38)

which is a contradiction Now by taking 119906 = 120596120582119897(119909lowast 119879119909lowast)

and V = 0 from (H2) we have

120596120582119897 (119909lowast 119879119909lowast) le 120595 (0) = 0 (39)

Hence 120596120582119897(119909lowast 119879119909lowast) = 0 that is 119909lowast = 119879119909lowast Similarly we can

deduce that 119879119909lowast = 119909lowast when 120578120582(119909119899+1 119879119909119899+1) le 120572120582(119909119899+1 119909

lowast)

By using Example 10 and Theorem 14 we can obtain thefollowing corollary

Corollary 15 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(40)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 120595(max120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(41)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

6 The Scientific World Journal

Corollary 16 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(42)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(43)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Corollary 17 Let119883120596 be a complete modular metric space Let119879 119883120596 rarr 119883120596 be self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(44)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(45)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Example 18 Let 119883120596 = [0 +infin) and 120596120582(119909 119910) = (1120582)|119909 minus 119910|Define 119879 119883120596 rarr 119883120596 120572 120578 119883120596 times 119883120596 rarr [0infin) and 120595

[0infin) rarr [0infin) by

119879119909 =

1

16

1199092 if 119909 isin [0 1]

sin2119909 + cos119909 + 1radic1199092+ 1

if 119909 isin (1 2]

7119909 + ln119909 if 119909 isin (2infin)

120572 (119909 119910) =

1

2

if 119909 119910 isin [0 1]

1

8

otherwise

120578 (119909 119910) =

1

4

120595 (119905) =

1

2

119905

(46)

Let 120572(119909 119910) ge 120578(119909 119910) then 119909 119910 isin [0 1] On the otherhand 119879119908 isin [0 1] for all 119908 isin [0 1] Then 120572(119879119909 119879119910) ge

120578(119879119909 119879119910) That is 119879 is an 120572-admissible mapping with respectto 120578 If 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then 119879119909119899 1198792119909119899 1198793119909119899 isin

[0 1] for all 119899 isin N That is

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909)

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(47)

hold for all 119899 isin N Clearly 120572(0 1198790) ge 120578(0 1198790) Let 120572(119909 119910) ge120578(119909 119879119909) Now if 119909 notin [0 1] or 119910 notin [0 1] then 18 ge 14which is a contradiction So 119909 119910 isin [0 1] Therefore

1205961205822 (119879119909 119879119910) =

2

120582

1

16

100381610038161003816100381610038161199092minus 119910210038161003816100381610038161003816

=

1

8120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 + 119910

1003816100381610038161003816le

1

4120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

1

120582 (12)

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

120596120582(12) (119909 119910) = 120595 (120596120582(12) (119909 119910))

le 120595(max120596120582119897 (119909 119910)

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(48)

Hence all conditions of Corollary 16 hold and 119879 has a uniquefixed point

Let (119883120596 ⪯) be a partially ordered modular metric spaceRecall that 119879 119883120596 rarr 119883120596 is nondecreasing if for all 119909119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) Fixed point theoremsfor monotone operators in ordered metric spaces are widely

The Scientific World Journal 7

investigated and have found various applications in differ-ential and integral equations (see [20 22ndash25] and referencestherein) From results proved above we derive the followingnew results in partially ordered modular metric spaces

Theorem 19 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfy-ing the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(49)

where 0 lt l lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 20 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(50)

holds for all 119899 isin N(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(51)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Corollary 21 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(52)

holds for all 119899 isin N

(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 wehave

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(53)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 22 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(54)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(55)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 23 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(56)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 119886120596120582119897 (119909 119910) + 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(57)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

8 The Scientific World Journal

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

3 Suzuki Type Fixed Point Results inModular Metric Spaces

In 2008 Suzuki proved a remarkable fixedpoint theorem thatis a generalization of the Banach contraction principle andcharacterizes themetric completeness Consequently a num-ber of extensions and generalizations of this result appearedin the literature (see [26ndash30] and references therein) As anapplication of our results proved abovewededuce Suzuki typefixed point theorems in the setting of modular metric spaces

Theorem 24 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

120596120582119897 (119909 119879119909) le 120596120582119897 (119909 119910)

997904rArr H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(58)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has afixed point Moreover if H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Define 120572 120578 (0infin) times 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 120596120582119897 (119909 119910) 120578 (119909 119910) = 120596120582119897 (119909 119910) (59)

Clearly 120578(119909 119910) le 120572(119909 119910) for all 119909 119910 isin 119883120596 and 120582 gt 0Since 119879 is continuous 119879 is 120572-120578-continuous Thus conditions(i)ndash(iii) of Theorem 12 hold Let 120578(119909 119879119909) le 120572(119909 119910) Then120596120582119897(119909 119879119909) le 120596120582119897(119909 119910) So from (58) we obtain

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(60)

Therefore all conditions of Theorem 12 hold and 119879 has aunique fixed point

Theorem 25 Let119883120596 be a complete modular metric space and119879 self-mapping on 119883 Assume that

1 minus 119887

1 minus 119887 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(61)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Proof Define 120572 120578 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 1205961205822119897 (119909 119910) 120578 (119909 119910) = 119896120596120582119897 (119909 119910) (62)

where 119896 = (1 minus 119887)(1 minus 119887 + 119886) Clearly 120578(119909 119910) le 120572(119909 119910) for all119909 119910 isin 119883120596 and 120582 gt 0 Then conditions (i)-(ii) of Corollary 17hold Let 119909119899 be a sequence such that 119909119899 rarr 119909 as 119899 rarr infinSince 119896120596120582119897(119879119909119899 119879

2119909119899) le 1205961205822119897(119879119909119899 119879

2119909119899) for all 119899 isin N from

(61) we obtain

120596120582119897 (1198792119909119899 1198793119909119899)

le 120596120582119888 (1198792119909119899 1198793119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899)

+ 119887

[1 + 120596120582119897 (119879119909119899 1198792119909119899)] 120596120582119897 (119879

2119909119899 1198793119909119899)

1 + 120596120582119897 (119879119909119899 1198792119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899) + 119887120596120582119897 (119879

2119909119899 1198793119909119899)

(63)

which implies

120596120582119897 (1198792119909119899 1198793119909119899) le

119886

1 minus 119887

120596120582119897 (119879119909119899 1198792119909119899) (64)

Suppose there exists 1198990 isin N such that

120578 (1198791199091198990 11987921199091198990

) gt 120572 (1198791199091198990 119909) or

120578 (11987921199091198990

11987931199091198990

) gt 120572 (11987921199091198990

119909)

(65)

then119896120596120582119897 (1198791199091198990

11987921199091198990

) gt 1205961205822119897 (1198791199091198990 119909) or

119896120596120582119897 (11987921199091198990

11987931199091198990

) gt 1205961205822119897 (11987921199091198990

119909)

(66)

Therefore

120596120582119897 (1198791199091198990 11987921199091198990

)

le 1205961205822119897 (1198791199091198990 119909) + 1205961205822119897 (119879

21199091198990

119909)

lt 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896120596120582119897 (11987921199091198990

11987931199091198990

)

le 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896

119886

1 minus 119887

120596120582119897 (1198791199091198990 11987921199091198990

)

le (119896 + 119896

119886

1 minus 119887

)120596120582119897 (1198791199091198990 11987921199091198990

)

= 120596120582119897 (1198791199091198990 11987921199091198990

)

(67)

which is a contradiction Hence (iii) of Corollary 17 holdsThus all conditions of Corollary 17 hold and 119879 has a uniquefixed point

Corollary 26 Let119883120596 be a complete modularmetric space and119879 self-mapping on119883 Assume that

1

1 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

(68)

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

The Scientific World Journal 5

Corollary 13 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 a self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 1(iii) 119879 is an 120572-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 120572(119909 119910) ge 1 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(32)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 1 le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then119879 has a unique fixed point

Theorem 14 Let 119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(33)

holds for all 119899 isin N(iv) condition (iv) of Theorem 12 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof As in proof of Theorem 12 we can deduce a sequence119909119899 such that 119909119899+1 = 119879119909119899 with 120572(119909119899 119909119899+1) ge 120578(119909119899 119909119899+1) andthere exists 119909lowast isin 119883120596 such that 119909119899 rarr 119909

lowast as 119899 rarr infin From(iii) either

120578 (119909119899 119879119909119899) le 120572 (119909119899 119909lowast) or

120578 (119909119899+1 119879119909119899+1) le 120572 (119909119899+1 119909lowast)

(34)

holds for all 119899 isin N Let 120578(119909119899 119879119909119899) le 120572(119909119899 119909lowast)Then by taking

119909 = 119909119899 and 119910 = 119909lowast in (iv) we have

H (120596120582119888 (119879119909119899 119879119909lowast) 120596120582119897 (119909119899 119909

lowast) 120596120582119897 (119909119899 119879119909119899)

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909119899 119879119909

lowast) 120596120582119897 (119909

lowast 119879119909119899)) le 0

(35)which implies

H (120596120582119888 (119909119899+1 119879119909lowast) 120596120582119897 (119909119899 119909

lowast)

120596120582119897 (119909119899 119909119899+1) 120596120582119897 (119909lowast 119879119909lowast)

1205962120582119897 (119909119899 119879119909lowast) 120596120582119897 (119909

lowast 119909119899+1)) le 0

(36)

Taking limit as 119899 rarr infin in the above inequality we obtain

H (120596120582119888 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 1205962120582119897 (119909

lowast 119879119909lowast) 0) le 0

(37)

Now since 1205962120582119897(119909lowast 119879119909lowast) lt 120596120582119897(119909

lowast 119879119909lowast) 120596120582119897(119909

lowast 119879119909lowast) lt

120596120582119888(119909lowast 119879119909lowast) H is increasing in its 1th variable and nonin-

creasing in its 5th variable so we obtain

H (120596120582119897 (119909lowast 119879119909lowast) 0 0

120596120582119897 (119909lowast 119879119909lowast) 120596120582119897 (119909

lowast 119879119909lowast) 0) le 0

(38)

which is a contradiction Now by taking 119906 = 120596120582119897(119909lowast 119879119909lowast)

and V = 0 from (H2) we have

120596120582119897 (119909lowast 119879119909lowast) le 120595 (0) = 0 (39)

Hence 120596120582119897(119909lowast 119879119909lowast) = 0 that is 119909lowast = 119879119909lowast Similarly we can

deduce that 119879119909lowast = 119909lowast when 120578120582(119909119899+1 119879119909119899+1) le 120572120582(119909119899+1 119909

lowast)

By using Example 10 and Theorem 14 we can obtain thefollowing corollary

Corollary 15 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(40)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 120595(max120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(41)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

6 The Scientific World Journal

Corollary 16 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(42)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(43)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Corollary 17 Let119883120596 be a complete modular metric space Let119879 119883120596 rarr 119883120596 be self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(44)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(45)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Example 18 Let 119883120596 = [0 +infin) and 120596120582(119909 119910) = (1120582)|119909 minus 119910|Define 119879 119883120596 rarr 119883120596 120572 120578 119883120596 times 119883120596 rarr [0infin) and 120595

[0infin) rarr [0infin) by

119879119909 =

1

16

1199092 if 119909 isin [0 1]

sin2119909 + cos119909 + 1radic1199092+ 1

if 119909 isin (1 2]

7119909 + ln119909 if 119909 isin (2infin)

120572 (119909 119910) =

1

2

if 119909 119910 isin [0 1]

1

8

otherwise

120578 (119909 119910) =

1

4

120595 (119905) =

1

2

119905

(46)

Let 120572(119909 119910) ge 120578(119909 119910) then 119909 119910 isin [0 1] On the otherhand 119879119908 isin [0 1] for all 119908 isin [0 1] Then 120572(119879119909 119879119910) ge

120578(119879119909 119879119910) That is 119879 is an 120572-admissible mapping with respectto 120578 If 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then 119879119909119899 1198792119909119899 1198793119909119899 isin

[0 1] for all 119899 isin N That is

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909)

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(47)

hold for all 119899 isin N Clearly 120572(0 1198790) ge 120578(0 1198790) Let 120572(119909 119910) ge120578(119909 119879119909) Now if 119909 notin [0 1] or 119910 notin [0 1] then 18 ge 14which is a contradiction So 119909 119910 isin [0 1] Therefore

1205961205822 (119879119909 119879119910) =

2

120582

1

16

100381610038161003816100381610038161199092minus 119910210038161003816100381610038161003816

=

1

8120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 + 119910

1003816100381610038161003816le

1

4120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

1

120582 (12)

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

120596120582(12) (119909 119910) = 120595 (120596120582(12) (119909 119910))

le 120595(max120596120582119897 (119909 119910)

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(48)

Hence all conditions of Corollary 16 hold and 119879 has a uniquefixed point

Let (119883120596 ⪯) be a partially ordered modular metric spaceRecall that 119879 119883120596 rarr 119883120596 is nondecreasing if for all 119909119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) Fixed point theoremsfor monotone operators in ordered metric spaces are widely

The Scientific World Journal 7

investigated and have found various applications in differ-ential and integral equations (see [20 22ndash25] and referencestherein) From results proved above we derive the followingnew results in partially ordered modular metric spaces

Theorem 19 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfy-ing the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(49)

where 0 lt l lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 20 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(50)

holds for all 119899 isin N(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(51)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Corollary 21 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(52)

holds for all 119899 isin N

(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 wehave

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(53)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 22 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(54)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(55)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 23 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(56)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 119886120596120582119897 (119909 119910) + 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(57)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

8 The Scientific World Journal

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

3 Suzuki Type Fixed Point Results inModular Metric Spaces

In 2008 Suzuki proved a remarkable fixedpoint theorem thatis a generalization of the Banach contraction principle andcharacterizes themetric completeness Consequently a num-ber of extensions and generalizations of this result appearedin the literature (see [26ndash30] and references therein) As anapplication of our results proved abovewededuce Suzuki typefixed point theorems in the setting of modular metric spaces

Theorem 24 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

120596120582119897 (119909 119879119909) le 120596120582119897 (119909 119910)

997904rArr H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(58)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has afixed point Moreover if H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Define 120572 120578 (0infin) times 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 120596120582119897 (119909 119910) 120578 (119909 119910) = 120596120582119897 (119909 119910) (59)

Clearly 120578(119909 119910) le 120572(119909 119910) for all 119909 119910 isin 119883120596 and 120582 gt 0Since 119879 is continuous 119879 is 120572-120578-continuous Thus conditions(i)ndash(iii) of Theorem 12 hold Let 120578(119909 119879119909) le 120572(119909 119910) Then120596120582119897(119909 119879119909) le 120596120582119897(119909 119910) So from (58) we obtain

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(60)

Therefore all conditions of Theorem 12 hold and 119879 has aunique fixed point

Theorem 25 Let119883120596 be a complete modular metric space and119879 self-mapping on 119883 Assume that

1 minus 119887

1 minus 119887 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(61)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Proof Define 120572 120578 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 1205961205822119897 (119909 119910) 120578 (119909 119910) = 119896120596120582119897 (119909 119910) (62)

where 119896 = (1 minus 119887)(1 minus 119887 + 119886) Clearly 120578(119909 119910) le 120572(119909 119910) for all119909 119910 isin 119883120596 and 120582 gt 0 Then conditions (i)-(ii) of Corollary 17hold Let 119909119899 be a sequence such that 119909119899 rarr 119909 as 119899 rarr infinSince 119896120596120582119897(119879119909119899 119879

2119909119899) le 1205961205822119897(119879119909119899 119879

2119909119899) for all 119899 isin N from

(61) we obtain

120596120582119897 (1198792119909119899 1198793119909119899)

le 120596120582119888 (1198792119909119899 1198793119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899)

+ 119887

[1 + 120596120582119897 (119879119909119899 1198792119909119899)] 120596120582119897 (119879

2119909119899 1198793119909119899)

1 + 120596120582119897 (119879119909119899 1198792119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899) + 119887120596120582119897 (119879

2119909119899 1198793119909119899)

(63)

which implies

120596120582119897 (1198792119909119899 1198793119909119899) le

119886

1 minus 119887

120596120582119897 (119879119909119899 1198792119909119899) (64)

Suppose there exists 1198990 isin N such that

120578 (1198791199091198990 11987921199091198990

) gt 120572 (1198791199091198990 119909) or

120578 (11987921199091198990

11987931199091198990

) gt 120572 (11987921199091198990

119909)

(65)

then119896120596120582119897 (1198791199091198990

11987921199091198990

) gt 1205961205822119897 (1198791199091198990 119909) or

119896120596120582119897 (11987921199091198990

11987931199091198990

) gt 1205961205822119897 (11987921199091198990

119909)

(66)

Therefore

120596120582119897 (1198791199091198990 11987921199091198990

)

le 1205961205822119897 (1198791199091198990 119909) + 1205961205822119897 (119879

21199091198990

119909)

lt 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896120596120582119897 (11987921199091198990

11987931199091198990

)

le 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896

119886

1 minus 119887

120596120582119897 (1198791199091198990 11987921199091198990

)

le (119896 + 119896

119886

1 minus 119887

)120596120582119897 (1198791199091198990 11987921199091198990

)

= 120596120582119897 (1198791199091198990 11987921199091198990

)

(67)

which is a contradiction Hence (iii) of Corollary 17 holdsThus all conditions of Corollary 17 hold and 119879 has a uniquefixed point

Corollary 26 Let119883120596 be a complete modularmetric space and119879 self-mapping on119883 Assume that

1

1 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

(68)

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

6 The Scientific World Journal

Corollary 16 Let 119883120596 be a complete modular metric spaceLet 119879 119883120596 rarr 119883120596 be self-mapping satisfying the followingassertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(42)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(43)

where 120595 isin Ψ and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Corollary 17 Let119883120596 be a complete modular metric space Let119879 119883120596 rarr 119883120596 be self-mapping satisfying the following asser-tions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(44)

holds for all 119899 isin N(iv) for all 119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we

have

120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(45)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) then 119879 has a unique fixed point

Example 18 Let 119883120596 = [0 +infin) and 120596120582(119909 119910) = (1120582)|119909 minus 119910|Define 119879 119883120596 rarr 119883120596 120572 120578 119883120596 times 119883120596 rarr [0infin) and 120595

[0infin) rarr [0infin) by

119879119909 =

1

16

1199092 if 119909 isin [0 1]

sin2119909 + cos119909 + 1radic1199092+ 1

if 119909 isin (1 2]

7119909 + ln119909 if 119909 isin (2infin)

120572 (119909 119910) =

1

2

if 119909 119910 isin [0 1]

1

8

otherwise

120578 (119909 119910) =

1

4

120595 (119905) =

1

2

119905

(46)

Let 120572(119909 119910) ge 120578(119909 119910) then 119909 119910 isin [0 1] On the otherhand 119879119908 isin [0 1] for all 119908 isin [0 1] Then 120572(119879119909 119879119910) ge

120578(119879119909 119879119910) That is 119879 is an 120572-admissible mapping with respectto 120578 If 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then 119879119909119899 1198792119909119899 1198793119909119899 isin

[0 1] for all 119899 isin N That is

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909)

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(47)

hold for all 119899 isin N Clearly 120572(0 1198790) ge 120578(0 1198790) Let 120572(119909 119910) ge120578(119909 119879119909) Now if 119909 notin [0 1] or 119910 notin [0 1] then 18 ge 14which is a contradiction So 119909 119910 isin [0 1] Therefore

1205961205822 (119879119909 119879119910) =

2

120582

1

16

100381610038161003816100381610038161199092minus 119910210038161003816100381610038161003816

=

1

8120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 + 119910

1003816100381610038161003816le

1

4120582

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

1

120582 (12)

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

=

1

2

120596120582(12) (119909 119910) = 120595 (120596120582(12) (119909 119910))

le 120595(max120596120582119897 (119909 119910)

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(48)

Hence all conditions of Corollary 16 hold and 119879 has a uniquefixed point

Let (119883120596 ⪯) be a partially ordered modular metric spaceRecall that 119879 119883120596 rarr 119883120596 is nondecreasing if for all 119909119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) Fixed point theoremsfor monotone operators in ordered metric spaces are widely

The Scientific World Journal 7

investigated and have found various applications in differ-ential and integral equations (see [20 22ndash25] and referencestherein) From results proved above we derive the followingnew results in partially ordered modular metric spaces

Theorem 19 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfy-ing the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(49)

where 0 lt l lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 20 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(50)

holds for all 119899 isin N(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(51)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Corollary 21 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(52)

holds for all 119899 isin N

(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 wehave

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(53)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 22 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(54)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(55)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 23 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(56)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 119886120596120582119897 (119909 119910) + 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(57)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

8 The Scientific World Journal

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

3 Suzuki Type Fixed Point Results inModular Metric Spaces

In 2008 Suzuki proved a remarkable fixedpoint theorem thatis a generalization of the Banach contraction principle andcharacterizes themetric completeness Consequently a num-ber of extensions and generalizations of this result appearedin the literature (see [26ndash30] and references therein) As anapplication of our results proved abovewededuce Suzuki typefixed point theorems in the setting of modular metric spaces

Theorem 24 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

120596120582119897 (119909 119879119909) le 120596120582119897 (119909 119910)

997904rArr H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(58)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has afixed point Moreover if H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Define 120572 120578 (0infin) times 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 120596120582119897 (119909 119910) 120578 (119909 119910) = 120596120582119897 (119909 119910) (59)

Clearly 120578(119909 119910) le 120572(119909 119910) for all 119909 119910 isin 119883120596 and 120582 gt 0Since 119879 is continuous 119879 is 120572-120578-continuous Thus conditions(i)ndash(iii) of Theorem 12 hold Let 120578(119909 119879119909) le 120572(119909 119910) Then120596120582119897(119909 119879119909) le 120596120582119897(119909 119910) So from (58) we obtain

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(60)

Therefore all conditions of Theorem 12 hold and 119879 has aunique fixed point

Theorem 25 Let119883120596 be a complete modular metric space and119879 self-mapping on 119883 Assume that

1 minus 119887

1 minus 119887 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(61)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Proof Define 120572 120578 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 1205961205822119897 (119909 119910) 120578 (119909 119910) = 119896120596120582119897 (119909 119910) (62)

where 119896 = (1 minus 119887)(1 minus 119887 + 119886) Clearly 120578(119909 119910) le 120572(119909 119910) for all119909 119910 isin 119883120596 and 120582 gt 0 Then conditions (i)-(ii) of Corollary 17hold Let 119909119899 be a sequence such that 119909119899 rarr 119909 as 119899 rarr infinSince 119896120596120582119897(119879119909119899 119879

2119909119899) le 1205961205822119897(119879119909119899 119879

2119909119899) for all 119899 isin N from

(61) we obtain

120596120582119897 (1198792119909119899 1198793119909119899)

le 120596120582119888 (1198792119909119899 1198793119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899)

+ 119887

[1 + 120596120582119897 (119879119909119899 1198792119909119899)] 120596120582119897 (119879

2119909119899 1198793119909119899)

1 + 120596120582119897 (119879119909119899 1198792119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899) + 119887120596120582119897 (119879

2119909119899 1198793119909119899)

(63)

which implies

120596120582119897 (1198792119909119899 1198793119909119899) le

119886

1 minus 119887

120596120582119897 (119879119909119899 1198792119909119899) (64)

Suppose there exists 1198990 isin N such that

120578 (1198791199091198990 11987921199091198990

) gt 120572 (1198791199091198990 119909) or

120578 (11987921199091198990

11987931199091198990

) gt 120572 (11987921199091198990

119909)

(65)

then119896120596120582119897 (1198791199091198990

11987921199091198990

) gt 1205961205822119897 (1198791199091198990 119909) or

119896120596120582119897 (11987921199091198990

11987931199091198990

) gt 1205961205822119897 (11987921199091198990

119909)

(66)

Therefore

120596120582119897 (1198791199091198990 11987921199091198990

)

le 1205961205822119897 (1198791199091198990 119909) + 1205961205822119897 (119879

21199091198990

119909)

lt 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896120596120582119897 (11987921199091198990

11987931199091198990

)

le 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896

119886

1 minus 119887

120596120582119897 (1198791199091198990 11987921199091198990

)

le (119896 + 119896

119886

1 minus 119887

)120596120582119897 (1198791199091198990 11987921199091198990

)

= 120596120582119897 (1198791199091198990 11987921199091198990

)

(67)

which is a contradiction Hence (iii) of Corollary 17 holdsThus all conditions of Corollary 17 hold and 119879 has a uniquefixed point

Corollary 26 Let119883120596 be a complete modularmetric space and119879 self-mapping on119883 Assume that

1

1 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

(68)

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

The Scientific World Journal 7

investigated and have found various applications in differ-ential and integral equations (see [20 22ndash25] and referencestherein) From results proved above we derive the followingnew results in partially ordered modular metric spaces

Theorem 19 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfy-ing the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(49)

where 0 lt l lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 20 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(50)

holds for all 119899 isin N(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we haveH (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(51)

where 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Corollary 21 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(52)

holds for all 119899 isin N

(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 wehave

120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) + 120596120582119897 (119910 119879119910)

2

1205962120582119897 (119909 119879119910) + 120596120582119897 (119910 119879119909)

2

)

(53)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 22 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(54)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 120595(max120596120582119897 (119909 119910) [1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

)

(55)

where 120595 isin Ψ and 0 lt 119897 lt 119888Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

Corollary 23 Let (119883120596 ⪯) be a complete partially orderedmodular metric space and 119879 119883120596 rarr 119883120596 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with

119909119899 rarr 119909 as 119899 rarr infin then either119879119909119899 ⪯ 119909 or 119879

2119909119899 ⪯ 119909

(56)

holds for all 119899 isin N(iv) assume that for all 119909 119910 isin 119883120596 and 120582 gt 0 with 119909 ⪯ 119910 we

have120596120582119888 (119879119909 119879119910)

le 119886120596120582119897 (119909 119910) + 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(57)

where 119886 + 119887 lt 1 and 0 lt 119897 lt 119888

8 The Scientific World Journal

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

3 Suzuki Type Fixed Point Results inModular Metric Spaces

In 2008 Suzuki proved a remarkable fixedpoint theorem thatis a generalization of the Banach contraction principle andcharacterizes themetric completeness Consequently a num-ber of extensions and generalizations of this result appearedin the literature (see [26ndash30] and references therein) As anapplication of our results proved abovewededuce Suzuki typefixed point theorems in the setting of modular metric spaces

Theorem 24 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

120596120582119897 (119909 119879119909) le 120596120582119897 (119909 119910)

997904rArr H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(58)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has afixed point Moreover if H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Define 120572 120578 (0infin) times 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 120596120582119897 (119909 119910) 120578 (119909 119910) = 120596120582119897 (119909 119910) (59)

Clearly 120578(119909 119910) le 120572(119909 119910) for all 119909 119910 isin 119883120596 and 120582 gt 0Since 119879 is continuous 119879 is 120572-120578-continuous Thus conditions(i)ndash(iii) of Theorem 12 hold Let 120578(119909 119879119909) le 120572(119909 119910) Then120596120582119897(119909 119879119909) le 120596120582119897(119909 119910) So from (58) we obtain

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(60)

Therefore all conditions of Theorem 12 hold and 119879 has aunique fixed point

Theorem 25 Let119883120596 be a complete modular metric space and119879 self-mapping on 119883 Assume that

1 minus 119887

1 minus 119887 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(61)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Proof Define 120572 120578 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 1205961205822119897 (119909 119910) 120578 (119909 119910) = 119896120596120582119897 (119909 119910) (62)

where 119896 = (1 minus 119887)(1 minus 119887 + 119886) Clearly 120578(119909 119910) le 120572(119909 119910) for all119909 119910 isin 119883120596 and 120582 gt 0 Then conditions (i)-(ii) of Corollary 17hold Let 119909119899 be a sequence such that 119909119899 rarr 119909 as 119899 rarr infinSince 119896120596120582119897(119879119909119899 119879

2119909119899) le 1205961205822119897(119879119909119899 119879

2119909119899) for all 119899 isin N from

(61) we obtain

120596120582119897 (1198792119909119899 1198793119909119899)

le 120596120582119888 (1198792119909119899 1198793119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899)

+ 119887

[1 + 120596120582119897 (119879119909119899 1198792119909119899)] 120596120582119897 (119879

2119909119899 1198793119909119899)

1 + 120596120582119897 (119879119909119899 1198792119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899) + 119887120596120582119897 (119879

2119909119899 1198793119909119899)

(63)

which implies

120596120582119897 (1198792119909119899 1198793119909119899) le

119886

1 minus 119887

120596120582119897 (119879119909119899 1198792119909119899) (64)

Suppose there exists 1198990 isin N such that

120578 (1198791199091198990 11987921199091198990

) gt 120572 (1198791199091198990 119909) or

120578 (11987921199091198990

11987931199091198990

) gt 120572 (11987921199091198990

119909)

(65)

then119896120596120582119897 (1198791199091198990

11987921199091198990

) gt 1205961205822119897 (1198791199091198990 119909) or

119896120596120582119897 (11987921199091198990

11987931199091198990

) gt 1205961205822119897 (11987921199091198990

119909)

(66)

Therefore

120596120582119897 (1198791199091198990 11987921199091198990

)

le 1205961205822119897 (1198791199091198990 119909) + 1205961205822119897 (119879

21199091198990

119909)

lt 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896120596120582119897 (11987921199091198990

11987931199091198990

)

le 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896

119886

1 minus 119887

120596120582119897 (1198791199091198990 11987921199091198990

)

le (119896 + 119896

119886

1 minus 119887

)120596120582119897 (1198791199091198990 11987921199091198990

)

= 120596120582119897 (1198791199091198990 11987921199091198990

)

(67)

which is a contradiction Hence (iii) of Corollary 17 holdsThus all conditions of Corollary 17 hold and 119879 has a uniquefixed point

Corollary 26 Let119883120596 be a complete modularmetric space and119879 self-mapping on119883 Assume that

1

1 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

(68)

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

8 The Scientific World Journal

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 then 119879 has a unique fixed point

3 Suzuki Type Fixed Point Results inModular Metric Spaces

In 2008 Suzuki proved a remarkable fixedpoint theorem thatis a generalization of the Banach contraction principle andcharacterizes themetric completeness Consequently a num-ber of extensions and generalizations of this result appearedin the literature (see [26ndash30] and references therein) As anapplication of our results proved abovewededuce Suzuki typefixed point theorems in the setting of modular metric spaces

Theorem 24 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

120596120582119897 (119909 119879119909) le 120596120582119897 (119909 119910)

997904rArr H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910)

120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(58)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has afixed point Moreover if H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Proof Define 120572 120578 (0infin) times 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 120596120582119897 (119909 119910) 120578 (119909 119910) = 120596120582119897 (119909 119910) (59)

Clearly 120578(119909 119910) le 120572(119909 119910) for all 119909 119910 isin 119883120596 and 120582 gt 0Since 119879 is continuous 119879 is 120572-120578-continuous Thus conditions(i)ndash(iii) of Theorem 12 hold Let 120578(119909 119879119909) le 120572(119909 119910) Then120596120582119897(119909 119879119909) le 120596120582119897(119909 119910) So from (58) we obtain

H (120596120582119888 (119879119909 119879119910) 120596120582119897 (119909 119910) 120596120582119897 (119909 119879119909) 120596120582119897 (119910 119879119910)

1205962120582119897 (119909 119879119910) 120596120582119897 (119910 119879119909)) le 0

(60)

Therefore all conditions of Theorem 12 hold and 119879 has aunique fixed point

Theorem 25 Let119883120596 be a complete modular metric space and119879 self-mapping on 119883 Assume that

1 minus 119887

1 minus 119887 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

+ 119887

[1 + 120596120582119897 (119909 119879119909)] 120596120582119897 (119910 119879119910)

1 + 120596120582119897 (119909 119910)

(61)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Proof Define 120572 120578 119883120596 times 119883120596 rarr [0infin) by

120572 (119909 119910) = 1205961205822119897 (119909 119910) 120578 (119909 119910) = 119896120596120582119897 (119909 119910) (62)

where 119896 = (1 minus 119887)(1 minus 119887 + 119886) Clearly 120578(119909 119910) le 120572(119909 119910) for all119909 119910 isin 119883120596 and 120582 gt 0 Then conditions (i)-(ii) of Corollary 17hold Let 119909119899 be a sequence such that 119909119899 rarr 119909 as 119899 rarr infinSince 119896120596120582119897(119879119909119899 119879

2119909119899) le 1205961205822119897(119879119909119899 119879

2119909119899) for all 119899 isin N from

(61) we obtain

120596120582119897 (1198792119909119899 1198793119909119899)

le 120596120582119888 (1198792119909119899 1198793119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899)

+ 119887

[1 + 120596120582119897 (119879119909119899 1198792119909119899)] 120596120582119897 (119879

2119909119899 1198793119909119899)

1 + 120596120582119897 (119879119909119899 1198792119909119899)

le 119886120596120582119897 (119879119909119899 1198792119909119899) + 119887120596120582119897 (119879

2119909119899 1198793119909119899)

(63)

which implies

120596120582119897 (1198792119909119899 1198793119909119899) le

119886

1 minus 119887

120596120582119897 (119879119909119899 1198792119909119899) (64)

Suppose there exists 1198990 isin N such that

120578 (1198791199091198990 11987921199091198990

) gt 120572 (1198791199091198990 119909) or

120578 (11987921199091198990

11987931199091198990

) gt 120572 (11987921199091198990

119909)

(65)

then119896120596120582119897 (1198791199091198990

11987921199091198990

) gt 1205961205822119897 (1198791199091198990 119909) or

119896120596120582119897 (11987921199091198990

11987931199091198990

) gt 1205961205822119897 (11987921199091198990

119909)

(66)

Therefore

120596120582119897 (1198791199091198990 11987921199091198990

)

le 1205961205822119897 (1198791199091198990 119909) + 1205961205822119897 (119879

21199091198990

119909)

lt 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896120596120582119897 (11987921199091198990

11987931199091198990

)

le 119896120596120582119897 (1198791199091198990 11987921199091198990

) + 119896

119886

1 minus 119887

120596120582119897 (1198791199091198990 11987921199091198990

)

le (119896 + 119896

119886

1 minus 119887

)120596120582119897 (1198791199091198990 11987921199091198990

)

= 120596120582119897 (1198791199091198990 11987921199091198990

)

(67)

which is a contradiction Hence (iii) of Corollary 17 holdsThus all conditions of Corollary 17 hold and 119879 has a uniquefixed point

Corollary 26 Let119883120596 be a complete modularmetric space and119879 self-mapping on119883 Assume that

1

1 + 119886

120596120582119897 (119909 119879119909) le 1205961205822119897 (119909 119910)

997904rArr 120596120582119888 (119879119909 119879119910) le 119886120596120582119897 (119909 119910)

(68)

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

The Scientific World Journal 9

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 lt 1 Then 119879has a unique fixed point

4 Fixed Point Results forIntegral Type Contractions

Recently Azadifar et al [31] and Razani and Moradi [32]proved common fixed point theorems of integral type inmodular metric spaces In this section we present moregeneral fixed point theorems for integral type contractions

Theorem 27 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) 119879 is an 120572-120578-continuous function

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(69)

where 0 lt 119897 lt 119888 120588 [0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfying int120576

0120588(119905)119889119905 gt 0 for 120576 gt 0

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 28 Let119883120596 be a complete modular metric space and119879 119883120596 rarr 119883120596 self-mapping satisfying the following assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578

(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(70)

holds for all 119899 isin N

(iv) condition (iv) of Theorem 27 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 29 Let119883120596 be a complete modular metric space and119879 continuous self-mapping on 119883 Assume that

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

997904rArr H(int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119910)

0

120588 (119905) 119889119905

int

1205962120582119897(119909119879119910)

0

120588 (119905) 119889119905 int

120596120582119897(119910119879119909)

0

120588 (119905) 119889119905) le 0

(71)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 120588

[0infin) rarr [0infin) is a Lebesgue-integrable mapping satisfyingint

120576

0120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a fixed point Moreover if

H(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 has a unique fixedpoint

Theorem 30 Let119883120596 be a complete modular metric space and119879 self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

int

120596120582119897(119909119879119909)

0

120588 (119905) 119889119905

le int

1205961205822119897(119909119910)

0

120588 (119905) 119889119905 997904rArr int

120596120582119888(119879119909119879119910)

0

120588 (119905) 119889119905

le 119886int

120596120582119897(119909119910)

0

120588 (119905) 119889119905

+ 119887

[1 + int

120596120582119897(119909119879119909)

0120588 (119905) 119889119905] int

120596120582119897(119910119879119910)

0120588 (119905) 119889119905

1 + int

120596120582119897(119909119910)

0120588 (119905) 119889119905

(72)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 119886 + 119887 lt 1 and120588 [0infin) rarr [0infin) is a Lebesgue-integrablemapping satisfy-ing int1205760120588(119905)119889119905 gt 0 for 120576 gt 0 Then 119879 has a unique fixed point

5 Modular Metric Spaces toFuzzy Metric Spaces

In 1988 Grabiec [33] defined contractivemappings on a fuzzymetric space and extended fixed point theorems of Banachand Edelstein in such spaces Successively George and Veera-mani [34] slightlymodified the notion of a fuzzymetric spaceintroduced by Kramosil and Michalek and then defined aHausdorff and first countable topology on it Since then thenotion of a complete fuzzy metric space presented by Georgeand Veeramani has emerged as another characterization ofcompleteness and many fixed point theorems have also beenproved (see for more details [35ndash39] and the referencestherein) In this section we develop an important relation

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

10 The Scientific World Journal

between modular metric and fuzzy metric and deduce infixed point results in a triangular fuzzy metric space

Definition 31 A 3-tuple (119883119872 lowast) is said to be a fuzzy metricspace if119883 is an arbitrary set lowast is a continuous 119905-norm and119872is fuzzy set on1198832times(0infin) satisfying the following conditionsfor all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(i) 119872(119909 119910 119905) gt 0(ii) 119872(119909 119910 119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910(iii) 119872(119909 119910 119905) = 119872(119910 119909 119905)(iv) 119872(119909 119910 119905) lowast 119872(119910 119911 119904) le 119872(119909 119911 119905 + 119904)(v) 119872(119909 119910 ) (0infin) rarr [0 1] is continuous

The function 119872(119909 119910 119905) denotes the degree of nearnessbetween 119909 and 119910 with respect to 119905

Definition 32 (see [36]) Let (119883119872 lowast) be a fuzzymetric spaceThe fuzzy metric119872 is called triangular whenever

1

119872(119909 119910 119905)

minus 1 le

1

119872 (119909 119911 119905)

minus 1 +

1

119872(119911 119910 119905)

minus 1 (73)

for all 119909 119910 119911 isin 119883 and all 119905 gt 0

Lemma 33 (see [33 35]) For all 119909 119910 isin 119883 119872(119909 119910 sdot) isnondecreasing on (0infin)

As an application of Lemma 33 we establish the followingimportant fact that each triangular fuzzymetric on119883 inducesa modular metric

Lemma 34 Let (119883119872 lowast) be a triangular fuzzy metric spaceDefine

120596120582 (119909 119910) =

1

119872(119909 119910 120582)

minus 1 (74)

for all 119909 119910 isin 119883 and all 120582 gt 0 Then 120596120582 is a modular metric on119883

Proof Let 119904 119905 gt 0 Then we get

119872(119909 119910 119904) = 119872(119909 119910 119904) lowast 1

= 119872(119909 119910 119904) lowast 119872 (119909 119909 119905) le 119872 (119909 119910 119904 + 119905)

(75)

for all119909 119910 isin 119883 and 119904 119905 gt 0 Now since (119883119872 lowast) is triangularthen we get

120596120582+120583 (119909 119910) =

1

119872(119909 119910 120583 + 120582)

minus 1

le

1

119872(119909 119911 120583 + 120582)

minus 1 +

1

119872(119911 119910 120583 + 120582)

minus 1

le

1

119872 (119909 119911 120582)

minus 1 +

1

119872(119911 119910 120583)

minus 1

= 120596120582 (119909 119911) + 120596120583 (119911 119910)

(76)

As an application of Lemma 34 and the results provedabove we deduce following new fixed point theorems intriangular fuzzy metric spaces

Theorem 35 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) 119879 is an 120572-120578-continuous function(iv) assume that there exists H isin ΔH such that for all

119909 119910 isin 119883 and 120582 gt 0 with 120578(119909 119879119909) le 120572(119909 119910) we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(77)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 36 Let (119883119872 lowast) be a complete triangular fuzzymetric space and 119879 119883 rarr 119883 self-mapping satisfying thefollowing assertions

(i) 119879 is an 120572-admissible mapping with respect to 120578(ii) there exists 1199090 isin 119883 such that 120572(1199090 1198791199090) ge 120578(1199090 1198791199090)(iii) if 119909119899 is a sequence in 119883 such that 120572(119909119899 119909119899+1) ge

120578(119909119899 119909119899+1) with 119909119899 rarr 119909 as 119899 rarr infin then either

120578 (119879119909119899 1198792119909119899) le 120572 (119879119909119899 119909) or

120578 (1198792119909119899 1198793119909119899) le 120572 (119879

2119909119899 119909)

(78)

holds for all 119899 isin N(iv) condition (iv) of Theorem 35 holds

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 120578(119909 119909) le 120572(119909 119910) andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0then 119879 has a unique fixed point

Theorem 37 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090(iii) 119879 is continuous function

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

The Scientific World Journal 11

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883 and 120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(79)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 38 Let (119883119872 lowast ⪯) be a partially ordered completetriangular fuzzy metric space and 119879 119883 rarr 119883 self-mappingsatisfying the following assertions

(i) 119879 is nondecreasing

(ii) there exists 1199090 isin 119883 such that 1199090 ⪯ 1198791199090

(iii) if 119909119899 is a sequence in 119883 such that 119909119899 ⪯ 119909119899+1 with119909119899 rarr 119909 as 119899 rarr infin then either

119879119909119899 ⪯ 119909 or 1198792119909119899 ⪯ 119909

(80)

holds for all 119899 isin N

(iv) assume that there exists H isin ΔH such that for all119909 119910 isin 119883120596 and 0120582 gt 0 with 119909 ⪯ 119910 we have

H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(81)

where 0 lt 119897 lt 119888

Then 119879 has a fixed point Moreover if for all 119909 119910 isin Fix(119879) wehave 119909 ⪯ 119910 andH(119906 119906 0 0 119906 119906) gt 0 for all 119906 gt 0 then 119879 hasa unique fixed point

Theorem 39 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1

119872 (119909 119879119909 120582119897)

minus 1

le

1

119872(119909 119910 120582119897)

minus 1

997904rArr H(

1

119872(119879119909 119879119910 120582119888)

minus 1

1

119872 (119909 119910 120582119897)

minus 1

1

119872 (119909 119879119909 120582119897)

minus 1

1

119872 (119910 119879119910 120582119897)

minus 1

1

119872 (119909 119879119910 2120582119897)

minus 1

1

119872 (119910 119879119909 120582119897)

minus 1) le 0

(82)

for all 119909 119910 isin 119883 and 120582 gt 0 where 0 lt 119897 lt 119888 Then 119879 has aunique fixed point

Theorem 40 Let (119883119872 lowast) be a complete triangular fuzzymetric space and119879 continuous self-mapping on119883 Assume that

1 minus 119887

1 minus 119887 + 119886

(

1

119872 (119909 119879119909 120582119897)

minus 1)

le

1

119872(119909 119910 1205822119897)

minus 1 997904rArr

1

119872(119879119909 119879119910 120582119888)

minus 1

le 119886(

1

119872(119909 119910 120582119897)

minus 1)

+ 119887

(1119872 (119909 119879119909 120582119897)) [(1119872 (119910 119879119910 120582119897)) minus 1]

1119872 (119909 119910 120582119897)

(83)

for all 119909 119910 isin 119883120596 and 120582 gt 0 where 0 lt 119897 lt 119888 and 119886 + 119887 lt 1Then 119879 has a unique fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR KAU for financial support

References

[1] V V Chistyakov ldquoModular metric spaces I basic conceptsrdquoNonlinear Analysis Theory Methods and Applications vol 72no 1 pp 1ndash14 2010

[2] V V Chistyakov ldquoModular metric spaces II application tosuperposition operatorsrdquo Nonlinear Analysis Theory Methodsand Applications vol 72 no 1 pp 15ndash30 2010

[3] H Nakano Modulared Semi-Ordered Linear Spaces TokyoMathematical Book Series Maruzen Tokyo Japan 1950

[4] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[5] W Orlicz Collected Papers Part I PWN Polish ScientificPublishers Warsaw 1988

[6] W OrliczCollected Papers Part II Polish Academy of SciencesWarsaw Poland 1988

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014

12 The Scientific World Journal

[7] L DieningTheoretical and numerical results for electrorheolog-ical fluids [PhD thesis] University of Freiburg Freiburg imBreisgau Germany 2002

[8] P Harjulehto P Hasto M Koskenoja and S Varonen ldquoThedirichlet energy integral and variable exponent Sobolev spaceswith zero boundary valuesrdquo Potential Analysis vol 25 no 3 pp205ndash222 2006

[9] J Heinonen T Kilpelinen and O Martio Nonlinear PotentialTheory of Degenerate Elliptic Equations Oxford UniversityPress Oxford UK 1993

[10] M A Khamsi W K Kozlowski and S Reich ldquoFixed pointtheory in modular function spacesrdquo Nonlinear Analysis vol 14no 11 pp 935ndash953 1990

[11] M A Khamsi and W A Kirk An Introduction to Metric Spacesand Fixed PointTheory JohnWileyampSonsNewYorkNYUSA2001

[12] N Hussain M A Khamsi and A Latif ldquoBanach operator pairsand common fixed pointsin modular function spacesrdquo FixedPoint Theory and Applications vol 2011 p 75 2011

[13] W M KozlowskiModular Function Spaces vol 122 of Series ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1988

[14] A A N Abdou and M A Khamsi ldquoOn the fixed points ofnonexpansive mappings in modular metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 1 p 229 2013

[15] W M Kozlowski ldquoNotes on modular function spaces Irdquo Com-ment Mathematica vol 28 pp 91ndash104 1988

[16] W M Kozlowski ldquoNotes on modular function spaces IIrdquo Com-ment Mathematica vol 28 pp 105ndash120 1988

[17] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[18] P Salimi A Latif and N Hussain ldquoModified 120572-120595-contractivemappings with applicationsrdquo Fixed Point Theory and Applica-tions vol 2013 no 1 p 151 2013

[19] N Hussain M A Kutbi S Khaleghizadeh and P SalimildquoDiscussions on recent results for 120572-120595 contractive mappingsrdquoAbstract and Applied Analysis vol 2014 Article ID 456482 13pages 2014

[20] N HussainM A Kutbi and P Salimi ldquoFixed point theory in 120572-complete metric spaces with applicationsrdquoAbstract and AppliedAnalysis vol 2014 Article ID 280817 11 pages 2014

[21] N Hussain E Karapinar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 no 1 p 114 2013

[22] R P Agarwal N Hussain and M-A Taoudi ldquoFixed point the-orems in ordered banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[23] N Hussain S Al-Mezel and Peyman Salimi ldquoFixed points for120595-graphic contractions with application to integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 575869 11pages 2013

[24] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

[25] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash229 2005

[26] N Hussain A Latif and P Salimi ldquoBest proximity point resultsfor modified Suzuki 120572-120595-proximal contractionsrdquo Fixed PointTheory and Applications vol 2014 no 1 p 10 2014

[27] T Suzuki ldquoA new type of fixed point theorem in metric spacesrdquoNonlinear Analysis Theory Methods and Applications vol 71no 11 pp 5313ndash5317 2009

[28] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[29] Nabi Shobe Shaban Sedghi J R Roshan and N HussainldquoSuzuki-type fixed point results in metric-like spacesrdquo Journalof Function Spaces andApplications vol 2013 Article ID 1436869 pages 2013

[30] N Hussain D Doric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 p 126 2012

[31] B Azadifar G Sadeghi R Saadati and C Park ldquoIntegral typecontractions in modular metric spacesrdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 483 2013

[32] A Razani and R Moradi ldquoCommon fixed point theoremsof integral type in modular spacesrdquo Bulletin of the IranianMathematical Society vol 35 no 2 pp 11ndash24 2009

[33] M Grabiec ldquoFixed points in fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 27 no 3 pp 385ndash389 1988

[34] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[35] I Altun and D Turkoglu ldquoSome fixed point theorems on fuzzymetric spaces with implicit relationsrdquo Communications of theKorean Mathematical Society vol 23 no 23 pp 111ndash124 2008

[36] C di Bari and C Vetro ldquoA fixed point theorem for a familyof mappings in a fuzzy metric spacerdquo Rendiconti del CircoloMatematico di Palermo vol 52 no 2 pp 315ndash321 2003

[37] D Gopal M Imdad C Vetro and M Hasan ldquoFixed pointtheory for cyclic weak 120601-contraction in fuzzy metric spacerdquoJournal of Nonlinear Analysis and Application vol 2012 ArticleID jnaa-00110 11 pages 2012

[38] NHussain S Khaleghizadeh P Salimi andANAfrahAbdouldquoA new approach to fixed point results in triangular intuitionis-tic fuzzymetric spacesrdquoAbstract andApplied Analysis vol 2014Article ID 690139 16 pages 2014

[39] M A Kutbi J Ahmad A Azam and N Hussain ldquoOn fuzzyfixed points for fuzzy maps with generalized weak propertyrdquoJournal of Applied Mathematics vol 2014 Article ID 549504 12pages 2014