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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 739319, 11 pages http://dx.doi.org/10.1155/2013/739319 Research Article Certain Sequence Spaces over the Non-Newtonian Complex Field Sebiha Tekin and Feyzi BaGar Department of Mathematics, Faculty of Arts and Sciences, Fatih University, e Hadimk¨ oy Campus, B¨ uy¨ ukc ¸ekmece, 34500 Istanbul, Turkey Correspondence should be addressed to Feyzi Bas ¸ar; [email protected] Received 4 February 2013; Accepted 5 April 2013 Academic Editor: Victor Kovtunenko Copyright © 2013 S. Tekin and F. Bas ¸ar. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. It is known from functional analysis that in classical calculus, the sets , ℓ ∞ , , 0 and ℓ of all bounded, convergent, null and - absolutely summable sequences are Banach spaces with their natural norms and they are complete according to the metric reduced from their norm, where 0<<∞. In this study, our main goal is to construct the spaces ∗ , ℓ ∗ ∞ , ∗ , ∗ 0 and ℓ ∗ over the non- Newtonian complex field C ∗ and to obtain the corresponding results for these spaces, where ̈ 0 ̈ < ̈ <∞. 1. Preliminaries, Background, and Notations A complete ordered field is a system consisting of a set , four binary operations ̇ +, ̇ −, ̇ ×, ̇ / for , and an ordering relation ̇ < for , all of which behave with respect to the set exactly as +, −, ×, /, < behave with respect to the set R of real numbers. We call the realm of the complete ordered field, [1, page 32]. A complete ordered field is called arithmetic if its realm is a subset of R. A bijective function with domain R and range a subset of R is called a generator. For example, the identity function , exponential function, and the function 3 are generators. Bashirov et al. [2] have recently emphasized on the multiplicative calculus and gave the results with applications corresponding to the well-known properties of derivative and integral in the classical calculus. Quite recently, Uzer [3] has extended the multiplicative calculus to the complex valued functions and gave the statements of some fundamental theorems and concepts of multiplicative complex calculus, and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples. Bashirov and Rıza [4] have studied the multiplicative differentiation for complex-valued functions and established the multiplicative Cauchy-Riemann condi- tions. Bashirov et al. [5] have investigated various problems from different fields which can be modelled more efficiently using multiplicative calculus, in place of Newtonian calculus. Let be a generator with range . An arithmetic with range , and operations and ordering relation defined as follows, is called -arithmetic. Let , ∈ . en, we define the operations -addition (̇+), -subtraction (̇−), -multiplication (̇×), -division ( ̇ /), and -ordering ( ̇ ≤) as follows: -addition : ̇+ = { −1 () + −1 ()} -subtraction : ̇− = { −1 () − −1 ()} -multiplication : ̇× = { −1 () × −1 ()} -division ( ̸ = ̇ 0) : ̇ /={ −1 () −1 () } -ordering : ̇ ≤ ⇐⇒ −1 () ≤ −1 () . (1) With the above new operations, (, ̇+, ̇−, ̇×, ̇ /, ̇ ≤) is an - arithmetic. In other words, one can easily show that (, ̇+, ̇−, ̇×, ̇ /, ̇ ≤) is a complete ordered field. As was seen, - generator generates -arithmetic. For example, the identity function generates classical arithmetic, and exponential func- tion generates geometric arithmetic. Each generator gener- ates exactly one arithmetic and each arithmetic is generated by exactly one generator. We denote -zero by ̇ 0 and -one by ̇ 1 which are obtained from (0) and (1), respectively. ̇ 0 and numbers obtained by successive -addition of ̇ 1 to ̇ 0 with

Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

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Page 1: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 739319 11 pageshttpdxdoiorg1011552013739319

Research ArticleCertain Sequence Spaces over the Non-Newtonian Complex Field

Sebiha Tekin and Feyzi BaGar

Department of Mathematics Faculty of Arts and Sciences Fatih University The Hadimkoy Campus Buyukcekmece34500 Istanbul Turkey

Correspondence should be addressed to Feyzi Basar feyzibasargmailcom

Received 4 February 2013 Accepted 5 April 2013

Academic Editor Victor Kovtunenko

Copyright copy 2013 S Tekin and F BasarThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is known from functional analysis that in classical calculus the sets 120596 ℓinfin 119888 1198880and ℓ

119901of all bounded convergent null and 119901-

absolutely summable sequences are Banach spaces with their natural norms and they are complete according to the metric reducedfrom their norm where 0 lt 119901 lt infin In this study our main goal is to construct the spaces 120596lowast ℓlowast

infin 119888lowast 119888lowast

0and ℓ

lowast

119901over the non-

Newtonian complex field Clowast and to obtain the corresponding results for these spaces where 0 lt 119901 ltinfin

1 Preliminaries Background and Notations

A complete ordered field is a system consisting of a set119883 fourbinary operations + minus times

for119883 and an ordering relation lt

for119883 all of which behave with respect to the set119883 exactly as+ minus times lt behave with respect to the setR of real numbersWe call 119883 the realm of the complete ordered field [1 page32] A complete ordered field is called arithmetic if its realmis a subset of R A bijective function with domain R andrange a subset of R is called a generator For example theidentity function 119868 exponential function and the function 119909

3

are generatorsBashirov et al [2] have recently emphasized on the

multiplicative calculus and gave the results with applicationscorresponding to thewell-known properties of derivative andintegral in the classical calculus Quite recently Uzer [3] hasextended the multiplicative calculus to the complex valuedfunctions and gave the statements of some fundamentaltheorems and concepts of multiplicative complex calculusand demonstrated some analogies between the multiplicativecomplex calculus and classical calculus by theoretical andnumerical examples Bashirov and Rıza [4] have studied themultiplicative differentiation for complex-valued functionsand established the multiplicative Cauchy-Riemann condi-tions Bashirov et al [5] have investigated various problemsfrom different fields which can be modelled more efficientlyusing multiplicative calculus in place of Newtonian calculus

Let 120572 be a generator with range 119860 An arithmetic withrange 119860 and operations and ordering relation defined asfollows is called 120572-arithmetic Let 119910 119911 isin 119860 Then wedefine the operations 120572-addition (+) 120572-subtraction (minus)120572-multiplication (times) 120572-division (

) and 120572-ordering (le) as

follows120572-addition 119910 + 119911 = 120572 120572

minus1

(119910) + 120572minus1

(119911)

120572-subtraction 119910 minus 119911 = 120572 120572minus1

(119910) minus 120572minus1

(119911)

120572-multiplication 119910 times 119911 = 120572 120572minus1

(119910) times 120572minus1

(119911)

120572-division (119911 = 0) 119910

119911 = 120572

120572minus1

(119910)

120572minus1

(119911)

120572-ordering 119910 le 119911 lArrrArr 120572minus1

(119910) le 120572minus1

(119911)

(1)

With the above new operations (119860 + minus times

le) is an 120572-arithmetic In other words one can easily show that(119860 + minus times

le) is a complete ordered field As was seen 120572-

generator generates 120572-arithmetic For example the identityfunction generates classical arithmetic and exponential func-tion generates geometric arithmetic Each generator gener-ates exactly one arithmetic and each arithmetic is generatedby exactly one generator We denote 120572-zero by 0 and 120572-oneby 1 which are obtained from 120572(0) and 120572(1) respectively 0and numbers obtained by successive 120572-addition of 1 to 0with

2 Abstract and Applied Analysis

numbers obtained by successive 120572-subtraction of 1 from 0 arecalled 120572-119894119899119905119890119892119890119903119904 Thus 120572-integers are given as follows

120572 (minus2) 120572 (minus1) 120572 (0) 120572 (1) 120572 (2) (2)

Thus we have for all 119899 isin Z that 119899 = 120572(119899) Let 119909 isin 119860 If119909 gt 0 then 119909 is called 120572-positive and if 119909 lt 0 then 119909 is called120572-119899119890119892119886119905119894V119890 The 120572-absolute value

|119909| of 119909 isin 119860 is defined by

|119909

| =

119909 119909 gt 0

0 119909 = 0

0 minus 119909 119909 lt 0

(3)

For any elements 119903 and 119904 in119860with 119903 lt 119904 the set of all elements119909 in119860 such that 119903 le 119909 le 119904 is called an 120572-119894119899119905119890119903V119886119897 is denoted by[119903 119904

] has 120572-extent of 119904 minus 119903 and has the 120572-interior consisting

of all elements 119909 in 119860 such that 119903 lt 119909 lt 119904 Let (119906119899) be an

infinite sequence of the elements in 119860 Then there is at mostone element 119906 in 119860 such that every 120572-interval with 119906 in its120572-interior contains all but finitely many terms of (119906

119899) If there

is such a number 119906 then (119906119899) is said to be 120572-119888119900119899V119890119903119892119890119899119905 to 119906

which is called the 120572-limit of (119906119899) In other words

lim119899rarrinfin

119906119899= 119906 (120572-convergent)

lArrrArr forall120576 gt 0 exist1198990isin N ni

|119906119899minus119906

| lt 120576

forall119899 ge 1198990and some 119906 isin 119860

(4)

Let 120572 and 120573 be two arbitrarily selected generators andlet lowast-(ldquostarrdquo) also be the ordered pair of arithmetics (120572-arithmetic120573-arithmetic) (119861 + minus times

le) is a 120573-arithmeticDefinitions given for 120572-arithmetic are also valid for120573-arithmetic For example 120573-convergence is defined bymeans of 120573-intervals and their 120573-interiors

In the lowast-calculus 120572-arithmetic is used for argumentsand 120573-arithmetic is used for values in particular changesin arguments and values are measured by 120572-differences and120573-differences respectivelyTheoperators of thelowast-calculus areapplied only to functions with arguments in 119860 and valuesin 119861 The lowast-119897119894119898119894119905 of a function 119891 at an element 119886 in 119860 isif it exists the unique number 119887 in 119861 such that for everysequence (119886

119899) of arguments of 119891 distinct from 119886 if (119886

119899)

is 120572-convergent to 119886 then 119891(119886119899) 120573-converges to 119887 and is

denoted by limlowast119909rarr119886

119891(119909) = 119887 That is

limlowast119909rarr119886

119891 (119909) = 119887 lArrrArr forall120598 gt 0 exist120575 gt 0 ni|119891 (119909) minus 119887

| lt 120598

forall119909 isin 119860 with |119909 minus 119886

| lt 120575

(5)

A function 119891 is lowast-119888119900119899119905119894119899119906119900119906119904 at a point 119886 in 119860 if and onlyif 119886 is an argument of 119891 and limlowast

119909rarr119886119891(119909) = 119891(119886) When

120572 and 120573 are the identity function 119868 the concepts of lowast-limitandlowast-continuity are identical with those of classical limit andclassical continuity

The isomorphism from 120572-arithmetic to 120573-arithmetic isthe unique function 120580(119894119900119905119886) that possesses the following threeproperties

(i) 120580 is one to one(ii) 120580 is from 119860 onto 119861(iii) For any numbers 119906 and V in 119860

120580 (119906 + V) = 120580 (119906) + 120580 (V)

120580 (119906 minus V) = 120580 (119906) minus 120580 (V)

120580 (119906 times V) = 120580 (119906) times 120580 (V)

120580 (119906

V) = 120580 (119906)

120580 (V) V = 0

119906 leV lArrrArr 120580 (119906) le 120580 (V)

(6)

It turns out that 120580(119909) = 120573120572minus1

(119909) for every 119909 in 119860 andthat 120580( 119899) = 119899 for every integer 119899 Since for example 119906 + V

= 120580minus1

120580(119906) + 120580(V) it should be clear that any statement in120572-arithmetic can readily be transformed into a statement in120573-arithmetic

2 Non-Newtonian Complex Field andSome Inequalities

Let 119886 isin (119860 + minus times

le) and 119887 isin (119861 + minus times

le) be arbitrarily

chosen elements from corresponding arithmetics Then theordered pair ( 119886

119887) is called as a lowast-pointThe set of all lowast-points

is called the set of non-Newtonian complex numbers and isdenoted by Clowast that is

Clowast

= 119911lowast

= 119886 oplus (119894lowast

⊙119887) 119886 119887 isin R 119894

lowast

= (0 1)

= 119911lowast

= ( 119886119887) 119886 isin 119860 sube R

119887 isin 119861 sube R

(7)

The binary operations addition (oplus) and multiplication (⊙) ofnon-Newtonian complex numbers 119911

lowast

1= ( 11988611198871) and 119911

lowast

2=

( 11988621198872) are defined as follows

oplus Clowast

times Clowast

997888rarr Clowast

(119911lowast

1 119911lowast

2) 997891997888rarr 119911

lowast

1oplus 119911lowast

2= ( 1198861+ 11988621198871+

1198872)

⊙ Clowast

times Clowast

997888rarr Clowast

(119911lowast

1 119911lowast

2) 997891997888rarr 119911

lowast

1⊙ 119911lowast

2= (120572 ( 119886

11198862minus

1198871

1198872)

120573 ( 1198861

1198872+

1198871

1198862))

(8)

where 1198861 1198862isin 119860 and

11988711198872isin 119861 with

1198861= 120572minus1

( 1198861) = 120572minus1

(120572 (1198861)) = 119886

1isin R

1198871= 120573minus1

(1198871) = 120573minus1

(120573 (1198871)) = 119887

1isin R

(9)

Then we have the following

Theorem 1 (Clowast oplus ⊙) is a field

Proof A straightforward calculation leads to the followingstatements

Abstract and Applied Analysis 3

(i) (Clowast oplus) is an Abelian group(ii) (Clowast 120579

lowast

⊙) is an Abelian group(iii) the operation ⊙ is distributive over the operationoplus

which conclude that (Clowast oplus ⊙) is a field

Let 119887 isin 119861 sube R Then the number

119887 times119887 is called the 120573-

square of 119887 and is denoted by 119887

2 Let 119887 be a non-negative

number in 119861 Then 120573[radic120573minus1(119887)] is called the 120573-square root

of 119887 and is denoted by radic

119887 The lowast-distance 119889lowast between two

arbitrarily elements 119911lowast1

= ( 11988611198871) and 119911

lowast

2= ( 11988621198872) of the set

Clowast is defined by

119889lowast

Clowast

times Clowast

997888rarr[0infin

) = 1198611015840

sub 119861

(119911lowast

1 119911lowast

2) 997888rarr 119889

lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

(10)

Up to now we know that Clowast is a field and the distancebetween two points in Clowast is computed by the function 119889

lowastdefined by (10) Now we define the lowast-norm and next derivesome required inequalities in the sense of non-Newtoniancomplex calculus

Let 119911lowast isin Clowast be an arbitrary element 119889lowast(119911lowast 120579lowast) is calledlowast-norm of 119911lowast and is denoted by

sdot In other words

119911lowast = 119889lowast

(119911lowast

120579lowast

)

=

radic[120580 ( 119886 minus 0)]

2

+ (119887 minus 0)

2

= 120573 (radic1198862+ 1198872)

(11)

where 119911lowast

= ( 119886119887) and 120579

lowast

= (0 0) Moreover since for all119911lowast

1 119911lowast

2isin Clowast we have 119889lowast(119911lowast

1 119911lowast

2) =

119911lowast

1⊖119911lowast

2

119889lowast is the induced

metric from the norm sdot

Lemma 2 (lowast-Triangle inequality) Let 119911lowast1 119911lowast

2isin Clowast Then

119911lowast

1oplus 119911lowast

2

le

119911lowast

1

+

119911lowast

2

(12)

Proof Let 119911lowast1 119911lowast

2isin Clowast Then a straightforward calculation

gives that

119911lowast

1oplus 119911lowast

2

=

radic[120580 ( 1198861+ 1198862)]2

+(1198871+

1198872)

2

= 120573 [radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]

le 120573 (radic1198862

1+ 1198872

1+ radic1198862

2+ 1198872

2)

= 120573120573minus1

[120573 (radic1198862

1+ 1198872

1)] + 120573

minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573 [120573minus1

(119911lowast

1

) + 120573

minus1

(119911lowast

2

)]

=119911lowast

1

+

119911lowast

2

(13)

Hence the inequality (12) holds

Lemma 3 119911lowast

1⊙ 119911lowast

2

=

119911lowast

1

times

119911lowast

2

for all 119911lowast

1 119911lowast

2isin Clowast

Proof Let 119911lowast1 119911lowast

2isin Clowast In this case one can observe by a

routine verification that

119911lowast

1⊙ 119911lowast

2

=

radic120580 [120572 ( 119886

11198862minus

1198871

1198872)]

2

+ [120573 ( 1198861

1198872+

1198871

1198862)]

2

= 120573 [radic(11988611198862minus 11988711198872)2

+ (11988611198872+ 11988711198862)2

]

= 120573(radic1198862

1+ 1198872

1radic1198862

2+ 1198872

2)

= 120573120573minus1

[120573 (radic1198862

1+ 1198872

1)] times 120573

minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573 [120573minus1

(119911lowast

1

) times 120573

minus1

(119911lowast

2

)]

=119911lowast

1

times

119911lowast

2

(14)

as required

Lemma 4 Let 119911lowast

1 119911lowast

2isin Clowast Then the following inequality

holds

119911lowast

1oplus 119911lowast

2

(1 +

119911lowast

1oplus 119911lowast

2

)

le119911lowast

1

(1 +

119911lowast

1

) +

119911lowast

2

(1 +

119911lowast

2

)

(15)

Proof Let 119911lowast1 119911lowast

2isin Clowast Then one can see that

119911lowast

1oplus 119911lowast

2

(1 +

119911lowast

1oplus 119911lowast

2

)

= 120573 [radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]

1 + 120573 [radic(119886

1+ 1198862)2

+ (1198871+ 1198872)2

]

= 120573[[

[

radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

1 + radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]]

]

le 120573[[

[

(

radic1198862

1+ 1198872

1

1 + radic1198862

1+ 1198872

1

) + (

radic1198862

2+ 1198872

2

1 + radic1198862

2+ 1198872

2

)]]

]

4 Abstract and Applied Analysis

= 120573

120573minus1

[120573 (radic1198862

1+ 1198872

1)]

120573minus1

120573 [120573minus1

(1) + 120573minus1

[120573 (radic1198862

1+ 1198872

1)]]

+

120573minus1

[120573 (radic1198862

2+ 1198872

2)]

120573minus1

120573120573minus1

(1) + 120573minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

] + [

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573120573minus1

120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

]

+ 120573minus1

120573[

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573 120573minus1

[119911lowast

1

(1 +

119911lowast

1

)] + 120573

minus1

[119911lowast

2

(1 +

119911lowast

2

)]

=119911lowast

1

(1 +

119911lowast

1

) +

119911lowast

2

(1 +

119911lowast

2

)

(16)

This means that the inequality (15) holds

Lemma 5 (lowast-Minkowski inequality) Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin

Clowast for all 119896 isin 1 2 3 119899 Then

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(17)

Proof Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin Clowast for all 119896 isin 1 2 3 119899

Then119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

0oplus 119905lowast

0

119901

+119911lowast

1oplus 119905lowast

1

119901

+ sdot sdot sdot +119911lowast

119899oplus 119905lowast

119899

119901

= 120573 [120573minus1

(119911lowast

0oplus 119905lowast

0

119901

) + 120573minus1

(119911lowast

1oplus 119905lowast

1

119901

)

+ sdot sdot sdot + 120573minus1

(119911lowast

119899oplus 119905lowast

119899

119901

)]

(18)

Let us take 119911lowast

119896= (119896 119910119896) 119911119896

= (119909119896 119910119896) 119905lowast

119896= (119896 V119896) 119905119896

=

(119906119896 V119896) Then since the equality

119911lowast

119896oplus 119905lowast

119896

=

radic[120580 (119896+ 119896)]2

+ ( 119910119896+ V119896)2

= 120573 [radic(119909119896+ 119906119896)2

+ (119910119896+ V119896)2

]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)

(19)

holds for every fixed 119896 we obtain

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

119896oplus 119905lowast

119896

times sdot sdot sdot times

119911lowast

119896oplus 119905lowast

119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 [120573minus1

(119911lowast

119896oplus 119905lowast

119896

) times sdot sdot sdot times 120573

minus1

(119911lowast

119896oplus 119905lowast

119896

)]

= 120573 120573minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)] times sdot sdot sdot times 120573

minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

(20)

by (18) and Minkowski inequality in the complex field leadsus to119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

= 120573 120573minus1

[120573 (10038161003816100381610038161199110oplus 1199050

1003816100381610038161003816

120573minus1

(119901)

)] + 120573minus1

[120573 (10038161003816100381610038161199111oplus 1199051

1003816100381610038161003816

120573minus1

(119901)

)]

+ sdot sdot sdot + 120573minus1

[120573 (1003816100381610038161003816119911119899oplus 119905119899

1003816100381610038161003816

120573minus1

(119901)

)]

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

le 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

+(

119899

sum

119896=0

1003816100381610038161003816119905119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(21)

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

= 120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdottimes[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

= 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

Abstract and Applied Analysis 5

=

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+ 120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

=

120573minus1[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

120573minus1

(119901)

(22)

On the other hand let us prove

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

(23)

Indeed

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

119901

=120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

times sdot sdot sdot times 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

times sdot sdot sdot times (

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

=

119899

sum

119896=0

119911lowast

119896

119901

(24)

Substituting the relation (24) in (22) we obtain

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

=[

[

(

119899

sum

119896=0

|119911119896|120573minus1

(119901)

)

1120573minus1

(119901)

+ (

119899

sum

119896=0

|119905119896|120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(25)

By using this equality in (21) we get

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(26)

as desired

Theorem 6 (Clowast 119889lowast) is a complete metric space where 119889lowast is

defined by (10)

Proof First we show that 119889lowast defined by (10) is a metric onClowast

It is immediate for 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] ge 0

(27)

(i) Now we show that 119889lowast(119911lowast1 119911lowast

2) = 0 if and only if 119911lowast

1=

119911lowast

2for 119911lowast1 119911lowast

2isin Clowast Indeed

119889lowast

(119911lowast

1 119911lowast

2) = 0 lArrrArr

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 0

lArrrArr 120573[radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] = 120573 (0)

lArrrArr (1198861minus 1198862)2

+ (1198871minus 1198872)2

= 0

lArrrArr 1198861minus 1198862= 0 119887

1minus 1198872= 0

lArrrArr 1198861= 1198862 119887

1= 1198872

lArrrArr 1198861= 1198862

1198871=

1198872

lArrrArr 119911lowast

1= ( 11988611198871) = ( 119886

21198872) = 119911lowast

2

(28)

6 Abstract and Applied Analysis

(ii) One can easily establish for all 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

= 120573 [radic(1198862minus 1198861)2

+ (1198872minus 1198871)2

]

= 119889lowast

(119911lowast

2 119911lowast

1)

(29)

(iii) We show that the inequality 119889lowast

(119911lowast

1 119911lowast

2) +119889lowast

(119911lowast

2

119911lowast

3) ge 119889lowast

(119911lowast

1 119911lowast

3) holds for all 119911lowast

1 119911lowast

2 119911lowast

3isin Clowast In fact

119889lowast

(119911lowast

1 119911lowast

2) + 119889lowast

(119911lowast

2 119911lowast

3)

=

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

+

radic[120580 ( 1198862minus 1198863)]2

+ (1198872minus

1198873)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

+ 120573 [radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

+ radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

ge120573 [radic(1198861minus 1198863)2

+ (1198871minus 1198873)2

]

=

radic[120580 ( 1198861minus 1198863)]2

+ (1198871minus

1198873)

2

= 119889lowast

(119911lowast

1 119911lowast

3)

(30)

Therefore 119889lowast is a metric over ClowastNow we can show that the metric space (Clowast 119889lowast) is

complete Let (119911lowast119899)119899isinN

be an arbitrary Cauchy sequence inClowastIn this case for all 120576 gt 0 there exists an 119899

0isin N such that

119889lowast

(119911lowast

119898 119911lowast

119899) lt 120576 for all 119898 119899 ge 119899

0 Let 119911lowast

119898= ( 119886119898119887119898) isin Clowast and

119898 isin N Then

119889lowast

(119911lowast

119898 119911lowast

119899) =

radic[120580 ( 119886119898minus 119886119899)]2

+ (119887119898minus

119887119899)

2

= 120573 [radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

]

lt 120576 = 120573 (1205761015840

)

(31)

Thus we obtain that

radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

lt 1205761015840

(32)

On the other hand since the following inequalities

1003816100381610038161003816119886119898minus 119886119899

1003816100381610038161003816= radic(119886

119898minus 119886119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

1003816100381610038161003816119887119898minus 119887119899

1003816100381610038161003816= radic(119887

119898minus 119887119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

(33)

hold we therefore have by (32) that |119886119898

minus 119886119899| lt 1205761015840 and |119887

119898minus

119887119899| lt 1205761015840 This means that (119886

119899) and (119887

119899) are Cauchy sequences

with real terms Since R is complete it is clear that for every1205761015840

gt 0 there exists an 1198991isin N such that |119886

119899minus 119886| lt 120576

1015840

2 for all119899 ge 119899

1and for every 120576

1015840

gt 0 there exists an 1198992isin N such that

|119887119899minus 119887| lt 120576

1015840

2 for all 119899 ge 1198992

Define 119911lowast isin Clowast by 119911lowast

= ( 119886119887) Then we have

119889lowast

(119911lowast

119899 119911lowast

) =

radic[120580 ( 119886119899minus 119886)]2

+ [119887119899minus

119887]

2

= 120573 [radic(119886119899minus 119886)2

+ (119887119899minus 119887)2

]

le 120573 [radic(119886119899minus 119886)2

+ radic(119887119899minus 119887)2

]

= 120573 (1003816100381610038161003816119886119899minus 119886

1003816100381610038161003816+1003816100381610038161003816119887119899minus 119887

1003816100381610038161003816)

le 120573(

1205761015840

2

+

1205761015840

2

)

= 120573 (1205761015840

) = 120576

(34)

Hence (Clowast 119889lowast) is a complete metric space

Since Clowast is a complete metric space with the metric119889lowast defined by (10) induced by the norm

sdot as a direct

consequence of Theorem 6 we have the following

Corollary 7 Clowast is a Banach space with the norm sdot

defined

by

119911lowast =

radic[120580 ( 119886)]

2

+ (119887)

2

119911lowast

= ( 119886119887) isin C

lowast

(35)

3 Sequence Spaces over Non-NewtonianComplex Field

In this section we define the sets 120596lowast ℓlowastinfin 119888lowast 119888lowast

0 and ℓ

lowast

119901of

all bounded convergent null and absolutely 119901-summablesequences over the non-Newtonian complex field Clowast whichcorrespond to the sets 120596 ℓ

infin 119888 1198880 and ℓ

119901over the complex

field C respectively That is to say that

120596lowast

= 119911lowast

= (119911lowast

119896) 119911lowast

119896isin Clowast

forall119896 isin N

ℓlowast

infin= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sup119896isinN

119911lowast

119896

ltinfin

119888lowast

= 119911lowast

= (119911lowast

119896) isin 120596lowast

exist119897lowast

isin Clowast

ni limlowast119896rarrinfin

119911lowast

119896= 119897lowast

119888lowast

0= 119911

lowast

= (119911lowast

119896) isin 120596lowast

limlowast119896rarrinfin

119911lowast

119896= 120579lowast

ℓlowast

119901= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sum

119896

119911lowast

119896

119901

ltinfin

(36)

For simplicity in notation here and in what follows thesummation without limits runs from 0 to infin One can easily

Abstract and Applied Analysis 7

see that the set 120596lowast forms a vector space over Clowast with respectto the algebraic operations addition (+) and scalar multipli-cation (times) defined on 120596

lowast as follows

+ 120596lowast

times 120596lowast

997888rarr 120596lowast

(119911lowast

119905lowast

) 997891997888rarr 119911lowast

+ 119905lowast

= (119911lowast

119896oplus 119905lowast

119896)

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast

times Clowast

times 120596lowast

997888rarr 120596lowast

(120572 119911lowast

) 997891997888rarr 120572 times 119911lowast

= (120572 ⊙ 119911lowast

119896)

119911lowast

= (119911lowast

119896) isin 120596lowast

120572 isin Clowast

(37)

Theorem 8 Define the function 119889120596lowast by

119889120596lowast 120596lowast

times 120596lowast

997888rarr 1198611015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 tlowast119896)]

(38)

where (120583119896) isin 1198611015840

sube 119861 such that sum119896120583119896is convergent with 120583

119896gt 0

for all 119896 isin N Then (120596lowast 119889120596lowast) is a metric space

Proof We show that 119889120596lowast satisfies the metric axioms on the

space 120596lowast of all non-Newtonian complex valued sequences(i) First we show that 119889

120596lowast(119911lowast

119905lowast

) ge 0 for all 119911lowast 119905lowast isin 120596lowast

Because (Clowast 119889lowast) is a metric space we have 119889lowast(119911lowast119896 119905lowast

119896) ge 0

119911lowast

119896 119905lowast

119896isin Clowast and 1 + 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 Hence we obtain that

119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 Moreover since 120583

119896gt 0 for all

119896 isin N we have

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 (39)

This means that

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0

(40)

(ii) We show that 119889120596lowast(119911lowast

119905lowast

) = 0 iff 119911lowast

= 119905lowast In this situ-

ation one can see that

119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

lArrrArr 120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

120583119896gt 0 forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896) = 0 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 forall119896 isin N

lArrrArr 119911lowast

119896= 119905lowast

119896 119889lowastmetric forall119896 isin N

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(41)

(iii) We show that 119889120596lowast(119911lowast

119905lowast

) = 119889120596lowast(119905lowast

119911lowast

) for 119911lowast

=

(119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast First we know that 119889lowast is a metric over

Clowast Thus

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119911lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119911lowast

119896)]

= 119889120596lowast (119905lowast

119911lowast

)

(42)

(iv) We show that 119889120596lowast(119911lowast

119905lowast

) + 119889120596lowast(119905lowast

119906lowast

) ge 119889120596lowast(119911lowast

119906lowast

)

holds for 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin 120596

lowast Againusing the fact that (Clowast 119889lowast) is a metric space it is easy to seeby Lemma 4 that

119889120596lowast (119911lowast

119906lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

1 + [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times 119889

lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+ 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

= 119889120596lowast (119911lowast

119905lowast

) + 119889120596lowast (119905lowast

119906lowast

)

(43)

as required

Theorem 9 The set ℓlowastinfin

is a sequence space

Proof It is trivial that the inclusion ℓlowast

infinsub 120596lowast holds

(i) We show that 119911lowast + 119905lowast

isin ℓlowast

infinfor 119911lowast = (119911

lowast

119896) 119905lowast

= (119905lowast

119896) isin

ℓlowast

infin Indeed combining the hypothesis

sup119896isinN

119911lowast

119896

ltinfin sup

119896isinN

119905lowast

119896

ltinfin (44)

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

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Stochastic AnalysisInternational Journal of

Page 2: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

2 Abstract and Applied Analysis

numbers obtained by successive 120572-subtraction of 1 from 0 arecalled 120572-119894119899119905119890119892119890119903119904 Thus 120572-integers are given as follows

120572 (minus2) 120572 (minus1) 120572 (0) 120572 (1) 120572 (2) (2)

Thus we have for all 119899 isin Z that 119899 = 120572(119899) Let 119909 isin 119860 If119909 gt 0 then 119909 is called 120572-positive and if 119909 lt 0 then 119909 is called120572-119899119890119892119886119905119894V119890 The 120572-absolute value

|119909| of 119909 isin 119860 is defined by

|119909

| =

119909 119909 gt 0

0 119909 = 0

0 minus 119909 119909 lt 0

(3)

For any elements 119903 and 119904 in119860with 119903 lt 119904 the set of all elements119909 in119860 such that 119903 le 119909 le 119904 is called an 120572-119894119899119905119890119903V119886119897 is denoted by[119903 119904

] has 120572-extent of 119904 minus 119903 and has the 120572-interior consisting

of all elements 119909 in 119860 such that 119903 lt 119909 lt 119904 Let (119906119899) be an

infinite sequence of the elements in 119860 Then there is at mostone element 119906 in 119860 such that every 120572-interval with 119906 in its120572-interior contains all but finitely many terms of (119906

119899) If there

is such a number 119906 then (119906119899) is said to be 120572-119888119900119899V119890119903119892119890119899119905 to 119906

which is called the 120572-limit of (119906119899) In other words

lim119899rarrinfin

119906119899= 119906 (120572-convergent)

lArrrArr forall120576 gt 0 exist1198990isin N ni

|119906119899minus119906

| lt 120576

forall119899 ge 1198990and some 119906 isin 119860

(4)

Let 120572 and 120573 be two arbitrarily selected generators andlet lowast-(ldquostarrdquo) also be the ordered pair of arithmetics (120572-arithmetic120573-arithmetic) (119861 + minus times

le) is a 120573-arithmeticDefinitions given for 120572-arithmetic are also valid for120573-arithmetic For example 120573-convergence is defined bymeans of 120573-intervals and their 120573-interiors

In the lowast-calculus 120572-arithmetic is used for argumentsand 120573-arithmetic is used for values in particular changesin arguments and values are measured by 120572-differences and120573-differences respectivelyTheoperators of thelowast-calculus areapplied only to functions with arguments in 119860 and valuesin 119861 The lowast-119897119894119898119894119905 of a function 119891 at an element 119886 in 119860 isif it exists the unique number 119887 in 119861 such that for everysequence (119886

119899) of arguments of 119891 distinct from 119886 if (119886

119899)

is 120572-convergent to 119886 then 119891(119886119899) 120573-converges to 119887 and is

denoted by limlowast119909rarr119886

119891(119909) = 119887 That is

limlowast119909rarr119886

119891 (119909) = 119887 lArrrArr forall120598 gt 0 exist120575 gt 0 ni|119891 (119909) minus 119887

| lt 120598

forall119909 isin 119860 with |119909 minus 119886

| lt 120575

(5)

A function 119891 is lowast-119888119900119899119905119894119899119906119900119906119904 at a point 119886 in 119860 if and onlyif 119886 is an argument of 119891 and limlowast

119909rarr119886119891(119909) = 119891(119886) When

120572 and 120573 are the identity function 119868 the concepts of lowast-limitandlowast-continuity are identical with those of classical limit andclassical continuity

The isomorphism from 120572-arithmetic to 120573-arithmetic isthe unique function 120580(119894119900119905119886) that possesses the following threeproperties

(i) 120580 is one to one(ii) 120580 is from 119860 onto 119861(iii) For any numbers 119906 and V in 119860

120580 (119906 + V) = 120580 (119906) + 120580 (V)

120580 (119906 minus V) = 120580 (119906) minus 120580 (V)

120580 (119906 times V) = 120580 (119906) times 120580 (V)

120580 (119906

V) = 120580 (119906)

120580 (V) V = 0

119906 leV lArrrArr 120580 (119906) le 120580 (V)

(6)

It turns out that 120580(119909) = 120573120572minus1

(119909) for every 119909 in 119860 andthat 120580( 119899) = 119899 for every integer 119899 Since for example 119906 + V

= 120580minus1

120580(119906) + 120580(V) it should be clear that any statement in120572-arithmetic can readily be transformed into a statement in120573-arithmetic

2 Non-Newtonian Complex Field andSome Inequalities

Let 119886 isin (119860 + minus times

le) and 119887 isin (119861 + minus times

le) be arbitrarily

chosen elements from corresponding arithmetics Then theordered pair ( 119886

119887) is called as a lowast-pointThe set of all lowast-points

is called the set of non-Newtonian complex numbers and isdenoted by Clowast that is

Clowast

= 119911lowast

= 119886 oplus (119894lowast

⊙119887) 119886 119887 isin R 119894

lowast

= (0 1)

= 119911lowast

= ( 119886119887) 119886 isin 119860 sube R

119887 isin 119861 sube R

(7)

The binary operations addition (oplus) and multiplication (⊙) ofnon-Newtonian complex numbers 119911

lowast

1= ( 11988611198871) and 119911

lowast

2=

( 11988621198872) are defined as follows

oplus Clowast

times Clowast

997888rarr Clowast

(119911lowast

1 119911lowast

2) 997891997888rarr 119911

lowast

1oplus 119911lowast

2= ( 1198861+ 11988621198871+

1198872)

⊙ Clowast

times Clowast

997888rarr Clowast

(119911lowast

1 119911lowast

2) 997891997888rarr 119911

lowast

1⊙ 119911lowast

2= (120572 ( 119886

11198862minus

1198871

1198872)

120573 ( 1198861

1198872+

1198871

1198862))

(8)

where 1198861 1198862isin 119860 and

11988711198872isin 119861 with

1198861= 120572minus1

( 1198861) = 120572minus1

(120572 (1198861)) = 119886

1isin R

1198871= 120573minus1

(1198871) = 120573minus1

(120573 (1198871)) = 119887

1isin R

(9)

Then we have the following

Theorem 1 (Clowast oplus ⊙) is a field

Proof A straightforward calculation leads to the followingstatements

Abstract and Applied Analysis 3

(i) (Clowast oplus) is an Abelian group(ii) (Clowast 120579

lowast

⊙) is an Abelian group(iii) the operation ⊙ is distributive over the operationoplus

which conclude that (Clowast oplus ⊙) is a field

Let 119887 isin 119861 sube R Then the number

119887 times119887 is called the 120573-

square of 119887 and is denoted by 119887

2 Let 119887 be a non-negative

number in 119861 Then 120573[radic120573minus1(119887)] is called the 120573-square root

of 119887 and is denoted by radic

119887 The lowast-distance 119889lowast between two

arbitrarily elements 119911lowast1

= ( 11988611198871) and 119911

lowast

2= ( 11988621198872) of the set

Clowast is defined by

119889lowast

Clowast

times Clowast

997888rarr[0infin

) = 1198611015840

sub 119861

(119911lowast

1 119911lowast

2) 997888rarr 119889

lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

(10)

Up to now we know that Clowast is a field and the distancebetween two points in Clowast is computed by the function 119889

lowastdefined by (10) Now we define the lowast-norm and next derivesome required inequalities in the sense of non-Newtoniancomplex calculus

Let 119911lowast isin Clowast be an arbitrary element 119889lowast(119911lowast 120579lowast) is calledlowast-norm of 119911lowast and is denoted by

sdot In other words

119911lowast = 119889lowast

(119911lowast

120579lowast

)

=

radic[120580 ( 119886 minus 0)]

2

+ (119887 minus 0)

2

= 120573 (radic1198862+ 1198872)

(11)

where 119911lowast

= ( 119886119887) and 120579

lowast

= (0 0) Moreover since for all119911lowast

1 119911lowast

2isin Clowast we have 119889lowast(119911lowast

1 119911lowast

2) =

119911lowast

1⊖119911lowast

2

119889lowast is the induced

metric from the norm sdot

Lemma 2 (lowast-Triangle inequality) Let 119911lowast1 119911lowast

2isin Clowast Then

119911lowast

1oplus 119911lowast

2

le

119911lowast

1

+

119911lowast

2

(12)

Proof Let 119911lowast1 119911lowast

2isin Clowast Then a straightforward calculation

gives that

119911lowast

1oplus 119911lowast

2

=

radic[120580 ( 1198861+ 1198862)]2

+(1198871+

1198872)

2

= 120573 [radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]

le 120573 (radic1198862

1+ 1198872

1+ radic1198862

2+ 1198872

2)

= 120573120573minus1

[120573 (radic1198862

1+ 1198872

1)] + 120573

minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573 [120573minus1

(119911lowast

1

) + 120573

minus1

(119911lowast

2

)]

=119911lowast

1

+

119911lowast

2

(13)

Hence the inequality (12) holds

Lemma 3 119911lowast

1⊙ 119911lowast

2

=

119911lowast

1

times

119911lowast

2

for all 119911lowast

1 119911lowast

2isin Clowast

Proof Let 119911lowast1 119911lowast

2isin Clowast In this case one can observe by a

routine verification that

119911lowast

1⊙ 119911lowast

2

=

radic120580 [120572 ( 119886

11198862minus

1198871

1198872)]

2

+ [120573 ( 1198861

1198872+

1198871

1198862)]

2

= 120573 [radic(11988611198862minus 11988711198872)2

+ (11988611198872+ 11988711198862)2

]

= 120573(radic1198862

1+ 1198872

1radic1198862

2+ 1198872

2)

= 120573120573minus1

[120573 (radic1198862

1+ 1198872

1)] times 120573

minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573 [120573minus1

(119911lowast

1

) times 120573

minus1

(119911lowast

2

)]

=119911lowast

1

times

119911lowast

2

(14)

as required

Lemma 4 Let 119911lowast

1 119911lowast

2isin Clowast Then the following inequality

holds

119911lowast

1oplus 119911lowast

2

(1 +

119911lowast

1oplus 119911lowast

2

)

le119911lowast

1

(1 +

119911lowast

1

) +

119911lowast

2

(1 +

119911lowast

2

)

(15)

Proof Let 119911lowast1 119911lowast

2isin Clowast Then one can see that

119911lowast

1oplus 119911lowast

2

(1 +

119911lowast

1oplus 119911lowast

2

)

= 120573 [radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]

1 + 120573 [radic(119886

1+ 1198862)2

+ (1198871+ 1198872)2

]

= 120573[[

[

radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

1 + radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]]

]

le 120573[[

[

(

radic1198862

1+ 1198872

1

1 + radic1198862

1+ 1198872

1

) + (

radic1198862

2+ 1198872

2

1 + radic1198862

2+ 1198872

2

)]]

]

4 Abstract and Applied Analysis

= 120573

120573minus1

[120573 (radic1198862

1+ 1198872

1)]

120573minus1

120573 [120573minus1

(1) + 120573minus1

[120573 (radic1198862

1+ 1198872

1)]]

+

120573minus1

[120573 (radic1198862

2+ 1198872

2)]

120573minus1

120573120573minus1

(1) + 120573minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

] + [

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573120573minus1

120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

]

+ 120573minus1

120573[

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573 120573minus1

[119911lowast

1

(1 +

119911lowast

1

)] + 120573

minus1

[119911lowast

2

(1 +

119911lowast

2

)]

=119911lowast

1

(1 +

119911lowast

1

) +

119911lowast

2

(1 +

119911lowast

2

)

(16)

This means that the inequality (15) holds

Lemma 5 (lowast-Minkowski inequality) Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin

Clowast for all 119896 isin 1 2 3 119899 Then

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(17)

Proof Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin Clowast for all 119896 isin 1 2 3 119899

Then119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

0oplus 119905lowast

0

119901

+119911lowast

1oplus 119905lowast

1

119901

+ sdot sdot sdot +119911lowast

119899oplus 119905lowast

119899

119901

= 120573 [120573minus1

(119911lowast

0oplus 119905lowast

0

119901

) + 120573minus1

(119911lowast

1oplus 119905lowast

1

119901

)

+ sdot sdot sdot + 120573minus1

(119911lowast

119899oplus 119905lowast

119899

119901

)]

(18)

Let us take 119911lowast

119896= (119896 119910119896) 119911119896

= (119909119896 119910119896) 119905lowast

119896= (119896 V119896) 119905119896

=

(119906119896 V119896) Then since the equality

119911lowast

119896oplus 119905lowast

119896

=

radic[120580 (119896+ 119896)]2

+ ( 119910119896+ V119896)2

= 120573 [radic(119909119896+ 119906119896)2

+ (119910119896+ V119896)2

]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)

(19)

holds for every fixed 119896 we obtain

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

119896oplus 119905lowast

119896

times sdot sdot sdot times

119911lowast

119896oplus 119905lowast

119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 [120573minus1

(119911lowast

119896oplus 119905lowast

119896

) times sdot sdot sdot times 120573

minus1

(119911lowast

119896oplus 119905lowast

119896

)]

= 120573 120573minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)] times sdot sdot sdot times 120573

minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

(20)

by (18) and Minkowski inequality in the complex field leadsus to119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

= 120573 120573minus1

[120573 (10038161003816100381610038161199110oplus 1199050

1003816100381610038161003816

120573minus1

(119901)

)] + 120573minus1

[120573 (10038161003816100381610038161199111oplus 1199051

1003816100381610038161003816

120573minus1

(119901)

)]

+ sdot sdot sdot + 120573minus1

[120573 (1003816100381610038161003816119911119899oplus 119905119899

1003816100381610038161003816

120573minus1

(119901)

)]

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

le 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

+(

119899

sum

119896=0

1003816100381610038161003816119905119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(21)

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

= 120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdottimes[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

= 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

Abstract and Applied Analysis 5

=

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+ 120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

=

120573minus1[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

120573minus1

(119901)

(22)

On the other hand let us prove

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

(23)

Indeed

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

119901

=120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

times sdot sdot sdot times 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

times sdot sdot sdot times (

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

=

119899

sum

119896=0

119911lowast

119896

119901

(24)

Substituting the relation (24) in (22) we obtain

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

=[

[

(

119899

sum

119896=0

|119911119896|120573minus1

(119901)

)

1120573minus1

(119901)

+ (

119899

sum

119896=0

|119905119896|120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(25)

By using this equality in (21) we get

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(26)

as desired

Theorem 6 (Clowast 119889lowast) is a complete metric space where 119889lowast is

defined by (10)

Proof First we show that 119889lowast defined by (10) is a metric onClowast

It is immediate for 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] ge 0

(27)

(i) Now we show that 119889lowast(119911lowast1 119911lowast

2) = 0 if and only if 119911lowast

1=

119911lowast

2for 119911lowast1 119911lowast

2isin Clowast Indeed

119889lowast

(119911lowast

1 119911lowast

2) = 0 lArrrArr

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 0

lArrrArr 120573[radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] = 120573 (0)

lArrrArr (1198861minus 1198862)2

+ (1198871minus 1198872)2

= 0

lArrrArr 1198861minus 1198862= 0 119887

1minus 1198872= 0

lArrrArr 1198861= 1198862 119887

1= 1198872

lArrrArr 1198861= 1198862

1198871=

1198872

lArrrArr 119911lowast

1= ( 11988611198871) = ( 119886

21198872) = 119911lowast

2

(28)

6 Abstract and Applied Analysis

(ii) One can easily establish for all 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

= 120573 [radic(1198862minus 1198861)2

+ (1198872minus 1198871)2

]

= 119889lowast

(119911lowast

2 119911lowast

1)

(29)

(iii) We show that the inequality 119889lowast

(119911lowast

1 119911lowast

2) +119889lowast

(119911lowast

2

119911lowast

3) ge 119889lowast

(119911lowast

1 119911lowast

3) holds for all 119911lowast

1 119911lowast

2 119911lowast

3isin Clowast In fact

119889lowast

(119911lowast

1 119911lowast

2) + 119889lowast

(119911lowast

2 119911lowast

3)

=

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

+

radic[120580 ( 1198862minus 1198863)]2

+ (1198872minus

1198873)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

+ 120573 [radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

+ radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

ge120573 [radic(1198861minus 1198863)2

+ (1198871minus 1198873)2

]

=

radic[120580 ( 1198861minus 1198863)]2

+ (1198871minus

1198873)

2

= 119889lowast

(119911lowast

1 119911lowast

3)

(30)

Therefore 119889lowast is a metric over ClowastNow we can show that the metric space (Clowast 119889lowast) is

complete Let (119911lowast119899)119899isinN

be an arbitrary Cauchy sequence inClowastIn this case for all 120576 gt 0 there exists an 119899

0isin N such that

119889lowast

(119911lowast

119898 119911lowast

119899) lt 120576 for all 119898 119899 ge 119899

0 Let 119911lowast

119898= ( 119886119898119887119898) isin Clowast and

119898 isin N Then

119889lowast

(119911lowast

119898 119911lowast

119899) =

radic[120580 ( 119886119898minus 119886119899)]2

+ (119887119898minus

119887119899)

2

= 120573 [radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

]

lt 120576 = 120573 (1205761015840

)

(31)

Thus we obtain that

radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

lt 1205761015840

(32)

On the other hand since the following inequalities

1003816100381610038161003816119886119898minus 119886119899

1003816100381610038161003816= radic(119886

119898minus 119886119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

1003816100381610038161003816119887119898minus 119887119899

1003816100381610038161003816= radic(119887

119898minus 119887119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

(33)

hold we therefore have by (32) that |119886119898

minus 119886119899| lt 1205761015840 and |119887

119898minus

119887119899| lt 1205761015840 This means that (119886

119899) and (119887

119899) are Cauchy sequences

with real terms Since R is complete it is clear that for every1205761015840

gt 0 there exists an 1198991isin N such that |119886

119899minus 119886| lt 120576

1015840

2 for all119899 ge 119899

1and for every 120576

1015840

gt 0 there exists an 1198992isin N such that

|119887119899minus 119887| lt 120576

1015840

2 for all 119899 ge 1198992

Define 119911lowast isin Clowast by 119911lowast

= ( 119886119887) Then we have

119889lowast

(119911lowast

119899 119911lowast

) =

radic[120580 ( 119886119899minus 119886)]2

+ [119887119899minus

119887]

2

= 120573 [radic(119886119899minus 119886)2

+ (119887119899minus 119887)2

]

le 120573 [radic(119886119899minus 119886)2

+ radic(119887119899minus 119887)2

]

= 120573 (1003816100381610038161003816119886119899minus 119886

1003816100381610038161003816+1003816100381610038161003816119887119899minus 119887

1003816100381610038161003816)

le 120573(

1205761015840

2

+

1205761015840

2

)

= 120573 (1205761015840

) = 120576

(34)

Hence (Clowast 119889lowast) is a complete metric space

Since Clowast is a complete metric space with the metric119889lowast defined by (10) induced by the norm

sdot as a direct

consequence of Theorem 6 we have the following

Corollary 7 Clowast is a Banach space with the norm sdot

defined

by

119911lowast =

radic[120580 ( 119886)]

2

+ (119887)

2

119911lowast

= ( 119886119887) isin C

lowast

(35)

3 Sequence Spaces over Non-NewtonianComplex Field

In this section we define the sets 120596lowast ℓlowastinfin 119888lowast 119888lowast

0 and ℓ

lowast

119901of

all bounded convergent null and absolutely 119901-summablesequences over the non-Newtonian complex field Clowast whichcorrespond to the sets 120596 ℓ

infin 119888 1198880 and ℓ

119901over the complex

field C respectively That is to say that

120596lowast

= 119911lowast

= (119911lowast

119896) 119911lowast

119896isin Clowast

forall119896 isin N

ℓlowast

infin= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sup119896isinN

119911lowast

119896

ltinfin

119888lowast

= 119911lowast

= (119911lowast

119896) isin 120596lowast

exist119897lowast

isin Clowast

ni limlowast119896rarrinfin

119911lowast

119896= 119897lowast

119888lowast

0= 119911

lowast

= (119911lowast

119896) isin 120596lowast

limlowast119896rarrinfin

119911lowast

119896= 120579lowast

ℓlowast

119901= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sum

119896

119911lowast

119896

119901

ltinfin

(36)

For simplicity in notation here and in what follows thesummation without limits runs from 0 to infin One can easily

Abstract and Applied Analysis 7

see that the set 120596lowast forms a vector space over Clowast with respectto the algebraic operations addition (+) and scalar multipli-cation (times) defined on 120596

lowast as follows

+ 120596lowast

times 120596lowast

997888rarr 120596lowast

(119911lowast

119905lowast

) 997891997888rarr 119911lowast

+ 119905lowast

= (119911lowast

119896oplus 119905lowast

119896)

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast

times Clowast

times 120596lowast

997888rarr 120596lowast

(120572 119911lowast

) 997891997888rarr 120572 times 119911lowast

= (120572 ⊙ 119911lowast

119896)

119911lowast

= (119911lowast

119896) isin 120596lowast

120572 isin Clowast

(37)

Theorem 8 Define the function 119889120596lowast by

119889120596lowast 120596lowast

times 120596lowast

997888rarr 1198611015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 tlowast119896)]

(38)

where (120583119896) isin 1198611015840

sube 119861 such that sum119896120583119896is convergent with 120583

119896gt 0

for all 119896 isin N Then (120596lowast 119889120596lowast) is a metric space

Proof We show that 119889120596lowast satisfies the metric axioms on the

space 120596lowast of all non-Newtonian complex valued sequences(i) First we show that 119889

120596lowast(119911lowast

119905lowast

) ge 0 for all 119911lowast 119905lowast isin 120596lowast

Because (Clowast 119889lowast) is a metric space we have 119889lowast(119911lowast119896 119905lowast

119896) ge 0

119911lowast

119896 119905lowast

119896isin Clowast and 1 + 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 Hence we obtain that

119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 Moreover since 120583

119896gt 0 for all

119896 isin N we have

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 (39)

This means that

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0

(40)

(ii) We show that 119889120596lowast(119911lowast

119905lowast

) = 0 iff 119911lowast

= 119905lowast In this situ-

ation one can see that

119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

lArrrArr 120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

120583119896gt 0 forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896) = 0 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 forall119896 isin N

lArrrArr 119911lowast

119896= 119905lowast

119896 119889lowastmetric forall119896 isin N

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(41)

(iii) We show that 119889120596lowast(119911lowast

119905lowast

) = 119889120596lowast(119905lowast

119911lowast

) for 119911lowast

=

(119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast First we know that 119889lowast is a metric over

Clowast Thus

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119911lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119911lowast

119896)]

= 119889120596lowast (119905lowast

119911lowast

)

(42)

(iv) We show that 119889120596lowast(119911lowast

119905lowast

) + 119889120596lowast(119905lowast

119906lowast

) ge 119889120596lowast(119911lowast

119906lowast

)

holds for 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin 120596

lowast Againusing the fact that (Clowast 119889lowast) is a metric space it is easy to seeby Lemma 4 that

119889120596lowast (119911lowast

119906lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

1 + [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times 119889

lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+ 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

= 119889120596lowast (119911lowast

119905lowast

) + 119889120596lowast (119905lowast

119906lowast

)

(43)

as required

Theorem 9 The set ℓlowastinfin

is a sequence space

Proof It is trivial that the inclusion ℓlowast

infinsub 120596lowast holds

(i) We show that 119911lowast + 119905lowast

isin ℓlowast

infinfor 119911lowast = (119911

lowast

119896) 119905lowast

= (119905lowast

119896) isin

ℓlowast

infin Indeed combining the hypothesis

sup119896isinN

119911lowast

119896

ltinfin sup

119896isinN

119905lowast

119896

ltinfin (44)

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

Abstract and Applied Analysis 3

(i) (Clowast oplus) is an Abelian group(ii) (Clowast 120579

lowast

⊙) is an Abelian group(iii) the operation ⊙ is distributive over the operationoplus

which conclude that (Clowast oplus ⊙) is a field

Let 119887 isin 119861 sube R Then the number

119887 times119887 is called the 120573-

square of 119887 and is denoted by 119887

2 Let 119887 be a non-negative

number in 119861 Then 120573[radic120573minus1(119887)] is called the 120573-square root

of 119887 and is denoted by radic

119887 The lowast-distance 119889lowast between two

arbitrarily elements 119911lowast1

= ( 11988611198871) and 119911

lowast

2= ( 11988621198872) of the set

Clowast is defined by

119889lowast

Clowast

times Clowast

997888rarr[0infin

) = 1198611015840

sub 119861

(119911lowast

1 119911lowast

2) 997888rarr 119889

lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

(10)

Up to now we know that Clowast is a field and the distancebetween two points in Clowast is computed by the function 119889

lowastdefined by (10) Now we define the lowast-norm and next derivesome required inequalities in the sense of non-Newtoniancomplex calculus

Let 119911lowast isin Clowast be an arbitrary element 119889lowast(119911lowast 120579lowast) is calledlowast-norm of 119911lowast and is denoted by

sdot In other words

119911lowast = 119889lowast

(119911lowast

120579lowast

)

=

radic[120580 ( 119886 minus 0)]

2

+ (119887 minus 0)

2

= 120573 (radic1198862+ 1198872)

(11)

where 119911lowast

= ( 119886119887) and 120579

lowast

= (0 0) Moreover since for all119911lowast

1 119911lowast

2isin Clowast we have 119889lowast(119911lowast

1 119911lowast

2) =

119911lowast

1⊖119911lowast

2

119889lowast is the induced

metric from the norm sdot

Lemma 2 (lowast-Triangle inequality) Let 119911lowast1 119911lowast

2isin Clowast Then

119911lowast

1oplus 119911lowast

2

le

119911lowast

1

+

119911lowast

2

(12)

Proof Let 119911lowast1 119911lowast

2isin Clowast Then a straightforward calculation

gives that

119911lowast

1oplus 119911lowast

2

=

radic[120580 ( 1198861+ 1198862)]2

+(1198871+

1198872)

2

= 120573 [radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]

le 120573 (radic1198862

1+ 1198872

1+ radic1198862

2+ 1198872

2)

= 120573120573minus1

[120573 (radic1198862

1+ 1198872

1)] + 120573

minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573 [120573minus1

(119911lowast

1

) + 120573

minus1

(119911lowast

2

)]

=119911lowast

1

+

119911lowast

2

(13)

Hence the inequality (12) holds

Lemma 3 119911lowast

1⊙ 119911lowast

2

=

119911lowast

1

times

119911lowast

2

for all 119911lowast

1 119911lowast

2isin Clowast

Proof Let 119911lowast1 119911lowast

2isin Clowast In this case one can observe by a

routine verification that

119911lowast

1⊙ 119911lowast

2

=

radic120580 [120572 ( 119886

11198862minus

1198871

1198872)]

2

+ [120573 ( 1198861

1198872+

1198871

1198862)]

2

= 120573 [radic(11988611198862minus 11988711198872)2

+ (11988611198872+ 11988711198862)2

]

= 120573(radic1198862

1+ 1198872

1radic1198862

2+ 1198872

2)

= 120573120573minus1

[120573 (radic1198862

1+ 1198872

1)] times 120573

minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573 [120573minus1

(119911lowast

1

) times 120573

minus1

(119911lowast

2

)]

=119911lowast

1

times

119911lowast

2

(14)

as required

Lemma 4 Let 119911lowast

1 119911lowast

2isin Clowast Then the following inequality

holds

119911lowast

1oplus 119911lowast

2

(1 +

119911lowast

1oplus 119911lowast

2

)

le119911lowast

1

(1 +

119911lowast

1

) +

119911lowast

2

(1 +

119911lowast

2

)

(15)

Proof Let 119911lowast1 119911lowast

2isin Clowast Then one can see that

119911lowast

1oplus 119911lowast

2

(1 +

119911lowast

1oplus 119911lowast

2

)

= 120573 [radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]

1 + 120573 [radic(119886

1+ 1198862)2

+ (1198871+ 1198872)2

]

= 120573[[

[

radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

1 + radic(1198861+ 1198862)2

+ (1198871+ 1198872)2

]]

]

le 120573[[

[

(

radic1198862

1+ 1198872

1

1 + radic1198862

1+ 1198872

1

) + (

radic1198862

2+ 1198872

2

1 + radic1198862

2+ 1198872

2

)]]

]

4 Abstract and Applied Analysis

= 120573

120573minus1

[120573 (radic1198862

1+ 1198872

1)]

120573minus1

120573 [120573minus1

(1) + 120573minus1

[120573 (radic1198862

1+ 1198872

1)]]

+

120573minus1

[120573 (radic1198862

2+ 1198872

2)]

120573minus1

120573120573minus1

(1) + 120573minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

] + [

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573120573minus1

120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

]

+ 120573minus1

120573[

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573 120573minus1

[119911lowast

1

(1 +

119911lowast

1

)] + 120573

minus1

[119911lowast

2

(1 +

119911lowast

2

)]

=119911lowast

1

(1 +

119911lowast

1

) +

119911lowast

2

(1 +

119911lowast

2

)

(16)

This means that the inequality (15) holds

Lemma 5 (lowast-Minkowski inequality) Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin

Clowast for all 119896 isin 1 2 3 119899 Then

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(17)

Proof Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin Clowast for all 119896 isin 1 2 3 119899

Then119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

0oplus 119905lowast

0

119901

+119911lowast

1oplus 119905lowast

1

119901

+ sdot sdot sdot +119911lowast

119899oplus 119905lowast

119899

119901

= 120573 [120573minus1

(119911lowast

0oplus 119905lowast

0

119901

) + 120573minus1

(119911lowast

1oplus 119905lowast

1

119901

)

+ sdot sdot sdot + 120573minus1

(119911lowast

119899oplus 119905lowast

119899

119901

)]

(18)

Let us take 119911lowast

119896= (119896 119910119896) 119911119896

= (119909119896 119910119896) 119905lowast

119896= (119896 V119896) 119905119896

=

(119906119896 V119896) Then since the equality

119911lowast

119896oplus 119905lowast

119896

=

radic[120580 (119896+ 119896)]2

+ ( 119910119896+ V119896)2

= 120573 [radic(119909119896+ 119906119896)2

+ (119910119896+ V119896)2

]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)

(19)

holds for every fixed 119896 we obtain

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

119896oplus 119905lowast

119896

times sdot sdot sdot times

119911lowast

119896oplus 119905lowast

119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 [120573minus1

(119911lowast

119896oplus 119905lowast

119896

) times sdot sdot sdot times 120573

minus1

(119911lowast

119896oplus 119905lowast

119896

)]

= 120573 120573minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)] times sdot sdot sdot times 120573

minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

(20)

by (18) and Minkowski inequality in the complex field leadsus to119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

= 120573 120573minus1

[120573 (10038161003816100381610038161199110oplus 1199050

1003816100381610038161003816

120573minus1

(119901)

)] + 120573minus1

[120573 (10038161003816100381610038161199111oplus 1199051

1003816100381610038161003816

120573minus1

(119901)

)]

+ sdot sdot sdot + 120573minus1

[120573 (1003816100381610038161003816119911119899oplus 119905119899

1003816100381610038161003816

120573minus1

(119901)

)]

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

le 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

+(

119899

sum

119896=0

1003816100381610038161003816119905119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(21)

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

= 120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdottimes[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

= 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

Abstract and Applied Analysis 5

=

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+ 120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

=

120573minus1[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

120573minus1

(119901)

(22)

On the other hand let us prove

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

(23)

Indeed

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

119901

=120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

times sdot sdot sdot times 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

times sdot sdot sdot times (

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

=

119899

sum

119896=0

119911lowast

119896

119901

(24)

Substituting the relation (24) in (22) we obtain

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

=[

[

(

119899

sum

119896=0

|119911119896|120573minus1

(119901)

)

1120573minus1

(119901)

+ (

119899

sum

119896=0

|119905119896|120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(25)

By using this equality in (21) we get

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(26)

as desired

Theorem 6 (Clowast 119889lowast) is a complete metric space where 119889lowast is

defined by (10)

Proof First we show that 119889lowast defined by (10) is a metric onClowast

It is immediate for 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] ge 0

(27)

(i) Now we show that 119889lowast(119911lowast1 119911lowast

2) = 0 if and only if 119911lowast

1=

119911lowast

2for 119911lowast1 119911lowast

2isin Clowast Indeed

119889lowast

(119911lowast

1 119911lowast

2) = 0 lArrrArr

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 0

lArrrArr 120573[radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] = 120573 (0)

lArrrArr (1198861minus 1198862)2

+ (1198871minus 1198872)2

= 0

lArrrArr 1198861minus 1198862= 0 119887

1minus 1198872= 0

lArrrArr 1198861= 1198862 119887

1= 1198872

lArrrArr 1198861= 1198862

1198871=

1198872

lArrrArr 119911lowast

1= ( 11988611198871) = ( 119886

21198872) = 119911lowast

2

(28)

6 Abstract and Applied Analysis

(ii) One can easily establish for all 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

= 120573 [radic(1198862minus 1198861)2

+ (1198872minus 1198871)2

]

= 119889lowast

(119911lowast

2 119911lowast

1)

(29)

(iii) We show that the inequality 119889lowast

(119911lowast

1 119911lowast

2) +119889lowast

(119911lowast

2

119911lowast

3) ge 119889lowast

(119911lowast

1 119911lowast

3) holds for all 119911lowast

1 119911lowast

2 119911lowast

3isin Clowast In fact

119889lowast

(119911lowast

1 119911lowast

2) + 119889lowast

(119911lowast

2 119911lowast

3)

=

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

+

radic[120580 ( 1198862minus 1198863)]2

+ (1198872minus

1198873)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

+ 120573 [radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

+ radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

ge120573 [radic(1198861minus 1198863)2

+ (1198871minus 1198873)2

]

=

radic[120580 ( 1198861minus 1198863)]2

+ (1198871minus

1198873)

2

= 119889lowast

(119911lowast

1 119911lowast

3)

(30)

Therefore 119889lowast is a metric over ClowastNow we can show that the metric space (Clowast 119889lowast) is

complete Let (119911lowast119899)119899isinN

be an arbitrary Cauchy sequence inClowastIn this case for all 120576 gt 0 there exists an 119899

0isin N such that

119889lowast

(119911lowast

119898 119911lowast

119899) lt 120576 for all 119898 119899 ge 119899

0 Let 119911lowast

119898= ( 119886119898119887119898) isin Clowast and

119898 isin N Then

119889lowast

(119911lowast

119898 119911lowast

119899) =

radic[120580 ( 119886119898minus 119886119899)]2

+ (119887119898minus

119887119899)

2

= 120573 [radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

]

lt 120576 = 120573 (1205761015840

)

(31)

Thus we obtain that

radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

lt 1205761015840

(32)

On the other hand since the following inequalities

1003816100381610038161003816119886119898minus 119886119899

1003816100381610038161003816= radic(119886

119898minus 119886119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

1003816100381610038161003816119887119898minus 119887119899

1003816100381610038161003816= radic(119887

119898minus 119887119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

(33)

hold we therefore have by (32) that |119886119898

minus 119886119899| lt 1205761015840 and |119887

119898minus

119887119899| lt 1205761015840 This means that (119886

119899) and (119887

119899) are Cauchy sequences

with real terms Since R is complete it is clear that for every1205761015840

gt 0 there exists an 1198991isin N such that |119886

119899minus 119886| lt 120576

1015840

2 for all119899 ge 119899

1and for every 120576

1015840

gt 0 there exists an 1198992isin N such that

|119887119899minus 119887| lt 120576

1015840

2 for all 119899 ge 1198992

Define 119911lowast isin Clowast by 119911lowast

= ( 119886119887) Then we have

119889lowast

(119911lowast

119899 119911lowast

) =

radic[120580 ( 119886119899minus 119886)]2

+ [119887119899minus

119887]

2

= 120573 [radic(119886119899minus 119886)2

+ (119887119899minus 119887)2

]

le 120573 [radic(119886119899minus 119886)2

+ radic(119887119899minus 119887)2

]

= 120573 (1003816100381610038161003816119886119899minus 119886

1003816100381610038161003816+1003816100381610038161003816119887119899minus 119887

1003816100381610038161003816)

le 120573(

1205761015840

2

+

1205761015840

2

)

= 120573 (1205761015840

) = 120576

(34)

Hence (Clowast 119889lowast) is a complete metric space

Since Clowast is a complete metric space with the metric119889lowast defined by (10) induced by the norm

sdot as a direct

consequence of Theorem 6 we have the following

Corollary 7 Clowast is a Banach space with the norm sdot

defined

by

119911lowast =

radic[120580 ( 119886)]

2

+ (119887)

2

119911lowast

= ( 119886119887) isin C

lowast

(35)

3 Sequence Spaces over Non-NewtonianComplex Field

In this section we define the sets 120596lowast ℓlowastinfin 119888lowast 119888lowast

0 and ℓ

lowast

119901of

all bounded convergent null and absolutely 119901-summablesequences over the non-Newtonian complex field Clowast whichcorrespond to the sets 120596 ℓ

infin 119888 1198880 and ℓ

119901over the complex

field C respectively That is to say that

120596lowast

= 119911lowast

= (119911lowast

119896) 119911lowast

119896isin Clowast

forall119896 isin N

ℓlowast

infin= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sup119896isinN

119911lowast

119896

ltinfin

119888lowast

= 119911lowast

= (119911lowast

119896) isin 120596lowast

exist119897lowast

isin Clowast

ni limlowast119896rarrinfin

119911lowast

119896= 119897lowast

119888lowast

0= 119911

lowast

= (119911lowast

119896) isin 120596lowast

limlowast119896rarrinfin

119911lowast

119896= 120579lowast

ℓlowast

119901= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sum

119896

119911lowast

119896

119901

ltinfin

(36)

For simplicity in notation here and in what follows thesummation without limits runs from 0 to infin One can easily

Abstract and Applied Analysis 7

see that the set 120596lowast forms a vector space over Clowast with respectto the algebraic operations addition (+) and scalar multipli-cation (times) defined on 120596

lowast as follows

+ 120596lowast

times 120596lowast

997888rarr 120596lowast

(119911lowast

119905lowast

) 997891997888rarr 119911lowast

+ 119905lowast

= (119911lowast

119896oplus 119905lowast

119896)

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast

times Clowast

times 120596lowast

997888rarr 120596lowast

(120572 119911lowast

) 997891997888rarr 120572 times 119911lowast

= (120572 ⊙ 119911lowast

119896)

119911lowast

= (119911lowast

119896) isin 120596lowast

120572 isin Clowast

(37)

Theorem 8 Define the function 119889120596lowast by

119889120596lowast 120596lowast

times 120596lowast

997888rarr 1198611015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 tlowast119896)]

(38)

where (120583119896) isin 1198611015840

sube 119861 such that sum119896120583119896is convergent with 120583

119896gt 0

for all 119896 isin N Then (120596lowast 119889120596lowast) is a metric space

Proof We show that 119889120596lowast satisfies the metric axioms on the

space 120596lowast of all non-Newtonian complex valued sequences(i) First we show that 119889

120596lowast(119911lowast

119905lowast

) ge 0 for all 119911lowast 119905lowast isin 120596lowast

Because (Clowast 119889lowast) is a metric space we have 119889lowast(119911lowast119896 119905lowast

119896) ge 0

119911lowast

119896 119905lowast

119896isin Clowast and 1 + 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 Hence we obtain that

119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 Moreover since 120583

119896gt 0 for all

119896 isin N we have

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 (39)

This means that

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0

(40)

(ii) We show that 119889120596lowast(119911lowast

119905lowast

) = 0 iff 119911lowast

= 119905lowast In this situ-

ation one can see that

119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

lArrrArr 120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

120583119896gt 0 forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896) = 0 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 forall119896 isin N

lArrrArr 119911lowast

119896= 119905lowast

119896 119889lowastmetric forall119896 isin N

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(41)

(iii) We show that 119889120596lowast(119911lowast

119905lowast

) = 119889120596lowast(119905lowast

119911lowast

) for 119911lowast

=

(119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast First we know that 119889lowast is a metric over

Clowast Thus

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119911lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119911lowast

119896)]

= 119889120596lowast (119905lowast

119911lowast

)

(42)

(iv) We show that 119889120596lowast(119911lowast

119905lowast

) + 119889120596lowast(119905lowast

119906lowast

) ge 119889120596lowast(119911lowast

119906lowast

)

holds for 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin 120596

lowast Againusing the fact that (Clowast 119889lowast) is a metric space it is easy to seeby Lemma 4 that

119889120596lowast (119911lowast

119906lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

1 + [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times 119889

lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+ 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

= 119889120596lowast (119911lowast

119905lowast

) + 119889120596lowast (119905lowast

119906lowast

)

(43)

as required

Theorem 9 The set ℓlowastinfin

is a sequence space

Proof It is trivial that the inclusion ℓlowast

infinsub 120596lowast holds

(i) We show that 119911lowast + 119905lowast

isin ℓlowast

infinfor 119911lowast = (119911

lowast

119896) 119905lowast

= (119905lowast

119896) isin

ℓlowast

infin Indeed combining the hypothesis

sup119896isinN

119911lowast

119896

ltinfin sup

119896isinN

119905lowast

119896

ltinfin (44)

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

4 Abstract and Applied Analysis

= 120573

120573minus1

[120573 (radic1198862

1+ 1198872

1)]

120573minus1

120573 [120573minus1

(1) + 120573minus1

[120573 (radic1198862

1+ 1198872

1)]]

+

120573minus1

[120573 (radic1198862

2+ 1198872

2)]

120573minus1

120573120573minus1

(1) + 120573minus1

[120573 (radic1198862

2+ 1198872

2)]

= 120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

] + [

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573120573minus1

120573[

120573minus1

(119911lowast

1

)

120573minus1

(1 +119911lowast

1

)

]

+ 120573minus1

120573[

120573minus1

(119911lowast

2

)

120573minus1

(1 +119911lowast

2

)

]

= 120573 120573minus1

[119911lowast

1

(1 +

119911lowast

1

)] + 120573

minus1

[119911lowast

2

(1 +

119911lowast

2

)]

=119911lowast

1

(1 +

119911lowast

1

) +

119911lowast

2

(1 +

119911lowast

2

)

(16)

This means that the inequality (15) holds

Lemma 5 (lowast-Minkowski inequality) Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin

Clowast for all 119896 isin 1 2 3 119899 Then

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(17)

Proof Let 119901 ge 1 and 119911lowast

119896 119905lowast

119896isin Clowast for all 119896 isin 1 2 3 119899

Then119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

0oplus 119905lowast

0

119901

+119911lowast

1oplus 119905lowast

1

119901

+ sdot sdot sdot +119911lowast

119899oplus 119905lowast

119899

119901

= 120573 [120573minus1

(119911lowast

0oplus 119905lowast

0

119901

) + 120573minus1

(119911lowast

1oplus 119905lowast

1

119901

)

+ sdot sdot sdot + 120573minus1

(119911lowast

119899oplus 119905lowast

119899

119901

)]

(18)

Let us take 119911lowast

119896= (119896 119910119896) 119911119896

= (119909119896 119910119896) 119905lowast

119896= (119896 V119896) 119905119896

=

(119906119896 V119896) Then since the equality

119911lowast

119896oplus 119905lowast

119896

=

radic[120580 (119896+ 119896)]2

+ ( 119910119896+ V119896)2

= 120573 [radic(119909119896+ 119906119896)2

+ (119910119896+ V119896)2

]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)

(19)

holds for every fixed 119896 we obtain

119911lowast

119896oplus 119905lowast

119896

119901

=119911lowast

119896oplus 119905lowast

119896

times sdot sdot sdot times

119911lowast

119896oplus 119905lowast

119896

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 [120573minus1

(119911lowast

119896oplus 119905lowast

119896

) times sdot sdot sdot times 120573

minus1

(119911lowast

119896oplus 119905lowast

119896

)]

= 120573 120573minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)] times sdot sdot sdot times 120573

minus1

[120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816)]

= 120573 (1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

(20)

by (18) and Minkowski inequality in the complex field leadsus to119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

= 120573 120573minus1

[120573 (10038161003816100381610038161199110oplus 1199050

1003816100381610038161003816

120573minus1

(119901)

)] + 120573minus1

[120573 (10038161003816100381610038161199111oplus 1199051

1003816100381610038161003816

120573minus1

(119901)

)]

+ sdot sdot sdot + 120573minus1

[120573 (1003816100381610038161003816119911119899oplus 119905119899

1003816100381610038161003816

120573minus1

(119901)

)]

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896+ 119905119896

1003816100381610038161003816

120573minus1

(119901)

)

le 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

+(

119899

sum

119896=0

1003816100381610038161003816119905119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(21)

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

= 120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdottimes[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

= 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times 120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

Abstract and Applied Analysis 5

=

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+ 120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

=

120573minus1[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

120573minus1

(119901)

(22)

On the other hand let us prove

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

(23)

Indeed

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

119901

=120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

times sdot sdot sdot times 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

times sdot sdot sdot times (

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

=

119899

sum

119896=0

119911lowast

119896

119901

(24)

Substituting the relation (24) in (22) we obtain

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

=[

[

(

119899

sum

119896=0

|119911119896|120573minus1

(119901)

)

1120573minus1

(119901)

+ (

119899

sum

119896=0

|119905119896|120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(25)

By using this equality in (21) we get

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(26)

as desired

Theorem 6 (Clowast 119889lowast) is a complete metric space where 119889lowast is

defined by (10)

Proof First we show that 119889lowast defined by (10) is a metric onClowast

It is immediate for 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] ge 0

(27)

(i) Now we show that 119889lowast(119911lowast1 119911lowast

2) = 0 if and only if 119911lowast

1=

119911lowast

2for 119911lowast1 119911lowast

2isin Clowast Indeed

119889lowast

(119911lowast

1 119911lowast

2) = 0 lArrrArr

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 0

lArrrArr 120573[radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] = 120573 (0)

lArrrArr (1198861minus 1198862)2

+ (1198871minus 1198872)2

= 0

lArrrArr 1198861minus 1198862= 0 119887

1minus 1198872= 0

lArrrArr 1198861= 1198862 119887

1= 1198872

lArrrArr 1198861= 1198862

1198871=

1198872

lArrrArr 119911lowast

1= ( 11988611198871) = ( 119886

21198872) = 119911lowast

2

(28)

6 Abstract and Applied Analysis

(ii) One can easily establish for all 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

= 120573 [radic(1198862minus 1198861)2

+ (1198872minus 1198871)2

]

= 119889lowast

(119911lowast

2 119911lowast

1)

(29)

(iii) We show that the inequality 119889lowast

(119911lowast

1 119911lowast

2) +119889lowast

(119911lowast

2

119911lowast

3) ge 119889lowast

(119911lowast

1 119911lowast

3) holds for all 119911lowast

1 119911lowast

2 119911lowast

3isin Clowast In fact

119889lowast

(119911lowast

1 119911lowast

2) + 119889lowast

(119911lowast

2 119911lowast

3)

=

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

+

radic[120580 ( 1198862minus 1198863)]2

+ (1198872minus

1198873)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

+ 120573 [radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

+ radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

ge120573 [radic(1198861minus 1198863)2

+ (1198871minus 1198873)2

]

=

radic[120580 ( 1198861minus 1198863)]2

+ (1198871minus

1198873)

2

= 119889lowast

(119911lowast

1 119911lowast

3)

(30)

Therefore 119889lowast is a metric over ClowastNow we can show that the metric space (Clowast 119889lowast) is

complete Let (119911lowast119899)119899isinN

be an arbitrary Cauchy sequence inClowastIn this case for all 120576 gt 0 there exists an 119899

0isin N such that

119889lowast

(119911lowast

119898 119911lowast

119899) lt 120576 for all 119898 119899 ge 119899

0 Let 119911lowast

119898= ( 119886119898119887119898) isin Clowast and

119898 isin N Then

119889lowast

(119911lowast

119898 119911lowast

119899) =

radic[120580 ( 119886119898minus 119886119899)]2

+ (119887119898minus

119887119899)

2

= 120573 [radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

]

lt 120576 = 120573 (1205761015840

)

(31)

Thus we obtain that

radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

lt 1205761015840

(32)

On the other hand since the following inequalities

1003816100381610038161003816119886119898minus 119886119899

1003816100381610038161003816= radic(119886

119898minus 119886119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

1003816100381610038161003816119887119898minus 119887119899

1003816100381610038161003816= radic(119887

119898minus 119887119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

(33)

hold we therefore have by (32) that |119886119898

minus 119886119899| lt 1205761015840 and |119887

119898minus

119887119899| lt 1205761015840 This means that (119886

119899) and (119887

119899) are Cauchy sequences

with real terms Since R is complete it is clear that for every1205761015840

gt 0 there exists an 1198991isin N such that |119886

119899minus 119886| lt 120576

1015840

2 for all119899 ge 119899

1and for every 120576

1015840

gt 0 there exists an 1198992isin N such that

|119887119899minus 119887| lt 120576

1015840

2 for all 119899 ge 1198992

Define 119911lowast isin Clowast by 119911lowast

= ( 119886119887) Then we have

119889lowast

(119911lowast

119899 119911lowast

) =

radic[120580 ( 119886119899minus 119886)]2

+ [119887119899minus

119887]

2

= 120573 [radic(119886119899minus 119886)2

+ (119887119899minus 119887)2

]

le 120573 [radic(119886119899minus 119886)2

+ radic(119887119899minus 119887)2

]

= 120573 (1003816100381610038161003816119886119899minus 119886

1003816100381610038161003816+1003816100381610038161003816119887119899minus 119887

1003816100381610038161003816)

le 120573(

1205761015840

2

+

1205761015840

2

)

= 120573 (1205761015840

) = 120576

(34)

Hence (Clowast 119889lowast) is a complete metric space

Since Clowast is a complete metric space with the metric119889lowast defined by (10) induced by the norm

sdot as a direct

consequence of Theorem 6 we have the following

Corollary 7 Clowast is a Banach space with the norm sdot

defined

by

119911lowast =

radic[120580 ( 119886)]

2

+ (119887)

2

119911lowast

= ( 119886119887) isin C

lowast

(35)

3 Sequence Spaces over Non-NewtonianComplex Field

In this section we define the sets 120596lowast ℓlowastinfin 119888lowast 119888lowast

0 and ℓ

lowast

119901of

all bounded convergent null and absolutely 119901-summablesequences over the non-Newtonian complex field Clowast whichcorrespond to the sets 120596 ℓ

infin 119888 1198880 and ℓ

119901over the complex

field C respectively That is to say that

120596lowast

= 119911lowast

= (119911lowast

119896) 119911lowast

119896isin Clowast

forall119896 isin N

ℓlowast

infin= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sup119896isinN

119911lowast

119896

ltinfin

119888lowast

= 119911lowast

= (119911lowast

119896) isin 120596lowast

exist119897lowast

isin Clowast

ni limlowast119896rarrinfin

119911lowast

119896= 119897lowast

119888lowast

0= 119911

lowast

= (119911lowast

119896) isin 120596lowast

limlowast119896rarrinfin

119911lowast

119896= 120579lowast

ℓlowast

119901= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sum

119896

119911lowast

119896

119901

ltinfin

(36)

For simplicity in notation here and in what follows thesummation without limits runs from 0 to infin One can easily

Abstract and Applied Analysis 7

see that the set 120596lowast forms a vector space over Clowast with respectto the algebraic operations addition (+) and scalar multipli-cation (times) defined on 120596

lowast as follows

+ 120596lowast

times 120596lowast

997888rarr 120596lowast

(119911lowast

119905lowast

) 997891997888rarr 119911lowast

+ 119905lowast

= (119911lowast

119896oplus 119905lowast

119896)

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast

times Clowast

times 120596lowast

997888rarr 120596lowast

(120572 119911lowast

) 997891997888rarr 120572 times 119911lowast

= (120572 ⊙ 119911lowast

119896)

119911lowast

= (119911lowast

119896) isin 120596lowast

120572 isin Clowast

(37)

Theorem 8 Define the function 119889120596lowast by

119889120596lowast 120596lowast

times 120596lowast

997888rarr 1198611015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 tlowast119896)]

(38)

where (120583119896) isin 1198611015840

sube 119861 such that sum119896120583119896is convergent with 120583

119896gt 0

for all 119896 isin N Then (120596lowast 119889120596lowast) is a metric space

Proof We show that 119889120596lowast satisfies the metric axioms on the

space 120596lowast of all non-Newtonian complex valued sequences(i) First we show that 119889

120596lowast(119911lowast

119905lowast

) ge 0 for all 119911lowast 119905lowast isin 120596lowast

Because (Clowast 119889lowast) is a metric space we have 119889lowast(119911lowast119896 119905lowast

119896) ge 0

119911lowast

119896 119905lowast

119896isin Clowast and 1 + 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 Hence we obtain that

119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 Moreover since 120583

119896gt 0 for all

119896 isin N we have

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 (39)

This means that

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0

(40)

(ii) We show that 119889120596lowast(119911lowast

119905lowast

) = 0 iff 119911lowast

= 119905lowast In this situ-

ation one can see that

119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

lArrrArr 120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

120583119896gt 0 forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896) = 0 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 forall119896 isin N

lArrrArr 119911lowast

119896= 119905lowast

119896 119889lowastmetric forall119896 isin N

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(41)

(iii) We show that 119889120596lowast(119911lowast

119905lowast

) = 119889120596lowast(119905lowast

119911lowast

) for 119911lowast

=

(119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast First we know that 119889lowast is a metric over

Clowast Thus

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119911lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119911lowast

119896)]

= 119889120596lowast (119905lowast

119911lowast

)

(42)

(iv) We show that 119889120596lowast(119911lowast

119905lowast

) + 119889120596lowast(119905lowast

119906lowast

) ge 119889120596lowast(119911lowast

119906lowast

)

holds for 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin 120596

lowast Againusing the fact that (Clowast 119889lowast) is a metric space it is easy to seeby Lemma 4 that

119889120596lowast (119911lowast

119906lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

1 + [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times 119889

lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+ 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

= 119889120596lowast (119911lowast

119905lowast

) + 119889120596lowast (119905lowast

119906lowast

)

(43)

as required

Theorem 9 The set ℓlowastinfin

is a sequence space

Proof It is trivial that the inclusion ℓlowast

infinsub 120596lowast holds

(i) We show that 119911lowast + 119905lowast

isin ℓlowast

infinfor 119911lowast = (119911

lowast

119896) 119905lowast

= (119905lowast

119896) isin

ℓlowast

infin Indeed combining the hypothesis

sup119896isinN

119911lowast

119896

ltinfin sup

119896isinN

119905lowast

119896

ltinfin (44)

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

Abstract and Applied Analysis 5

=

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

times sdot sdot sdot times

120573minus1 [

[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+ 120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

=

120573minus1[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

]]

]

+120573minus1 [

[

[

(

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

120573minus1

(119901)

(22)

On the other hand let us prove

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

(23)

Indeed

120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

119901

=120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

times sdot sdot sdot times 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

times sdot sdot sdot times (

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

= 120573

[

[

(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

= 120573(

119899

sum

119896=0

1003816100381610038161003816119911119896

1003816100381610038161003816

120573minus1

(119901)

)

=

119899

sum

119896=0

119911lowast

119896

119901

(24)

Substituting the relation (24) in (22) we obtain

120573minus1

[[

[

(

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

]]

]

119901

=[

[

(

119899

sum

119896=0

|119911119896|120573minus1

(119901)

)

1120573minus1

(119901)

+ (

119899

sum

119896=0

|119905119896|120573minus1

(119901)

)

1120573minus1

(119901)

]

]

120573minus1

(119901)

(25)

By using this equality in (21) we get

(

119899

sum

119896=0

119911lowast

119896oplus 119905lowast

119896

119901

)

1119901

le (

119899

sum

119896=0

119911lowast

119896

119901

)

1119901

+ (

119899

sum

119896=0

119905lowast

119896

119901

)

1119901

(26)

as desired

Theorem 6 (Clowast 119889lowast) is a complete metric space where 119889lowast is

defined by (10)

Proof First we show that 119889lowast defined by (10) is a metric onClowast

It is immediate for 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] ge 0

(27)

(i) Now we show that 119889lowast(119911lowast1 119911lowast

2) = 0 if and only if 119911lowast

1=

119911lowast

2for 119911lowast1 119911lowast

2isin Clowast Indeed

119889lowast

(119911lowast

1 119911lowast

2) = 0 lArrrArr

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 0

lArrrArr 120573[radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

] = 120573 (0)

lArrrArr (1198861minus 1198862)2

+ (1198871minus 1198872)2

= 0

lArrrArr 1198861minus 1198862= 0 119887

1minus 1198872= 0

lArrrArr 1198861= 1198862 119887

1= 1198872

lArrrArr 1198861= 1198862

1198871=

1198872

lArrrArr 119911lowast

1= ( 11988611198871) = ( 119886

21198872) = 119911lowast

2

(28)

6 Abstract and Applied Analysis

(ii) One can easily establish for all 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

= 120573 [radic(1198862minus 1198861)2

+ (1198872minus 1198871)2

]

= 119889lowast

(119911lowast

2 119911lowast

1)

(29)

(iii) We show that the inequality 119889lowast

(119911lowast

1 119911lowast

2) +119889lowast

(119911lowast

2

119911lowast

3) ge 119889lowast

(119911lowast

1 119911lowast

3) holds for all 119911lowast

1 119911lowast

2 119911lowast

3isin Clowast In fact

119889lowast

(119911lowast

1 119911lowast

2) + 119889lowast

(119911lowast

2 119911lowast

3)

=

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

+

radic[120580 ( 1198862minus 1198863)]2

+ (1198872minus

1198873)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

+ 120573 [radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

+ radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

ge120573 [radic(1198861minus 1198863)2

+ (1198871minus 1198873)2

]

=

radic[120580 ( 1198861minus 1198863)]2

+ (1198871minus

1198873)

2

= 119889lowast

(119911lowast

1 119911lowast

3)

(30)

Therefore 119889lowast is a metric over ClowastNow we can show that the metric space (Clowast 119889lowast) is

complete Let (119911lowast119899)119899isinN

be an arbitrary Cauchy sequence inClowastIn this case for all 120576 gt 0 there exists an 119899

0isin N such that

119889lowast

(119911lowast

119898 119911lowast

119899) lt 120576 for all 119898 119899 ge 119899

0 Let 119911lowast

119898= ( 119886119898119887119898) isin Clowast and

119898 isin N Then

119889lowast

(119911lowast

119898 119911lowast

119899) =

radic[120580 ( 119886119898minus 119886119899)]2

+ (119887119898minus

119887119899)

2

= 120573 [radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

]

lt 120576 = 120573 (1205761015840

)

(31)

Thus we obtain that

radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

lt 1205761015840

(32)

On the other hand since the following inequalities

1003816100381610038161003816119886119898minus 119886119899

1003816100381610038161003816= radic(119886

119898minus 119886119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

1003816100381610038161003816119887119898minus 119887119899

1003816100381610038161003816= radic(119887

119898minus 119887119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

(33)

hold we therefore have by (32) that |119886119898

minus 119886119899| lt 1205761015840 and |119887

119898minus

119887119899| lt 1205761015840 This means that (119886

119899) and (119887

119899) are Cauchy sequences

with real terms Since R is complete it is clear that for every1205761015840

gt 0 there exists an 1198991isin N such that |119886

119899minus 119886| lt 120576

1015840

2 for all119899 ge 119899

1and for every 120576

1015840

gt 0 there exists an 1198992isin N such that

|119887119899minus 119887| lt 120576

1015840

2 for all 119899 ge 1198992

Define 119911lowast isin Clowast by 119911lowast

= ( 119886119887) Then we have

119889lowast

(119911lowast

119899 119911lowast

) =

radic[120580 ( 119886119899minus 119886)]2

+ [119887119899minus

119887]

2

= 120573 [radic(119886119899minus 119886)2

+ (119887119899minus 119887)2

]

le 120573 [radic(119886119899minus 119886)2

+ radic(119887119899minus 119887)2

]

= 120573 (1003816100381610038161003816119886119899minus 119886

1003816100381610038161003816+1003816100381610038161003816119887119899minus 119887

1003816100381610038161003816)

le 120573(

1205761015840

2

+

1205761015840

2

)

= 120573 (1205761015840

) = 120576

(34)

Hence (Clowast 119889lowast) is a complete metric space

Since Clowast is a complete metric space with the metric119889lowast defined by (10) induced by the norm

sdot as a direct

consequence of Theorem 6 we have the following

Corollary 7 Clowast is a Banach space with the norm sdot

defined

by

119911lowast =

radic[120580 ( 119886)]

2

+ (119887)

2

119911lowast

= ( 119886119887) isin C

lowast

(35)

3 Sequence Spaces over Non-NewtonianComplex Field

In this section we define the sets 120596lowast ℓlowastinfin 119888lowast 119888lowast

0 and ℓ

lowast

119901of

all bounded convergent null and absolutely 119901-summablesequences over the non-Newtonian complex field Clowast whichcorrespond to the sets 120596 ℓ

infin 119888 1198880 and ℓ

119901over the complex

field C respectively That is to say that

120596lowast

= 119911lowast

= (119911lowast

119896) 119911lowast

119896isin Clowast

forall119896 isin N

ℓlowast

infin= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sup119896isinN

119911lowast

119896

ltinfin

119888lowast

= 119911lowast

= (119911lowast

119896) isin 120596lowast

exist119897lowast

isin Clowast

ni limlowast119896rarrinfin

119911lowast

119896= 119897lowast

119888lowast

0= 119911

lowast

= (119911lowast

119896) isin 120596lowast

limlowast119896rarrinfin

119911lowast

119896= 120579lowast

ℓlowast

119901= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sum

119896

119911lowast

119896

119901

ltinfin

(36)

For simplicity in notation here and in what follows thesummation without limits runs from 0 to infin One can easily

Abstract and Applied Analysis 7

see that the set 120596lowast forms a vector space over Clowast with respectto the algebraic operations addition (+) and scalar multipli-cation (times) defined on 120596

lowast as follows

+ 120596lowast

times 120596lowast

997888rarr 120596lowast

(119911lowast

119905lowast

) 997891997888rarr 119911lowast

+ 119905lowast

= (119911lowast

119896oplus 119905lowast

119896)

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast

times Clowast

times 120596lowast

997888rarr 120596lowast

(120572 119911lowast

) 997891997888rarr 120572 times 119911lowast

= (120572 ⊙ 119911lowast

119896)

119911lowast

= (119911lowast

119896) isin 120596lowast

120572 isin Clowast

(37)

Theorem 8 Define the function 119889120596lowast by

119889120596lowast 120596lowast

times 120596lowast

997888rarr 1198611015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 tlowast119896)]

(38)

where (120583119896) isin 1198611015840

sube 119861 such that sum119896120583119896is convergent with 120583

119896gt 0

for all 119896 isin N Then (120596lowast 119889120596lowast) is a metric space

Proof We show that 119889120596lowast satisfies the metric axioms on the

space 120596lowast of all non-Newtonian complex valued sequences(i) First we show that 119889

120596lowast(119911lowast

119905lowast

) ge 0 for all 119911lowast 119905lowast isin 120596lowast

Because (Clowast 119889lowast) is a metric space we have 119889lowast(119911lowast119896 119905lowast

119896) ge 0

119911lowast

119896 119905lowast

119896isin Clowast and 1 + 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 Hence we obtain that

119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 Moreover since 120583

119896gt 0 for all

119896 isin N we have

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 (39)

This means that

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0

(40)

(ii) We show that 119889120596lowast(119911lowast

119905lowast

) = 0 iff 119911lowast

= 119905lowast In this situ-

ation one can see that

119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

lArrrArr 120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

120583119896gt 0 forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896) = 0 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 forall119896 isin N

lArrrArr 119911lowast

119896= 119905lowast

119896 119889lowastmetric forall119896 isin N

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(41)

(iii) We show that 119889120596lowast(119911lowast

119905lowast

) = 119889120596lowast(119905lowast

119911lowast

) for 119911lowast

=

(119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast First we know that 119889lowast is a metric over

Clowast Thus

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119911lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119911lowast

119896)]

= 119889120596lowast (119905lowast

119911lowast

)

(42)

(iv) We show that 119889120596lowast(119911lowast

119905lowast

) + 119889120596lowast(119905lowast

119906lowast

) ge 119889120596lowast(119911lowast

119906lowast

)

holds for 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin 120596

lowast Againusing the fact that (Clowast 119889lowast) is a metric space it is easy to seeby Lemma 4 that

119889120596lowast (119911lowast

119906lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

1 + [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times 119889

lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+ 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

= 119889120596lowast (119911lowast

119905lowast

) + 119889120596lowast (119905lowast

119906lowast

)

(43)

as required

Theorem 9 The set ℓlowastinfin

is a sequence space

Proof It is trivial that the inclusion ℓlowast

infinsub 120596lowast holds

(i) We show that 119911lowast + 119905lowast

isin ℓlowast

infinfor 119911lowast = (119911

lowast

119896) 119905lowast

= (119905lowast

119896) isin

ℓlowast

infin Indeed combining the hypothesis

sup119896isinN

119911lowast

119896

ltinfin sup

119896isinN

119905lowast

119896

ltinfin (44)

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

6 Abstract and Applied Analysis

(ii) One can easily establish for all 119911lowast1 119911lowast

2isin Clowast that

119889lowast

(119911lowast

1 119911lowast

2) =

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

= 120573 [radic(1198862minus 1198861)2

+ (1198872minus 1198871)2

]

= 119889lowast

(119911lowast

2 119911lowast

1)

(29)

(iii) We show that the inequality 119889lowast

(119911lowast

1 119911lowast

2) +119889lowast

(119911lowast

2

119911lowast

3) ge 119889lowast

(119911lowast

1 119911lowast

3) holds for all 119911lowast

1 119911lowast

2 119911lowast

3isin Clowast In fact

119889lowast

(119911lowast

1 119911lowast

2) + 119889lowast

(119911lowast

2 119911lowast

3)

=

radic[120580 ( 1198861minus 1198862)]2

+ (1198871minus

1198872)

2

+

radic[120580 ( 1198862minus 1198863)]2

+ (1198872minus

1198873)

2

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

]

+ 120573 [radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

= 120573 [radic(1198861minus 1198862)2

+ (1198871minus 1198872)2

+ radic(1198862minus 1198863)2

+ (1198872minus 1198873)2

]

ge120573 [radic(1198861minus 1198863)2

+ (1198871minus 1198873)2

]

=

radic[120580 ( 1198861minus 1198863)]2

+ (1198871minus

1198873)

2

= 119889lowast

(119911lowast

1 119911lowast

3)

(30)

Therefore 119889lowast is a metric over ClowastNow we can show that the metric space (Clowast 119889lowast) is

complete Let (119911lowast119899)119899isinN

be an arbitrary Cauchy sequence inClowastIn this case for all 120576 gt 0 there exists an 119899

0isin N such that

119889lowast

(119911lowast

119898 119911lowast

119899) lt 120576 for all 119898 119899 ge 119899

0 Let 119911lowast

119898= ( 119886119898119887119898) isin Clowast and

119898 isin N Then

119889lowast

(119911lowast

119898 119911lowast

119899) =

radic[120580 ( 119886119898minus 119886119899)]2

+ (119887119898minus

119887119899)

2

= 120573 [radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

]

lt 120576 = 120573 (1205761015840

)

(31)

Thus we obtain that

radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

lt 1205761015840

(32)

On the other hand since the following inequalities

1003816100381610038161003816119886119898minus 119886119899

1003816100381610038161003816= radic(119886

119898minus 119886119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

1003816100381610038161003816119887119898minus 119887119899

1003816100381610038161003816= radic(119887

119898minus 119887119899)2

le radic(119886119898minus 119886119899)2

+ (119887119898minus 119887119899)2

(33)

hold we therefore have by (32) that |119886119898

minus 119886119899| lt 1205761015840 and |119887

119898minus

119887119899| lt 1205761015840 This means that (119886

119899) and (119887

119899) are Cauchy sequences

with real terms Since R is complete it is clear that for every1205761015840

gt 0 there exists an 1198991isin N such that |119886

119899minus 119886| lt 120576

1015840

2 for all119899 ge 119899

1and for every 120576

1015840

gt 0 there exists an 1198992isin N such that

|119887119899minus 119887| lt 120576

1015840

2 for all 119899 ge 1198992

Define 119911lowast isin Clowast by 119911lowast

= ( 119886119887) Then we have

119889lowast

(119911lowast

119899 119911lowast

) =

radic[120580 ( 119886119899minus 119886)]2

+ [119887119899minus

119887]

2

= 120573 [radic(119886119899minus 119886)2

+ (119887119899minus 119887)2

]

le 120573 [radic(119886119899minus 119886)2

+ radic(119887119899minus 119887)2

]

= 120573 (1003816100381610038161003816119886119899minus 119886

1003816100381610038161003816+1003816100381610038161003816119887119899minus 119887

1003816100381610038161003816)

le 120573(

1205761015840

2

+

1205761015840

2

)

= 120573 (1205761015840

) = 120576

(34)

Hence (Clowast 119889lowast) is a complete metric space

Since Clowast is a complete metric space with the metric119889lowast defined by (10) induced by the norm

sdot as a direct

consequence of Theorem 6 we have the following

Corollary 7 Clowast is a Banach space with the norm sdot

defined

by

119911lowast =

radic[120580 ( 119886)]

2

+ (119887)

2

119911lowast

= ( 119886119887) isin C

lowast

(35)

3 Sequence Spaces over Non-NewtonianComplex Field

In this section we define the sets 120596lowast ℓlowastinfin 119888lowast 119888lowast

0 and ℓ

lowast

119901of

all bounded convergent null and absolutely 119901-summablesequences over the non-Newtonian complex field Clowast whichcorrespond to the sets 120596 ℓ

infin 119888 1198880 and ℓ

119901over the complex

field C respectively That is to say that

120596lowast

= 119911lowast

= (119911lowast

119896) 119911lowast

119896isin Clowast

forall119896 isin N

ℓlowast

infin= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sup119896isinN

119911lowast

119896

ltinfin

119888lowast

= 119911lowast

= (119911lowast

119896) isin 120596lowast

exist119897lowast

isin Clowast

ni limlowast119896rarrinfin

119911lowast

119896= 119897lowast

119888lowast

0= 119911

lowast

= (119911lowast

119896) isin 120596lowast

limlowast119896rarrinfin

119911lowast

119896= 120579lowast

ℓlowast

119901= 119911

lowast

= (119911lowast

119896) isin 120596lowast

sum

119896

119911lowast

119896

119901

ltinfin

(36)

For simplicity in notation here and in what follows thesummation without limits runs from 0 to infin One can easily

Abstract and Applied Analysis 7

see that the set 120596lowast forms a vector space over Clowast with respectto the algebraic operations addition (+) and scalar multipli-cation (times) defined on 120596

lowast as follows

+ 120596lowast

times 120596lowast

997888rarr 120596lowast

(119911lowast

119905lowast

) 997891997888rarr 119911lowast

+ 119905lowast

= (119911lowast

119896oplus 119905lowast

119896)

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast

times Clowast

times 120596lowast

997888rarr 120596lowast

(120572 119911lowast

) 997891997888rarr 120572 times 119911lowast

= (120572 ⊙ 119911lowast

119896)

119911lowast

= (119911lowast

119896) isin 120596lowast

120572 isin Clowast

(37)

Theorem 8 Define the function 119889120596lowast by

119889120596lowast 120596lowast

times 120596lowast

997888rarr 1198611015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 tlowast119896)]

(38)

where (120583119896) isin 1198611015840

sube 119861 such that sum119896120583119896is convergent with 120583

119896gt 0

for all 119896 isin N Then (120596lowast 119889120596lowast) is a metric space

Proof We show that 119889120596lowast satisfies the metric axioms on the

space 120596lowast of all non-Newtonian complex valued sequences(i) First we show that 119889

120596lowast(119911lowast

119905lowast

) ge 0 for all 119911lowast 119905lowast isin 120596lowast

Because (Clowast 119889lowast) is a metric space we have 119889lowast(119911lowast119896 119905lowast

119896) ge 0

119911lowast

119896 119905lowast

119896isin Clowast and 1 + 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 Hence we obtain that

119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 Moreover since 120583

119896gt 0 for all

119896 isin N we have

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 (39)

This means that

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0

(40)

(ii) We show that 119889120596lowast(119911lowast

119905lowast

) = 0 iff 119911lowast

= 119905lowast In this situ-

ation one can see that

119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

lArrrArr 120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

120583119896gt 0 forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896) = 0 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 forall119896 isin N

lArrrArr 119911lowast

119896= 119905lowast

119896 119889lowastmetric forall119896 isin N

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(41)

(iii) We show that 119889120596lowast(119911lowast

119905lowast

) = 119889120596lowast(119905lowast

119911lowast

) for 119911lowast

=

(119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast First we know that 119889lowast is a metric over

Clowast Thus

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119911lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119911lowast

119896)]

= 119889120596lowast (119905lowast

119911lowast

)

(42)

(iv) We show that 119889120596lowast(119911lowast

119905lowast

) + 119889120596lowast(119905lowast

119906lowast

) ge 119889120596lowast(119911lowast

119906lowast

)

holds for 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin 120596

lowast Againusing the fact that (Clowast 119889lowast) is a metric space it is easy to seeby Lemma 4 that

119889120596lowast (119911lowast

119906lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

1 + [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times 119889

lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+ 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

= 119889120596lowast (119911lowast

119905lowast

) + 119889120596lowast (119905lowast

119906lowast

)

(43)

as required

Theorem 9 The set ℓlowastinfin

is a sequence space

Proof It is trivial that the inclusion ℓlowast

infinsub 120596lowast holds

(i) We show that 119911lowast + 119905lowast

isin ℓlowast

infinfor 119911lowast = (119911

lowast

119896) 119905lowast

= (119905lowast

119896) isin

ℓlowast

infin Indeed combining the hypothesis

sup119896isinN

119911lowast

119896

ltinfin sup

119896isinN

119905lowast

119896

ltinfin (44)

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

Abstract and Applied Analysis 7

see that the set 120596lowast forms a vector space over Clowast with respectto the algebraic operations addition (+) and scalar multipli-cation (times) defined on 120596

lowast as follows

+ 120596lowast

times 120596lowast

997888rarr 120596lowast

(119911lowast

119905lowast

) 997891997888rarr 119911lowast

+ 119905lowast

= (119911lowast

119896oplus 119905lowast

119896)

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast

times Clowast

times 120596lowast

997888rarr 120596lowast

(120572 119911lowast

) 997891997888rarr 120572 times 119911lowast

= (120572 ⊙ 119911lowast

119896)

119911lowast

= (119911lowast

119896) isin 120596lowast

120572 isin Clowast

(37)

Theorem 8 Define the function 119889120596lowast by

119889120596lowast 120596lowast

times 120596lowast

997888rarr 1198611015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 tlowast119896)]

(38)

where (120583119896) isin 1198611015840

sube 119861 such that sum119896120583119896is convergent with 120583

119896gt 0

for all 119896 isin N Then (120596lowast 119889120596lowast) is a metric space

Proof We show that 119889120596lowast satisfies the metric axioms on the

space 120596lowast of all non-Newtonian complex valued sequences(i) First we show that 119889

120596lowast(119911lowast

119905lowast

) ge 0 for all 119911lowast 119905lowast isin 120596lowast

Because (Clowast 119889lowast) is a metric space we have 119889lowast(119911lowast119896 119905lowast

119896) ge 0

119911lowast

119896 119905lowast

119896isin Clowast and 1 + 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 Hence we obtain that

119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 Moreover since 120583

119896gt 0 for all

119896 isin N we have

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0 (39)

This means that

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] ge 0

(40)

(ii) We show that 119889120596lowast(119911lowast

119905lowast

) = 0 iff 119911lowast

= 119905lowast In this situ-

ation one can see that

119889120596lowast (119911lowast

119905lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

lArrrArr 120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)] = 0

120583119896gt 0 forall119896 isin N

lArrrArr 119889lowast

(119911lowast

119896 119905lowast

119896) = 0 119889

lowast

(119911lowast

119896 119905lowast

119896) ge 0 forall119896 isin N

lArrrArr 119911lowast

119896= 119905lowast

119896 119889lowastmetric forall119896 isin N

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(41)

(iii) We show that 119889120596lowast(119911lowast

119905lowast

) = 119889120596lowast(119905lowast

119911lowast

) for 119911lowast

=

(119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 120596lowast First we know that 119889lowast is a metric over

Clowast Thus

119889120596lowast (119911lowast

119905lowast

) =

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119911lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119911lowast

119896)]

= 119889120596lowast (119905lowast

119911lowast

)

(42)

(iv) We show that 119889120596lowast(119911lowast

119905lowast

) + 119889120596lowast(119905lowast

119906lowast

) ge 119889120596lowast(119911lowast

119906lowast

)

holds for 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin 120596

lowast Againusing the fact that (Clowast 119889lowast) is a metric space it is easy to seeby Lemma 4 that

119889120596lowast (119911lowast

119906lowast

)

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

1 + [119889

lowast

(119911lowast

119896 119905lowast

119896) + 119889lowast

(119905lowast

119896 119906lowast

119896)]

le

sum

119896

120583119896times 119889

lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+ 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

=

sum

119896

120583119896times 119889lowast

(119911lowast

119896 119905lowast

119896)

[1 + 119889

lowast

(119911lowast

119896 119905lowast

119896)]

+

sum

119896

120583119896times 119889lowast

(119905lowast

119896 119906lowast

119896)

[1 + 119889

lowast

(119905lowast

119896 119906lowast

119896)]

= 119889120596lowast (119911lowast

119905lowast

) + 119889120596lowast (119905lowast

119906lowast

)

(43)

as required

Theorem 9 The set ℓlowastinfin

is a sequence space

Proof It is trivial that the inclusion ℓlowast

infinsub 120596lowast holds

(i) We show that 119911lowast + 119905lowast

isin ℓlowast

infinfor 119911lowast = (119911

lowast

119896) 119905lowast

= (119905lowast

119896) isin

ℓlowast

infin Indeed combining the hypothesis

sup119896isinN

119911lowast

119896

ltinfin sup

119896isinN

119905lowast

119896

ltinfin (44)

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

8 Abstract and Applied Analysis

with the fact 119911lowast

119896oplus 119905lowast

119896

le

119911lowast

119896

+

119905lowast

119896

obtained from Lemma 2

we can easily derive that

sup119896isinN

119911lowast

119896oplus 119905lowast

119896

le sup119896isinN

119911lowast

119896

+ sup119896isinN

119905lowast

119896

ltinfin (45)

Hence 119911lowast + 119905lowast

isin ℓlowast

infin

(ii) We show that 120572 times 119911lowast

isin ℓlowast

infinfor any 120572 isin Clowast and for

119911lowast

= (119911lowast

119896) isin ℓlowast

infin

Since 120572⊙119911lowast

119896

=

120572

times

119911lowast

119896

by Lemma 3 and sup

119896isinN119911lowast

119896

lt

infin it is immediate that

sup119896isinN

120572 ⊙ 119911

lowast

119896

=

120572

times sup119896isinN

119911lowast

119896

ltinfin (46)

which means that 120572 times 119911lowast

isin ℓlowast

infin

Therefore we have proved that ℓlowastinfin

is a subspace of thespace 120596lowast

Theorem 10 Define the relation 119889lowast

infinby

119889lowast

infin ℓlowast

infintimes ℓlowast

infin997888rarr 119861

1015840

sube 119861

(119911lowast

119905lowast

) 997891997888rarr 119889lowast

infin(119911lowast

119905lowast

) = sup119896isinN

119911lowast

119896⊖ 119905lowast

119896

(47)

Then (ℓlowastinfin 119889lowast

infin) is a complete metric space

Proof One can easily show by a routine verification that 119889lowastinfin

satisfies the metric axioms on the space ℓlowast

infin So we omit the

detailsNow we prove the second part of the theorem Let (119911lowast

119898)

be a Cauchy sequence in ℓlowast

infin where 119911

lowast

119898= (119911lowast

119896

119898

)119896isinN

Thenthere exists a positive integer 119896

0such that 119889

lowast

infin(119911lowast

119898 119911lowast

119903) =

sup119896isinN

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 for all 119898 119903 isin N with 119898 119903 gt 119896

0 For

any fixed 119896 if119898 119903 gt 1198960then

119911lowast

119896

119898

⊖ 119911lowast

119896

119903 lt 120576 (48)

In this case for any fixed 119896 (119911lowast

119896

0

119911lowast

119896

1

) is a Cauchysequence of non-Newtonian complex numbers and sinceClowast is complete it converges to a 119911

lowast

119896isin Clowast Define 119911

lowast

=

(119911lowast

0 119911lowast

1 )with infinitely many limits 119911lowast

0 119911lowast

1 Let us show

119911lowast

isin ℓlowast

infinand 119911

lowast

119898rarr 119911lowast as 119898 rarr infin Indeed by (48) by

letting 119903 rarr infin for119898 gt 1198960we obtain that

119911lowast

119896

119898

⊖ 119911lowast

119896

le 120576 (49)

On the other hand since 119911lowast

119898= (119911lowast

119896

119898

)119896isinN

isin ℓlowast

infin there exists

119905119898

isin 119861 sube R such that 119911lowast

119896

119898 le 119905119898for all 119896 isin N Hence by

triangle inequality (12) the inequality

119911lowast

119896

le

119911lowast

119896⊖ 119911lowast

119896

119898 +

119911lowast

119896

119898 le 120576 + 119905

119898(50)

holds for all 119896 isin N which is independent of 119896 Hence 119911lowast =

(119911lowast

119896)119896isinN isin ℓ

lowast

infin By (49) since119898 gt 119896

0 we obtain 119889

lowast

infin(119911lowast

119898 119911lowast

) =

sup119896isinN

119911lowast

119896

119898

⊖119911lowast

119896

le 120576Therefore the sequence (119911lowast

119898) converges

to 119911lowast which means that ℓlowast

infinis complete

Since it is known by Theorem 10 that ℓinfin

is a completemetric space with the metric 119889lowast

infininduced by the norm

sdotinfin

defined by

119911lowast infin

= sup119896isinN

119911lowast

119896

119911

lowast

= (119911lowast

119896) isin ℓlowast

infin (51)

we have the following

Corollary 11 ℓlowast

infinis a Banach space with the norm

sdotinfin

defined by (51)

Nowwe give the following lemma required in proving thefact that ℓlowast

119901is a sequence space in the case 0 lt 119901 lt 1

Lemma 12 Let 0 lt 119901 lt 1 Then the inequality ( 119886 +119887)

119901

lt 119886119901

+119887119901 holds for all 119886

119887 gt 0

Proof Let 0 lt 119901 lt 1 and 119886119887 gt 0 Then one can easily see that

( 119886 +119887)

119901

= ( 119886 +119887) times sdot sdot sdot times ( 119886 +

119887)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119901-times

= 120573 119886 + 119887 times sdot sdot sdot times 120573 119886 + 119887

= 120573

120573minus1

[120573 (119886 + 119887)] times sdot sdot sdot times 120573minus1

[120573 (119886 + 119887)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120573minus1

(119901)-times

= 120573(119886 + 119887)120573minus1

(119901)

lt120573 119886120573minus1

(119901)

+ 119887120573minus1

(119901)

= 120573 120573minus1

( 119886119901

) + 120573minus1

(119887119901

)

= 119886119901

+119887119901

(52)

as desired

Theorem 13 The sets 119888lowast

119888lowast

0 and ℓ

lowast

119901are sequence spaces

where 0 lt 119901 ltinfin

Proof It is not hard to establish by the similar way that 119888lowast0and

ℓlowast

119901are the sequence spaces So to avoid the repetition of the

similar statements we consider only the set 119888lowastIt is obvious that the inclusion 119888

lowast

sub 120596lowast strictly holds

(i) Let 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin 119888lowast Then there exist 119897lowast

1 119897lowast

2isin

Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast

1and limlowast

119896rarrinfin119905lowast

119896= 119897lowast

2Thus there

exist 1198961 1198962isin N such that

forall120576 gt 0119911lowast

119896⊖ 119897lowast

1

le 120576

2 forall119896 ge 119896

1

forall120576 gt 0119905lowast

119896⊖ 119897lowast

2

le 120576

2 forall119896 ge 119896

2

(53)

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

Abstract and Applied Analysis 9

Thus if we set 1198960= max119896

1 1198962 by (53) we obtain for all

119896 ge 1198960that

(119911lowast

119896oplus 119905lowast

119896) ⊖ (119897lowast

1oplus 119897lowast

2)

= (119911lowast

119896⊖ 119897lowast

1) oplus (119905lowast

119896⊖ 119897lowast

2)

le119911lowast

119896⊖ 119897lowast

1

+

119905lowast

119896⊖ 119897lowast

2

le 120576

2 + 120576

2

= 120576

(54)

which means thatlimlowast119896rarrinfin

(119911lowast

119896oplus 119905lowast

119896) = 119897lowast

1oplus 119897lowast

2= limlowast119896rarrinfin

119911lowast

119896oplus limlowast119896rarrinfin

119905lowast

119896 (55)

Therefore 119911lowast + 119905lowast

isin 119888lowast

(ii) Let 119911lowast = (119911lowast

119896) isin 119888lowast and 120572 isin Clowast 120579

lowast

Since 119911lowast

isin 119888lowast

there exists an 119897lowast

isin Clowast such that limlowast119896rarrinfin

119911lowast

119896= 119897lowast we have

forall120576 gt 0 exist1198960isin N such that

119911lowast

119896⊖ 119897lowast le 120576

120572

forall119896 ge 119896

0

(56)

Thus for 119896 ge 1198960 we have (120572 ⊙ 119911

lowast

119896) ⊖ (120572 ⊙ ℓ

lowast

)

=120572 ⊙ (119911

lowast

119896⊖ ℓlowast

)

=120572

times

119911lowast

119896⊖ ℓlowast

le120572

times 120576

120572

= 120576

(57)

which implies that limlowast119896rarrinfin

(120572 ⊙ 119911lowast

119896) = 120572 ⊙ 119897

lowast

= 120572 ⊙

limlowast119896rarrinfin

119911lowast

119896Hence 120572 times 119911

lowast

isin 119888lowast

That is to say that 119888lowast is a subspace of 120596lowast

Theorem 14 (119888lowast

119889lowast

infin) (119888lowast

0 119889lowast

infin) and (ℓ

lowast

119901 119889lowast

119901) are complete

metric spaces where 119889lowast119901is defined as follows

119889lowast

119901(119911lowast

119905lowast

) =

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

0lt119901lt1

(

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

119901ge1

119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901

(58)

Proof We consider only the space ℓlowast119901with 119901 ge 1

(i) For 119911lowast = (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓlowast

119901 we establish that the two

sided implication 119889lowast

119901(119911lowast

119905lowast

) = 0 hArr 119911lowast

= 119905lowast holds In fact

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 0

lArrrArr

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

119901

= 0

lArrrArr forall119896 isin N119911lowast

119896⊖ 119905lowast

119896

= 0

lArrrArr forall119896 isin N 119911lowast

119896= 119905lowast

119896

lArrrArr 119911lowast

= (119911lowast

119896) = (119905

lowast

119896) = 119905lowast

(59)

(ii) For 119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) isin ℓ

lowast

119901 we show that

119889lowast

119901(119911lowast

119905lowast

) = 119889lowast

119901(119905lowast

119911lowast

) In this situation since 119911lowast

119896⊖ 119905lowast

119896

=

119905lowast

119896⊖ 119911lowast

119896

holds for every fixed 119896 isin N it is immediate that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= (

sum

119896

119905lowast

119896⊖ 119911lowast

119896

119901

)

1119901

= 119889lowast

119901(119905lowast

119911lowast

)

(60)

(iii) By Minkowski inequality in Lemma 5 we have for119911lowast

= (119911lowast

119896) 119905lowast

= (119905lowast

119896) 119906lowast

= (119906lowast

119896) isin ℓlowast

119901that

119889lowast

119901(119911lowast

119905lowast

) = (

sum

119896

119911lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= [

sum

119896

(119911lowast

119896⊖ 119906lowast

119896) oplus (119906

lowast

119896⊖ 119905lowast

119896)

119901

]

1119901

le(

sum

119896

119911lowast

119896⊖ 119906lowast

119896

119901

)

1119901

+ (

sum

119896

119906lowast

119896⊖ 119905lowast

119896

119901

)

1119901

= 119889lowast

119901(119911lowast

119906lowast

) + 119889lowast

119901(119906lowast

119905lowast

)

(61)

Hence triangle inequality is satisfied by 119889lowast

119901on the space

ℓlowast

119901 Therefore the function 119889

lowast

119901is a metric over the space ℓ

lowast

119901

Now we show that the metric space (ℓlowast

119901 119889lowast

119901) is complete

Let (119911lowast119898)119898isinN be an arbitrary Cauchy sequence in the space ℓlowast

119901

where 119911lowast119898

= (119911lowast

1

119898

119911lowast

2

119898

) Then for any 120576 gt 0 there exists an1198990isin N such that

119889lowast

119901(119911lowast

119898 119911lowast

119899) = (

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

)

1119901

lt 120576 (62)

for all 119898 119899 ge 1198990 Hence for 119898 119899 ge 119899

0and every fixed 119896 isin N

we obtain

119911lowast

119896

119898

⊖ 119911lowast

119896

119899 lt 120576 (63)

If we set 119896 fixed then it follows by (63) that (119911lowast

119896

0

119911lowast

119896

1

)

is a Cauchy sequence Since Clowast is complete this sequenceconverges to a point say 119911

lowast

119896 Let us define the sequence 119911

lowast

=

(119911lowast

0 119911lowast

1 )with these limits and show 119911

lowast

isin ℓlowast

119901and 119911lowast

119898rarr 119911lowast

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

10 Abstract and Applied Analysis

as 119898 rarr infin Indeed by (62) we obtain the inequality for all119898 119899 isin N with119898 119899 ge 119899

0that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119899

119901

lt 120576119901

forall119895 isin N (64)

and thus we have by letting 119899 rarr infin and119898 gt 1198990that

119895

sum

119896=0

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

forall119895 isin N (65)

which gives as 119895 rarr infin and for all119898 gt 1198990that

[119889lowast

119901(119911lowast

119898 119911lowast

)]

119901

=

sum

119896

119911lowast

119896

119898

⊖ 119911lowast

119896

119901

lt 120576119901

(66)

Setting 119911lowast

119896= 119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

) and applying Lemma 5 weobtain by (66) and the fact (119911lowast

119896

119898

) isin ℓlowast

119901that

(

sum

119896

119911lowast

119896

119901

)

1119901

= (

sum

119896

119911lowast

119896

119898

oplus (119911lowast

119896⊖ 119911lowast

119896

119898

)

119901

)

1119901

le (

sum

119896

119911lowast

119896

119898

119901

)

1119901

+ (

sum

119896

119911lowast

119896⊖ 119911lowast

119896

119898

119901

)

1119901

ltinfin

(67)

whichmeans that 119911lowast = (119911lowast

119896) isin ℓlowast

119901Therefore we see from (66)

that 119911lowast119898

rarr 119911lowast Since the arbitrary Cauchy sequence (119911

lowast

119898) =

(119911lowast

119896

119898

)119896119898isinN

isin ℓlowast

119901is convergent the space ℓ

lowast

119901is complete

Corollary 15 119888lowast and 119888

lowast

0are the Banach spaces equipped with

the norm sdot

infin

defined in (51)Since it is known byTheorem 14 that ℓlowast

119901is a completemetric

space with the metric 119889lowast119901induced in the case 119901 ge 1 by the norm

sdot

119901and in the case 0 lt 119901 lt 1 by the 119901-norm

sdot119901 defined for

119911lowast

= (119911lowast

119896) isin ℓlowast

119901by

119911lowast 119901= (

sum

119896

119911lowast

119896

119901

)

1119901

119911lowast119901=

sum

119896

119911lowast

119896

119901

(68)

we have the following

Corollary 16 The space ℓlowast

119901is a Banach space with the norm

sdot

119901and 119901-norm

sdot119901defined by (68)

4 Conclusion

We present some important inequalities such as triangleMinkowski and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used Westate the classical sequence spaces over the non-Newtonian

complex field Clowast and try to understand their structure ofbeing non-Newtonian complex vector space There are lotsof techniques that have been developed in the sense of non-Newtonian complex calculus If the non-Newtonian complexcalculus is employed instead of the classical calculus in theformulations then many of the complicated phenomena inphysics or engineering may be analyzed more easily

As an alternative to the classical (additive) calculusGrossman and Katz [1] introduced some new kind of calcu-lus named as non-Newtonian calculus geometric calculusanageometric calculus and bigeometric calculus Turkmenand Basar [6 7] have recentlystudied the classical sequencespaces and related topics in the sense of geometric calculusQuite recently Cakmak andBasar [8] have alsoworked on thesame subject by using non-Newtonian calculus In the presentpaper we use the non-Newtonian complex calculus insteadof non-Newtonian real calculus and geometric calculus It istrivial that in the special cases of the generators 120572 and 120573 thenon-Newtonian complex calculus gives the special kind of thefollowing calculus

(i) if 120572 = 120573 = 119868 the identity function then the non-New-tonian complex calculus is reduced to the classicalcalculus

(ii) if 120572 = 119868 and 120573 = exp then the non-Newtoniancomplex calculus is reduced to the geometric calculus

(iii) if 120572 = exp and 120573 = 119868 then the non-Newtoniancomplex calculus is reduced to the anageometriccalculus

(iv) if 120572 = 120573 = exp then the non-Newtonian complexcalculus is reduced to the bigeometric calculus

Since our results are obtained by using the non-Newtoniancomplex calculus they are much more general and compre-hensive than those of Turkmen and Basar [6 7] Cakmak andBasar [8]

Quite recently Talo and Basar have studied the certainsets of sequences of fuzzy numbers and introduced theclassical sets ℓ

infin(119865) 119888(119865) 119888

0(119865) and ℓ

119901(119865) consisting of

the bounded convergent null and absolutely 119901-summablesequences of fuzzy numbers in [9] Nextly they have definedthe alpha- beta- and gamma-duals of a set of sequencesof fuzzy numbers and gave the duals of the classical sets ofsequences of fuzzy numbers together with the characteriza-tion of the classes of infinite matrices of fuzzy numbers trans-forming one of the classical set into another one FollowingBashirov et al [2] and Uzer [3] we give the correspondingresults for multiplicative calculus to the results derived forthe sets of fuzzy valued sequences in Talo and Basar [9] as abeginning As a natural continuation of this paper we shouldrecord from now on that it is meaningful to define the alpha-beta- and gamma-duals of a set of sequences over the non-Newtonian complex field Clowast and to determine the duals ofclassical spaces ℓ

lowast

infin 119888lowast 119888lowast

0 and ℓ

lowast

119901 Further one can obtain

the similar results by using another type of calculus insteadof non-Newtonian complex calculus

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

Abstract and Applied Analysis 11

Acknowledgment

The authors are very grateful to the referees for many helpfulsuggestions and comments about the paper which improvedthe presentation and its readability

References

[1] M Grossman and R Katz Non-Newtonian Calculus LowellTechnological Institute 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[4] A Bashirov andM Rıza ldquoOn complexmultiplicative differenti-ationrdquo TWMS Journal of Applied and Engineering Mathematicsvol 1 no 1 pp 75ndash85 2011

[5] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[6] C Turkmen and F Basar ldquoSome basic results on the sets ofsequences with geometric calculusrdquo AIP Conference Proceed-ings vol 1470 pp 95ndash98 2012

[7] C Turkmen and F Basar ldquoSome basic results on the geo-metric calculusrdquo Communications de la Faculte des Sciences delrsquoUniversite drsquoAnkara A

1 vol 61 no 2 pp 17ndash34 2012

[8] A F Cakmak and F Basar ldquoOn the classical sequence spacesand non-Newtonian calculusrdquo Journal of Inequalities and Appli-cations vol 2012 Article ID 932734 13 pages 2012

[9] O Talo and F Basar ldquoDetermination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tionsrdquo Computers amp Mathematics with Applications vol 58 no4 pp 717ndash733 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Certain Sequence Spaces over the Non ...=8f8 1 g8M 0 G 0 1 Nh+8 1 g8M 1 G 1 1 Nh +ZZZ+8 1 g8M G 1 Nhi =8 =0 + 1 8 # % & j l =0 1 1/ 1 + =0 s 1 1/ 1 m o rs t u, 8 1

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


SWAN Continuous Water Quality Monitoring Cuyler Smith 1 1 1 11 1 1 1 11 11 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0

SWAN Continuous Water Quality Monitoring Cuyler Smith 1 1 1 11 1 1 1 11 11 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0

Documents
New  · 2019. 10. 24. · 9 0 0 10 0 8 0 0 7 0 0 1 0 6 0 0 1 2 0 0 1 3 0 0 50 0 1 4 0 4 5 0 15 0 1 6 0 0 8 00 1 0 0 0 9 0 0 1 1 0 0 8 0 0 1 3 0 0 900 1 5 0 0 1 2 0 0 1 0 0 0 1 1 0

New  · 2019. 10. 24. · 9 0 0 10 0 8 0 0 7 0 0 1 0 6 0 0 1 2 0 0 1 3 0 0 50 0 1 4 0 4 5 0 15 0 1 6 0 0 8 00 1 0 0 0 9 0 0 1 1 0 0 8 0 0 1 3 0 0 900 1 5 0 0 1 2 0 0 1 0 0 0 1 1 0

Documents
Number System (1A) · Data Type 5 Young Won Lim 7/7/10 Number Systems 0 0 1 1 3 0 3 0 0 1 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 7 0 7 0 1 1 0 6 0 6 0 0 1 0 0 4 0 4 0 1 0 1 5

Number System (1A) · Data Type 5 Young Won Lim 7/7/10 Number Systems 0 0 1 1 3 0 3 0 0 1 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 7 0 7 0 1 1 0 6 0 6 0 0 1 0 0 4 0 4 0 1 0 1 5

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0 1 0 1 0 1 INTELIGÊNCIA 1 0 0 1 0 1 1 0 1 0 1 0 ATIVA

0 1 0 1 0 1 INTELIGÊNCIA 1 0 0 1 0 1 1 0 1 0 1 0 ATIVA

Documents
0 1 0 1 ACTIVE 1 0 0 1 0 1 0 1 1 0 1 0 1 0

0 1 0 1 ACTIVE 1 0 0 1 0 1 0 1 1 0 1 0 1 0

Documents
Option 1A Option 1 - transport.gov.scot · Loch Vaa 1 1 0 0 0. 0 0 0 1 8 0 0. 0 0 0 1 1 8 0 0. 0 0 0 1 1 6 0 0. 0 0 0 1 1 2 0 0. 0 0 1 1 4. 0 A9P11-AMJ-EBD-Z_ZZZZZ_XX-DR-GI-0192 Doc

Option 1A Option 1 - transport.gov.scot · Loch Vaa 1 1 0 0 0. 0 0 0 1 8 0 0. 0 0 0 1 1 8 0 0. 0 0 0 1 1 6 0 0. 0 0 0 1 1 2 0 0. 0 0 1 1 4. 0 A9P11-AMJ-EBD-Z_ZZZZZ_XX-DR-GI-0192 Doc

Documents
CONFERÈNCIES, CINEMA, ACCIONS I MOSTRES … · 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 1 0 1 0

CONFERÈNCIES, CINEMA, ACCIONS I MOSTRES … · 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 1 0 1 0

Documents
ATC Visual Display Solutions - eizoglobal.com · 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0

ATC Visual Display Solutions - eizoglobal.com · 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0

Documents
economicsinsight.files.wordpress.com · Web view1 1 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 1 0

economicsinsight.files.wordpress.com · Web view1 1 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 1 0

Documents
 · XLS file · Web view2016-08-03 · 0 0 2 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0

 · XLS file · Web view2016-08-03 · 0 0 2 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0 0. 0 0 1 0

Documents
Boolean Functions 1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 x 2 x 3 x f mapping truth table

Boolean Functions 1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 x 2 x 3 x f mapping truth table

Documents
0 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 Q

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