Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions

$Page 1: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions$
Research ArticleExistence Results for a Coupled System of NonlinearFractional Hybrid Differential Equations with HomogeneousBoundary Conditions

Josefa Caballero1 Mohamed Abdalla Darwish23

Kishin Sadarangani1 and Wafa M Shammakh2

1 Department of Mathematics University of Las Palmas de Gran Canaria Campus de Tafira Baja35017 Las Palmas de Gran Canaria Spain

2Department of Mathematics Sciences Faculty for Girls King Abdulaziz University Jeddah Saudi Arabia3 Department of Mathematics Faculty of Science Damanhour University Damanhour Egypt

Correspondence should be addressed to Mohamed Abdalla Darwish drmadarwishgmailcom

Received 7 April 2014 Accepted 16 June 2014 Published 14 July 2014

Academic Editor Ljubisa Kocinac

Copyright copy 2014 Josefa Caballero et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study an existence result for the following coupled system of nonlinear fractional hybrid differential equations withhomogeneous boundary conditions 119863

120572

0+ [119909(119905)119891(119905 119909(119905) 119910(119905))] = 119892(119905 119909(119905) 119910(119905)) 119863

120572

0+ [119910(119905)119891(119905 119910(119905) 119909(119905))] = 119892(119905 119910(119905) 119909(119905)) 0 lt

119905 lt 1 and 119909(0) = 119910(0) = 0 where 120572 isin (0 1) and 119863120572

0+ denotes the Riemann-Liouville fractional derivative The main tools in our

study are the techniques associated to measures of noncompactness in the Banach algebras and a fixed point theorem of Darbotype

1 Introduction

Fractional differential equations arise in many engineeringand scientific disciplines as the mathematical modelling ofa great number of processes which appear in physics chem-istry aerodynamics and so forth and involve also derivativesof fractional order For details see [1ndash5] and the referencestherein

On the other hand about the theory of hybrid differentialequations we refer to the paper [6] where the authors studiedthe hybrid differential equation of first order

119889

119889119905[

119909 (119905)

119891 (119905 119909 (119905))] = 119892 (119905 119909 (119905)) 119905 isin 119869 = [0 119879)

119909 (1199050) = 1199090

isin R

(1)

where 119891 isin 119862(119869 times RR 0) and 119892 isin 119862(119869 times RR)

In [7] the authors studied the fractional version of theabovementioned problem that is

119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905))] = 119892 (119905 119909 (119905)) 119905 isin 119869 0 lt 120572 lt 1

119909 (0) = 0

(2)

under the same assumptions on 119891 and 119892 in [6]Recently in [8] the authors studied the following frac-

tional hybrid initial value problem with supremum

119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) max0le120591le119905

|119909 (120591)|)] = 119892 (119905 119909 (119905))

0 lt 119905 lt 1

119909 (0) = 0

(3)

where 0 lt 120572 lt 1 119891 isin 119862([0 1] times R times RR 0) and 119892 isin

119862([0 1] times RR)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 672167 10 pageshttpdxdoiorg1011552014672167

2 Abstract and Applied Analysis

The coupled systems involving fractional differentialequations are very important because they occur in numerousproblems of applied nature for instance see [9ndash13]

In this paper we consider the following coupled system

119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905))

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905))

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(4)

where 120572 isin (0 1) and 119863120572

0+ is the standard Riemann-Liouville

fractional derivativeThe main tool in our study is a fixed point theorem of

Darbo type associated to measures of noncompactness

2 Preliminaries

We begin this section with some definitions and results aboutfractional calculus

Let 120572 gt 0 and 119899 = [120572] + 1 where [120572] denotes the integerpart of 120572 For a function 119891 (0 infin) rarr R the Riemann-Liouville fractional integral of order 120572 gt 0 of 119891 is defined as

119868120572

0+119891 (119905) =

1

Γ (120572)int

119905

0

119891 (119904)

(119905 minus 119904)1minus120572

119889119904 (5)

provided that the right side is pointwise defined on (0 infin)The Riemann-Liouville fractional derivative of order 120572 of

a continuous function 119891 is defined by

119863120572

0+119891 (119909) =

1

Γ (119899 minus 120572)

119889119899

119889119909119899int

119909

0

119891 (119904)

(119909 minus 119904)120572minus119899+1

119889119904 (6)

provided that the right side is pointwise defined on (0 infin)The following lemma will be useful for our study [14]

Lemma 1 Let ℎ isin 1198711

(0 1) and 0 lt 120572 lt 1 Then

(a)

119863120572

0+119868120572

0+ℎ (119909) = ℎ (119909) (7)

(b)

119868120572

0+119863120572

0+ℎ (119909) = ℎ (119909) minus

1198681minus120572

0+ ℎ (119909)

10038161003816100381610038161003816119909=0

Γ (120572)119909120572minus1

119886119890 on (0 1)

(8)

Lemma 2 Let 0 lt 120572 lt 1 and suppose that 119891 isin 119862([0 1]R

0) and119910 isin 119862[0 1]Then the unique solution of the fractionalhybrid initial value problem

119863120572

0+ [

119909 (119905)

119891 (119905)] = 119910 (119905) 0 lt 119905 lt 1

119909 (0) = 0

(9)

is given by

119909 (119905) =119891 (119905)

Γ (120572)int

119905

0

119910 (119904)

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1] (10)

Proof Suppose that 119909(119905) is a solution of problem (9) Usingthe operator 119868

120572

0+ and taking into account Lemma 1 we get

119868120572

0+119863120572

0+ [

119909 (119905)

119891 (119905)] = 119868120572

0+119910 (119905) (11)

or equivalently

119909 (119905)

119891 (119905)minus

1198681minus120572

0+ (119909 (119905) 119891 (119905))

10038161003816100381610038161003816119905=0

Γ (120572)119905120572minus1

= 119868120572

0+119910 (119905) (12)

Since 119909(119905)119891(119905)|119905=0

= 119909(0)119891(0) = 0119891(0) = 0 (because119891(0) = 0) we have

119909 (119905) = 119891 (119905) 119868120572

0+119910 (119905) (13)

This means that

119909 (119905) =119891 (119905)

Γ (120572)int

119905

0

119910 (119904)

(119905 minus 119904)1minus120572

119889119904 (14)

Conversely suppose that 119909(119905) is given by

119909 (119905) =119891 (119905)

Γ (120572)int

119905

0

119910 (119904)

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1] (15)

This means that

119909 (119905) = 119891 (119905) 119868120572

0+119910 (119905) 119905 isin [0 1] (16)

Applying 119863120572

0+ and taking into account Lemma 1 and that

119891(119905) = 0 for 119905 isin [0 1] we obtain

119863120572

0+ [

119909 (119905)

119891 (119905)] = 119863

120572

0+119868120572

0+119910 (119905) = 119910 (119905) 0 lt 119905 lt 1 (17)

Moreover for 119905 = 0 in (16) we have 119909(0) = 119891(0) sdot 0 = 0 Thiscompletes the proof

In the sequel we recall some definitions and basic factsabout measures of noncompactness

Assume that 119864 is a real Banach space with norm sdot

and zero element 120579 By 119861(119909 119903) we denote the closed ball in 119864

centered at 119909with radius 119903 By 119861119903we denote the ball 119861(120579 119903) If

119883 is a nonempty subset of119864 by the symbols119883 andConv119883wedenote the closure and the convex closure of 119883 respectivelyBy 119883 we denote the quantity 119883 = sup119909 119909 isin 119883Finally by M

119864we will denote the family of all nonempty

and bounded subsets of 119864 and byN119864we denote its subfamily

consisting of all relatively compact subsets of 119864

Definition 3 Amapping120583 M119864

rarr R+

= [0 infin) is said to bea measure of noncompactness in 119864 if it satisfies the followingconditions

(a) The family ker 120583 = 119883 isin M119864

120583(119883) = 0 is nonemptyand ker 120583 sub N

119864

Abstract and Applied Analysis 3

(b) 119883 sub 119884 rArr 120583(119883) le 120583(119884)(c) 120583(119883) = 120583(Conv119883) = 120583(119883)(d) 120583(120582119883 + (1 minus 120582)119884) le 120582120583(119883) + (1 minus 120582)120583(119884) for 120582 isin [0 1](e) If (119883

119899) is a sequence of closed subsets fromM

119864such

that 119883119899+1

sub 119883119899

(119899 ge 1) and lim119899rarrinfin

120583(119883119899) = 0 then

119883infin

= capinfin

119899=1119883119899

= 120601

The family ker 120583 appearing in (a) is called the kernel ofthe measure of noncompactness 120583 Notice that the set 119883

infin

appearing in (e) is an element of ker120583 Indeed since 120583(119883infin

) le

120583(119883119899) for 119899 = 1 2 we infer that 120583(119883

infin) = 0 and this says

that 119883infin

isin ker120583An important theorem about fixed point theorem in

the context of measures of noncompactness is the followingDarborsquos fixed point theorem [15]

Theorem4 Let119862 be a nonempty bounded closed and convexsubset of a Banach space 119864 and let 119879 119862 rarr 119862 be a continuousmapping Suppose that there exists a constant 119896 isin [0 1) suchthat

120583 (119879 (119883)) le 119896120583 (119883) (18)

for any nonempty subset 119883 of 119862Then 119879 has a fixed point

A generalization of Theorem 4 which will be very usefulin our study is the following theorem due to Sadovskiı [16]

Theorem5 Let119862 be a nonempty bounded closed and convexsubset of a Banach space 119864 and let 119879 119862 rarr 119862 be a continuousoperator satisfying

120583 (119879 (119883)) lt 120583 (119883) (19)

for any nonempty subset 119883 of 119862 with 120583(119883) gt 0Then 119879 has a fixed point

Next we will assume that the space 119864 has structure ofBanach algebra By 119909119910 we will denote the product of twoelements 119909 119910 isin 119883 and by 119883119884 we will denote the set definedby 119883119884 = 119909119910 119909 isin 119883 119910 isin 119884

Definition 6 Let 119864 be a Banach algebra We will say thata measure of noncompactness 120583 defined on 119864 satisfiescondition (119898) if

120583 (119883119884) le 119883 120583 (119884) + 119884 120583 (119883) (20)

for any 119883 119884 isin M119864

This definition appears in [17]In this paper we will work in the space 119862[0 1] consisting

of all real functions defined and continuous on [0 1] with thestandard supremum norm

119909 = sup |119909 (119905)| 119905 isin [0 1] (21)

for119909 isin 119862[0 1] It is clear that (119862[0 1] sdot) is a Banach algebrawhere the multiplication is defined as the usual product ofreal functions

Next we present the measure of noncompactness in119862[0 1] which will be used later Let us fix 119883 isin M

119862[01]and

120576 gt 0 For 119909 isin 119883 we denote by 120596(119909 120576) the modulus ofcontinuity of 119909 that is

120596 (119909 120576) = sup |119909 (119905) minus 119909 (119904)| 119905 119904 isin [0 1] |119905 minus 119904| le 120576 (22)

Put

120596 (119883 120576) = sup 120596 (119909 120576) 119909 isin 119883

1205960

(119883) = lim120576rarr0

120596 (119883 120576) (23)

In [15] it is proved that 1205960(119883) is a measure of noncompact-

ness in 119862[0 1]

Proposition 7 Themeasure of noncompactness 1205960on 119862[0 1]

satisfies condition (119898)

Proof Fix 119883 119884 isin M119862[01]

120576 gt 0 and 119905 119904 isin [0 1] with |119905 minus 119904| le

120576 Then we have1003816100381610038161003816119909 (119905) 119910 (119905) minus 119909 (119904) 119910 (119904)

1003816100381610038161003816 le1003816100381610038161003816119909 (119905) 119910 (119905) minus 119909 (119905) 119910 (119904)

1003816100381610038161003816

+1003816100381610038161003816119909 (119905) 119910 (119904) minus 119909 (119904) 119910 (119904)

1003816100381610038161003816

= |119909 (119905)|1003816100381610038161003816119910 (119905) minus 119910 (119904)

1003816100381610038161003816

+1003816100381610038161003816119910 (119904)

1003816100381610038161003816 |119909 (119905) minus 119909 (119904)|

le 119909 120596 (119910 120576) +1003817100381710038171003817119910

1003817100381710038171003817 120596 (119909 120576)

(24)

This means that

120596 (119909119910 120576) le 119909 120596 (119910 120576) +1003817100381710038171003817119910

1003817100381710038171003817 120596 (119909 120576) (25)

and therefore

120596 (119883119884 120576) le 119883 120596 (119884 120576) + 119884 120596 (119883 120576) (26)

Taking 120576 rarr 0 we get

1205960

(119883119884) le 119883 1205960

(119884) + 119884 1205960

(119883) (27)

This completes the proof

Proposition 7 appears in [17] and we have given the prooffor the paper is self-contained

3 Main Results

We begin this section introducing the following class A offunctions

A = 120593 R+

997888rarr R+

120593 is nondecreasing

and lim119899rarrinfin

120593119899

(119905) = 0 for any 119905 gt 0

(28)

where 120593119899 denotes the 119899-iteration of 120593


Remark 8 Notice that if 120593 isin A then 120593(119905) lt 119905 for any 119905 gt 0Indeed in contrary case we can find 119905

0gt 0 and 119905

0le 120593(119905

0)

Since 120593 is nondecreasing we have

0 lt 1199050

le 120593 (1199050) le 1205932

(1199050) le sdot sdot sdot le 120593

119899

(1199050) le sdot sdot sdot (29)

and therefore 0 lt 1199050

le lim119899rarrinfin

120593119899

(1199050) and this contradicts

the fact that 120593 isin AMoreover the fact that 120593(119905) lt 119905 for any 119905 gt 0 proves that

if 120593 isin A then 120593 is continuous at 1199050

= 0

Using Remark 8 and Theorem 5 we have the followingfixed point theorem


120583 (119879 (119883)) le 120593 (120583 (119883)) (30)

for any nonempty subset 119883 of 119862 where 120593 isin AThen 119879 has a fixed point

Theorem 9 appears in [18] where the authors present aproof without usingTheorem 5

The following result which appears in [19] will be inter-esting in our study

Theorem 10 Let 1205831 1205832 120583

119899be measures of noncompact-

ness in the Banach spaces 1198641 1198642 119864

119899 respectively Suppose

that 119865 [0 infin)119899

rarr [0 infin) is a convex function such that119865(1199091 1199092 119909

119899) = 0 if and only if 119909

119894= 0 for 119894 = 1 2 119899

Then

120583 (119883) = 119865 (1205831

(1198831) 1205832

(1198832) 120583

119899(119883119899)) (31)

defines a measure compactness in 1198641

times 1198642

times sdot sdot sdot times 119864119899 where 119883

119894

denotes the natural projection of 119883 into 119864119894 for 119894 = 1 2 119899

Remark 11 As a consequence ofTheorem 10 we have that if 120583

is a measure of noncompactness on a Banach space 119864 and weconsider the function 119865 [0 infin) times [0 infin) rarr [0 infin) definedby 119865(119909 119910) = max(119909 119910) then since 119865 is convex and 119865(119909 119910) =

0 if and only if 119909 = 119910 = 0 120583(119883) = max120583(1198831) 120583(119883

2) defines

a measure of noncompactness in the space 119864 times 119864

Next we present the definition of a coupled fixed point

Definition 12 An element (119909 119910) isin 119883 times 119883 is said to be acoupled fixed point of a mapping 119866 119883 times 119883 rarr 119883 if119866(119909 119910) = 119909 and 119866(119910 119909) = 119910

The following result is crucial for our study

Theorem 13 Let Ω be a nonempty bounded closed andconvex subset of a Banach space 119864 and let 120583 be a measure ofnoncompactness in 119864 Suppose that 119866 Ω times Ω rarr Ω is acontinuous operator satisfying

120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (32)

for all nonempty subsets 1198831and 119883

2of Ω where 120593 isin A

Then 119866 has at least a coupled fixed point

Proof Notice that by Remark 11 120583(119883) = max120583(1198831) 120583(119883

2)

is a measure of noncompactness in the space 119864 times 119864 where 1198831

and 1198832are the projections of 119883 into 119864

Now we consider the mapping 119866 Ω times Ω rarr Ω times Ω

defined by 119866(119909 119910) = (119866(119909 119910) 119866(119910 119909)) It is easily seen thatΩ times Ω is a nonempty bounded closed and convex subsetof 119864 times 119864 Since 119866 is continuous it is clear that 119866 is alsocontinuous

Next we take a nonempty 119883 of Ω times Ω Then

120583 (119866 (119883)) = 120583 (119866 (1198831

times 1198832))

= 120583 (119866 (1198831

times 1198832) times 119866 (119883

2times 1198831))

= max 120583 (119866 (1198831

times 1198832)) 120583 (119866 (119883

2times 1198831))

le max 120593 (max (120583 (1198831) 120583 (119883

2)))

120593 (max (120583 (1198832) 120583 (119883

1)))

= 120593 (max (120583 (1198831) 120583 (119883

2)))

= 120593 (120583 (1198831

times 1198832))

(33)

Since 120593 isin A by Theorem 9 the mapping 119866 has at least onefixed pointThismeans that there exists (119909

0 1199100) isin ΩtimesΩ such

that 119866(1199090 1199100) = (119909

0 1199100) or equivalently 119866(119909

0 1199100) = 1199090and

119866(1199100 1199090) = 1199100 This proves that 119866 has at least a coupled fixed

point

Now we consider the coupled system of integral equa-tions

119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1]

(34)

Lemma 14 Assume that 119891 isin 119862([0 1] times R times RR 0) and119892 isin 119862([0 1] times R times RR) Then (119909 119910) isin 119862[0 1] times 119862[0 1] isa solution of (34) if and only if (119909 119910) isin 119862[0 1] times 119862[0 1] is asolution of (4)

Proof The proof is an immediate consequence of Lemma 2so we omit it

Next we will study problem (4) under the followingassumptions

(H1) 119891 isin 119862([0 1] times R times RR 0) and 119892 isin 119862([0 1] times R times

RR)(H2) The functions 119891 and 119892 satisfy

1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le 1205931

(max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

1003816100381610038161003816119892 (119905 1199091 1199102) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le 1205932

(max (10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 10038161003816100381610038161199092 minus 119910

2

1003816100381610038161003816))

(35)


respectively for any 119905 isin [0 1] and 1199091 1199092 1199101 1199102

isin Rwhere 120593

1 1205932

isin A and 1205931is continuous

Notice that assumption (H1) gives us the existence

of two nonnegative constants 1198961and 119896

2such that

|119891(119905 0 0)| le 1198961and |119892(119905 0 0)| le 119896

2 for any 119905 isin [0 1]

(H3) There exists 119903

0gt 0 satisfying the inequalities

(1205931

(119903) + 1198961) sdot (1205932

(119903) + 1198962) le 119903Γ (120572 + 1)

1205932

(119903) + 1198962

le Γ (120572 + 1)

(36)

Theorem 15 Under assumptions (1198671)ndash(1198673) problem (4) has

at least one solution in 119862[0 1] times 119862[0 1]

Proof In virtue of Lemma 14 a solution (119909 119910) isin 119862[0 1] times

119862[0 1] of problem (4) satisfies

119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(37)

We consider the space 119862[0 1] times 119862[0 1] equipped with thenorm (119909 119910)

119862[01]times119862[01]= max119909 119910 for any (119909 119910) isin

119862[0 1] times 119862[0 1]In 119862[0 1] times 119862[0 1] we define the operator

119866 (119909 119910) (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(38)

LetF andG be the operators given by

F (119909 119910) (119905) = 119891 (119905 119909 (119905) 119910 (119905))

G (119909 119910) (119905) =1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

(39)

for any (119909 119910) isin 119862[0 1] times 119862[0 1] and any 119905 isin [0 1] Then

119866 (119909 119910) = F (119909 119910) sdot G (119909 119910) (40)

Firstly we will prove that 119866 applies 119862[0 1] times 119862[0 1] into119862[0 1] To do this it is sufficient to prove that F(119909 119910)G(119909 119910) isin 119862[0 1] for any (119909 119910) isin 119862[0 1] times 119862[0 1] since theproduct of continuous functions is continuous

In virtue of assumption (H1) it is clear that F(119909 119910) isin

119862[0 1] for (119909 119910) isin 119862[0 1] times 119862[0 1] In order to prove thatG(119909 119910) isin 119862[0 1] for (119909 119910) isin 119862[0 1] times 119862[0 1] we fix 119905

0isin

[0 1] and we consider a sequence (119905119899) isin [0 1] such that 119905

119899rarr

1199050 and we have to prove that G(119909 119910)(119905

119899) rarr G(119909 119910)(119905

0)

Without loss of generality we can suppose that 119905119899

gt 1199050 Then

we have

1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899

minus 119904)1minus120572

119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904

+1

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))1003816100381610038161003816 119889119904

(41)

By assumption (H1) since 119892 isin 119862([0 1] times R times RR) 119892 is

bounded on the compact set [0 1]times[minus119909 119909]times[minus119910 119910]Denote by

119872 = sup 1003816100381610038161003816119892 (119904 119909

1 1199101)1003816100381610038161003816 119904 isin [0 1] 119909

1isin [minus 119909 119909]

1199101

isin [minus1003817100381710038171003817119910

1003817100381710038171003817 1003817100381710038171003817119910

1003817100381710038171003817]

(42)

From the last estimate we obtain

1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+119872

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

(43)


As 0 lt 120572 lt 1 and 119905119899

gt 1199050 we infer that

1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)[int

1199050

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+ int

119905119899

1199050

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

=119872

Γ (120572)[ int

1199050

0

[(1199050


minus (119905119899


] 119889119904

+ int

119905119899

1199050

119889119904

(119905119899


+ int

119905119899

1199050

119889119904

(119904 minus 1199050)1minus120572

]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

le119872

Γ (120572 + 1)[(119905119899

minus 1199050)120572

+ 119905120572

0minus 119905120572

119899

+ (119905119899

minus 1199050)120572

+ (119905119899

minus 1199050)120572

]

+119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

=4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

+119872

Γ (120572 + 1)(119905120572

0minus 119905120572

119899)

lt4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

(44)

where the last inequality has been obtained by using the factthat 119905120572

0minus 119905120572

119899lt 0

Therefore since 119905119899

rarr 1199050 from the last estimate we

deduce that G(119909 119910)(119905119899) rarr G(119909 119910)(119905

0) This proves that

G(119909 119910) isin 119862[0 1] Consequently G 119862[0 1] times 119862[0 1] rarr

119862[0 1] On the other hand for (119909 119910) isin 119862[0 1] times 119862[0 1] and119905 isin 119862[0 1] we have

1003816100381610038161003816119866 (119909 119910) (119905)1003816100381610038161003816

=1003816100381610038161003816F (119909 119910) (119905) sdot G (119909 119910) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905))

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le [1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 0 0)

1003816100381610038161003816 +1003816100381610038161003816119891 (119905 0 0)

1003816100381610038161003816]

times [1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le1

Γ (120572)[1205931

(max (|119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816)) + 1198961]

times [int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

1003816100381610038161003816119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

times [int

119905

0

1205932

(max (|119909 (119904)| 1003816100381610038161003816119910 (119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

+ 1198962

int

119905

0

119889119904

(119905 minus 119904)1minus120572

]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

sdot [1205932

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198962] int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)(1205931

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198961)

sdot (1205932

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198962)

(45)

Now taking into account assumption (H3) we infer that the

operator 119866 applies 1198611199030

times 1198611199030

into 1198611199030

Moreover from the lastestimates it follows that

10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le 1205931

(1199030) + 1198961

10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le1205932

(1199030) + 1198962

Γ (120572 + 1)

(46)

Next we will prove that the operators F and G arecontinuous on the ball 119861

1199030

times 1198611199030

and consequently 119866 will bealso continuous

In fact we fix 120576 gt 0 and we take (1199090 1199100) (119909 119910) isin 119861

1199030

times 1198611199030

with (119909 119910)minus(1199090 1199100) = (119909minus119909

0 119910minus1199100) =max119909minus119909

0 119910minus

1199100 le 120576 Then for 119905 isin [0 1] we have

1003816100381610038161003816F (119909 119910) (119905) minus F (1199090 1199100) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 119909

0(119905) 1199100

(119905))1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (119905) minus 119909

0(119905)

1003816100381610038161003816 1003816100381610038161003816119910 (119905) 119910

0(119905)

1003816100381610038161003816))

le 1205931

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817))

le 1205931

(120576) lt 120576

(47)

where we have used Remark 8 This proves the continuity ofF on 119861

1199030

times 1198611199030


In order to prove the continuity ofG on 1198611199030

times1198611199030

we have1003816100381610038161003816G (119909 119910) (119905) minus G (119909

0 1199100) (119905)

1003816100381610038161003816

=1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

minus int

119905

0

119892 (119904 1199090

(119904) 1199100

(119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 1199090

(119904) 1199100

(119904))1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)int

119905

0

1205932

(max (1003816100381610038161003816119909 (119904) minus 119909

0(119904)

1003816100381610038161003816 1003816100381610038161003816119910 (119904) minus 119910

0(119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)1205932

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817)) int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)1205932

(120576)

lt120576

Γ (120572 + 1)

(48)

Therefore1003817100381710038171003817G (119909 119910) minus G (119909

0 1199100)1003817100381710038171003817 lt

120576

Γ (120572 + 1)(49)

and consequentlyG is a continuous operator on 1198611199030

times 1198611199030

In order to prove that G satisfies assumptions of

Theorem 13 only we have to check the condition

120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (50)

for any subsets 1198831and 119883

2of 1198611199030

To do this we fix 120576 gt 0 119905

1 1199052

isin [0 1] with |1199051minus 1199052| le 120576 and

(119909 119910) isin 1198831

times 1198832 then we have

1003816100381610038161003816F (119909 119910) (1199051) minus F (119909 119910) (119905

2)1003816100381610038161003816

=1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

1 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

+1003816100381610038161003816119891 (1199051 119909 (1199052) 119910 (119905

2)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (1199051) minus 119909 (119905

2)1003816100381610038161003816

1003816100381610038161003816119910 (1199051) minus 119910 (119905

2)1003816100381610038161003816))

+ 120596 (119891 120576)

le 1205931

(max (120596 (119909 120576) 120596 (119910 120576))) + 120596 (119891 120576)

(51)

where 120596(119891 120576) denotes the quantity

120596 (119891 120576) = sup 1003816100381610038161003816119891 (119905 119909 119910) minus 119891 (119904 119909 119910)

1003816100381610038161003816 119905 119904 isin [0 1]

|119905 minus 119904| le 120576 119909 119910 isin [minus1199030 1199030]

(52)

From the last estimate we infer that120596 (F (119883

1times 1198832) 120576)

le 1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576))) + 120596 (119891 120576)

(53)

Since 119891(119905 119909 119910) is uniformly continuous on bounded subsetsof [0 1] times R times R we deduce that 120596(119891 120576) rarr 0 as 120576 rarr 0 andtherefore

1205960

(F (1198831

times 1198832)) le lim120576rarr0

1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576)))

(54)

By assumption (H2) since 120593

1is continuous we infer

1205960

(F (1198831

times 1198832)) le 120593

1(max (120596

0(1198831) 1205960

(1198832))) (55)

Now we estimate the quantity 1205960(G(1198831

times 1198832))

Fix 120576 gt 0 1199051 1199052

isin [0 1] with |1199051

minus 1199052| le 120576 and (119909 119910) isin 119883

1times

1198832 Without loss of generality we can suppose that 119905

1lt 1199052

then we have1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

0

119892 (119904 119909 (119904) 119910 (119904))

(1199052


119889119904

minus int

1199051

0

119892 (119904 119909 (119904) 119910 (119904))

(1199051


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)[int

1199051

0

10038161003816100381610038161003816(1199052


minus (1199051

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

le1

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


]

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

(56)

Since 119892 isin 119862([0 1] times R times R timesR) is bounded on the compactsubsets of [0 1] times R times R particularly on [0 1] times [minus119903

0 1199030] times

[minus1199030 1199030] Put 119871 = sup|119892(119905 119909 119910)| 119905 isin [0 1] 119909 119910 isin [minus119903

0 1199030]

Then from the last inequality we infer that1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

le119871

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


] 119889119904

+ int

1199052

1199051

(1199052


119889119904]

le119871

Γ (120572 + 1)[(1199052

minus 1199051)120572

+ 119905120572

1minus 119905120572

2+ (1199052

minus 1199051)120572

]

le2119871

Γ (120572 + 1)(1199052

minus 1199051)120572

le2119871

Γ (120572 + 1)120576120572

(57)


where we have used the fact that 119905120572

1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)

From this it follows that

1205960

(G (1198831

times 1198832)) = 0 (59)

Next by Proposition 7 (46) (55) and (59) we have

1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962

le Γ(120572 + 1) and since itis easily proved that if 120572 isin [0 1] and 120593 isin A then 120572120593 isin A wededuce that

1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)

where 120593 isin AFinally by Theorem 13 the operator G has at least a

coupled fixed point and this is the desired result Thiscompletes the proof

The nonoscillary character of the solutions of problem(4) seems to be an interesting question from the practicalpoint of view This means that the solutions of problem (4)have a constant sign on the interval (0 1) In connection withthis question we notice that if 119891(119905 119909 119910) and 119892(119905 119909 119910) haveconstant sign and are equal (this means that 119891(119905 119909 119910) gt 0

and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0

for any 119905 isin [0 1] and 119909 119910 isin R) and under assumptionsof Theorem 15 then the solution (119909 119910) isin 119862[0 1] times 119862[0 1]

of problem (4) given by Theorem 15 satisfies 119909(119905) ge 0 and119910(119905) ge 0 for 119905 isin [0 1] since the solution (119909 119910) satisfies thesystem of integral equations

119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)

On the other hand if we perturb the data function in problem(4) of the following manner

119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)

where 0 lt 120572 lt 1 119891 isin 119862([0 1] times R times RR 0)119892 isin 119862([0 1] times R times RR) and 120578 isin 119862[0 1] then underassumptions of Theorem 15 problem (63) can be studiedby using Theorem 15 where assumptions (H

1) and (H

2) are

automatically satisfied and we only have to check assumption(H3) This fact gives a great applicability to Theorem 15Before presenting an example illustrating our results we

need some facts about the functions involving this exampleThe following lemma appears in [18]

Lemma 16 Let 120593 R+

rarr R+be a nondecreasing and upper

semicontinuous function Then the following conditions areequivalent

(i) lim119899rarrinfin

120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0

Particularly the functions 1205721 1205722

R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572

2(119905) = 119905(1 + 119905) belong to the class

A since they are nondecreasing and continuous and as itis easily seen they satisfy (ii) of Lemma 16

On the other hand since the function 1205721(119905) = arctan 119905 is

concave (because 12057210158401015840

(119905) le 0) and 1205721(0) = 0 we infer that 120572

1

is subadditive and therefore for any 119905 1199051015840

isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816

=10038161003816100381610038161003816arctan 119905 minus arctan 119905

101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)

Moreover it is easily seen that max(1205721 1205722) is a nondecreasing

and continuous function because 1205721and 120572

2are nondecreas-

ing and continuous andmax(1205721 1205722) satisfies (ii) of Lemma 16

Therefore max(1205721 1205722) isin A

Nowwe are ready to present an examplewhere our resultscan be applied

Example 17 Consider the following coupled system of frac-tional hybrid differential equations

11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)

Notice that problem (17) is a particular case of problem (4)where 120572 = 12 119891(119905 119909 119910) = 14 + (110) arctan |119909| +

(120)(|119910|(1 + |119910|)) and 119892(119905 119909 119910) = 17 + (19)119909 + (110)119910It is clear that 119891 isin 119862([0 1] times R times RR 0) and 119892 isin

119862([0 1] times R times RR) and moreover 1198961

= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962

= sup|119892(119905 0 0)| 119905 isin [0 1] = 17Therefore assumption (H

1) of Theorem 15 is satisfied

Moreover for 119905 isin [0 1] and 1199091 1199092 1199101 1199102

isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)

and 1205932(119905) = (29)119905 It is clear that 120593

2isin A Therefore

assumption (H2) of Theorem 15 is satisfied

In our case the inequality appearing in assumption (H3)

of Theorem 15 has the expression

[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)

It is easily seen that 1199030

= 1 satisfies the last inequalityMoreover

2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)

Finally Theorem 15 says that problem (17) has at least onesolution (119909 119910) isin 119862[0 1] such that max(119909 119910) le 1

Conflict of Interests

The authors declare that there is no conflict of interests in thesubmitted paper

References

[1] W Deng ldquoNumerical algorithm for the time fractional Fokker-Planck equationrdquo Journal of Computational Physics vol 227 no2 pp 1510ndash1522 2007

[2] S Z Rida H M El-Sherbiny and A A M Arafa ldquoOn thesolution of the fractional nonlinear Schrodinger equationrdquoPhysics Letters A vol 372 no 5 pp 553ndash558 2008

[3] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modeling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII pp 217ndash224 Springer Berlin Germany 1999

[4] F Mainardi ldquoFractional calculus some basic proble ms incontinuum and statistical mechanicsrdquo in Fractals and FractionalCalculus in Continuum Mechanics (Udine 1996) vol 378 ofCISM Courses and Lectures pp 291ndash348 Springer ViennaAustria 1997

[5] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[6] B C Dhage and V Lakshmikantham ldquoBasic results on hybriddifferential equationsrdquo Nonlinear Analysis Hybrid Systems vol4 no 3 pp 414ndash424 2010


[7] Y Zhao S Sun Z Han and Q Li ldquoTheory of fractionalhybrid differential equationsrdquo Computers amp Mathematics withApplications vol 62 no 3 pp 1312ndash1324 2011

[8] J Caballero M A Darwish and K Sadarangani ldquoSolvabilityof a fractional hybrid initial value problem with supremum byusingmeasures of noncompactness in Banach algebrasrdquoAppliedMathematics and Computation vol 224 pp 553ndash563 2013

[9] C Bai and J Fang ldquoThe existence of a positive solution fora singular coupled system of nonlinear fractional differentialequationsrdquo Applied Mathematics and Computation vol 150 no3 pp 611ndash621 2004

[10] Y Chen and H An ldquoNumerical solutions of coupled Burgersequations with time- and space-fractional derivativesrdquo AppliedMathematics and Computation vol 200 no 1 pp 87ndash95 2008

[11] M P Lazarevic ldquoFinite time stability analysis of PD120572 fractionalcontrol of robotic time-delay systemsrdquo Mechanics ResearchCommunications vol 33 no 2 pp 269ndash279 2006

[12] V Gafiychuk B Datsko V Meleshko and D Blackmore ldquoAnal-ysis of the solutions of coupled nonlinear fractional reaction-diffusion equationsrdquo Chaos Solitons and Fractals vol 41 no 3pp 1095ndash1104 2009

[13] B Ahmad and J J Nieto ldquoExistence results for a coupledsystem of nonlinear fractional differential equations with three-point boundary conditionsrdquo Computers amp Mathematics withApplications vol 58 no 9 pp 1838ndash1843 2009

[14] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[15] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces Lecture Notes in Pure andAppliedMathematicsMarcelDekker New York NY USA 1980

[16] B N Sadovskii ldquoOn a fixed point principlerdquo Functional Analysisand Its Applications vol 1 pp 151ndash153 1967

[17] J Banas and L Olszowy ldquoOn a class of measures of noncom-pactness in Banach algebras and their application to nonlinearintegral equationsrdquo Journal of Analysis and its Applications vol28 no 4 pp 475ndash498 2009

[18] A Aghajani J Banas and N Sabzali ldquoSome generalizationsof Darbo fixed point theorem and applicationsrdquo Bulletin of theBelgian Mathematical Society vol 20 no 2 pp 345ndash358 2013

[19] R R Akhmerov M I Kamenskii A S Potapov A E Rod-kina and B N Sadovskii Measures of Noncompactness andCondensing Operators vol 55 ofOperatorTheory Advances andApplications Birkhaauser Basel Switzerland 1992

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


The coupled systems involving fractional differentialequations are very important because they occur in numerousproblems of applied nature for instance see [9ndash13]

In this paper we consider the following coupled system

119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905))

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905))

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(4)

where 120572 isin (0 1) and 119863120572

0+ is the standard Riemann-Liouville

fractional derivativeThe main tool in our study is a fixed point theorem of

Darbo type associated to measures of noncompactness

2 Preliminaries

We begin this section with some definitions and results aboutfractional calculus

Let 120572 gt 0 and 119899 = [120572] + 1 where [120572] denotes the integerpart of 120572 For a function 119891 (0 infin) rarr R the Riemann-Liouville fractional integral of order 120572 gt 0 of 119891 is defined as

119868120572

0+119891 (119905) =

1

Γ (120572)int

119905

0

119891 (119904)

(119905 minus 119904)1minus120572

119889119904 (5)

provided that the right side is pointwise defined on (0 infin)The Riemann-Liouville fractional derivative of order 120572 of

a continuous function 119891 is defined by

119863120572

0+119891 (119909) =

1

Γ (119899 minus 120572)

119889119899

119889119909119899int

119909

0

119891 (119904)

(119909 minus 119904)120572minus119899+1

119889119904 (6)

provided that the right side is pointwise defined on (0 infin)The following lemma will be useful for our study [14]

Lemma 1 Let ℎ isin 1198711

(0 1) and 0 lt 120572 lt 1 Then

(a)

119863120572

0+119868120572

0+ℎ (119909) = ℎ (119909) (7)

(b)

119868120572

0+119863120572

0+ℎ (119909) = ℎ (119909) minus

1198681minus120572

0+ ℎ (119909)

10038161003816100381610038161003816119909=0

Γ (120572)119909120572minus1

119886119890 on (0 1)

(8)

Lemma 2 Let 0 lt 120572 lt 1 and suppose that 119891 isin 119862([0 1]R

0) and119910 isin 119862[0 1]Then the unique solution of the fractionalhybrid initial value problem

119863120572

0+ [

119909 (119905)

119891 (119905)] = 119910 (119905) 0 lt 119905 lt 1

119909 (0) = 0

(9)

is given by

119909 (119905) =119891 (119905)

Γ (120572)int

119905

0

119910 (119904)

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1] (10)

Proof Suppose that 119909(119905) is a solution of problem (9) Usingthe operator 119868

120572

0+ and taking into account Lemma 1 we get

119868120572

0+119863120572

0+ [

119909 (119905)

119891 (119905)] = 119868120572

0+119910 (119905) (11)

or equivalently

119909 (119905)

119891 (119905)minus

1198681minus120572

0+ (119909 (119905) 119891 (119905))

10038161003816100381610038161003816119905=0

Γ (120572)119905120572minus1

= 119868120572

0+119910 (119905) (12)

Since 119909(119905)119891(119905)|119905=0

= 119909(0)119891(0) = 0119891(0) = 0 (because119891(0) = 0) we have

119909 (119905) = 119891 (119905) 119868120572

0+119910 (119905) (13)

This means that

119909 (119905) =119891 (119905)

Γ (120572)int

119905

0

119910 (119904)

(119905 minus 119904)1minus120572

119889119904 (14)

Conversely suppose that 119909(119905) is given by

119909 (119905) =119891 (119905)

Γ (120572)int

119905

0

119910 (119904)

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1] (15)

This means that

119909 (119905) = 119891 (119905) 119868120572

0+119910 (119905) 119905 isin [0 1] (16)

Applying 119863120572

0+ and taking into account Lemma 1 and that

119891(119905) = 0 for 119905 isin [0 1] we obtain

119863120572

0+ [

119909 (119905)

119891 (119905)] = 119863

120572

0+119868120572

0+119910 (119905) = 119910 (119905) 0 lt 119905 lt 1 (17)

Moreover for 119905 = 0 in (16) we have 119909(0) = 119891(0) sdot 0 = 0 Thiscompletes the proof

In the sequel we recall some definitions and basic factsabout measures of noncompactness

Assume that 119864 is a real Banach space with norm sdot

and zero element 120579 By 119861(119909 119903) we denote the closed ball in 119864

centered at 119909with radius 119903 By 119861119903we denote the ball 119861(120579 119903) If

119883 is a nonempty subset of119864 by the symbols119883 andConv119883wedenote the closure and the convex closure of 119883 respectivelyBy 119883 we denote the quantity 119883 = sup119909 119909 isin 119883Finally by M

119864we will denote the family of all nonempty

and bounded subsets of 119864 and byN119864we denote its subfamily

consisting of all relatively compact subsets of 119864

Definition 3 Amapping120583 M119864

rarr R+

= [0 infin) is said to bea measure of noncompactness in 119864 if it satisfies the followingconditions

(a) The family ker 120583 = 119883 isin M119864

120583(119883) = 0 is nonemptyand ker 120583 sub N

119864




119864such

that 119883119899+1

sub 119883119899


120583(119883119899) = 0 then

119883infin

= capinfin

119899=1119883119899

= 120601


infin


) le

120583(119883119899) for 119899 = 1 2 we infer that 120583(119883


that 119883infin




120583 (119879 (119883)) le 119896120583 (119883) (18)




120583 (119879 (119883)) lt 120583 (119883) (19)




120583 (119883119884) le 119883 120583 (119884) + 119884 120583 (119883) (20)

for any 119883 119884 isin M119864



119909 = sup |119909 (119905)| 119905 isin [0 1] (21)



119862[01]and



Put

120596 (119883 120576) = sup 120596 (119909 120576) 119909 isin 119883

1205960

(119883) = lim120576rarr0

120596 (119883 120576) (23)


ness in 119862[0 1]



Proof Fix 119883 119884 isin M119862[01]



1003816100381610038161003816 le1003816100381610038161003816119909 (119905) 119910 (119905) minus 119909 (119905) 119910 (119904)

1003816100381610038161003816

+1003816100381610038161003816119909 (119905) 119910 (119904) minus 119909 (119904) 119910 (119904)

1003816100381610038161003816

= |119909 (119905)|1003816100381610038161003816119910 (119905) minus 119910 (119904)

1003816100381610038161003816

+1003816100381610038161003816119910 (119904)

1003816100381610038161003816 |119909 (119905) minus 119909 (119904)|

le 119909 120596 (119910 120576) +1003817100381710038171003817119910

1003817100381710038171003817 120596 (119909 120576)

(24)

This means that

120596 (119909119910 120576) le 119909 120596 (119910 120576) +1003817100381710038171003817119910

1003817100381710038171003817 120596 (119909 120576) (25)

and therefore

120596 (119883119884 120576) le 119883 120596 (119884 120576) + 119884 120596 (119883 120576) (26)


1205960

(119883119884) le 119883 1205960

(119884) + 119884 1205960

(119883) (27)



3 Main Results


A = 120593 R+

997888rarr R+



120593119899

(119905) = 0 for any 119905 gt 0

(28)




0gt 0 and 119905

0le 120593(119905

0)


0 lt 1199050

le 120593 (1199050) le 1205932


119899




120593119899




= 0



120583 (119879 (119883)) le 120593 (120583 (119883)) (30)




Theorem 10 Let 1205831 1205832 120583




that 119865 [0 infin)119899


119899) = 0 if and only if 119909

119894= 0 for 119894 = 1 2 119899

Then

120583 (119883) = 119865 (1205831

(1198831) 1205832

(1198832) 120583

119899(119883119899)) (31)


times 1198642


119894




0 if and only if 119909 = 119910 = 0 120583(119883) = max120583(1198831) 120583(119883

2) defines






120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (32)





2)






120583 (119866 (119883)) = 120583 (119866 (1198831

times 1198832))

= 120583 (119866 (1198831

times 1198832) times 119866 (119883

2times 1198831))

= max 120583 (119866 (1198831

times 1198832)) 120583 (119866 (119883

2times 1198831))

le max 120593 (max (120583 (1198831) 120583 (119883

2)))

120593 (max (120583 (1198832) 120583 (119883

1)))

= 120593 (max (120583 (1198831) 120583 (119883

2)))

= 120593 (120583 (1198831

times 1198832))

(33)



that 119866(1199090 1199100) = (119909

0 1199100) or equivalently 119866(119909

0 1199100) = 1199090and


point


119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1]

(34)






1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le 1205931

(max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

1003816100381610038161003816119892 (119905 1199091 1199102) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le 1205932

(max (10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 10038161003816100381610038161199092 minus 119910

2

1003816100381610038161003816))

(35)



isin Rwhere 120593

1 1205932




2such that

|119891(119905 0 0)| le 1198961and |119892(119905 0 0)| le 119896

2 for any 119905 isin [0 1]



(1205931

(119903) + 1198961) sdot (1205932

(119903) + 1198962) le 119903Γ (120572 + 1)

1205932

(119903) + 1198962

le Γ (120572 + 1)

(36)





119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(37)




119866 (119909 119910) (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(38)


F (119909 119910) (119905) = 119891 (119905 119909 (119905) 119910 (119905))

G (119909 119910) (119905) =1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

(39)


119866 (119909 119910) = F (119909 119910) sdot G (119909 119910) (40)




0isin


119899rarr


119899) rarr G(119909 119910)(119905

0)


gt 1199050 Then

we have

1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904

+1

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))1003816100381610038161003816 119889119904

(41)



119872 = sup 1003816100381610038161003816119892 (119904 119909

1 1199101)1003816100381610038161003816 119904 isin [0 1] 119909

1isin [minus 119909 119909]

1199101

isin [minus1003817100381710038171003817119910

1003817100381710038171003817 1003817100381710038171003817119910

1003817100381710038171003817]

(42)


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+119872

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

(43)


As 0 lt 120572 lt 1 and 119905119899


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)[int

1199050

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+ int

119905119899

1199050

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

=119872

Γ (120572)[ int

1199050

0

[(1199050


minus (119905119899


] 119889119904

+ int

119905119899

1199050

119889119904

(119905119899


+ int

119905119899

1199050

119889119904

(119904 minus 1199050)1minus120572

]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

le119872

Γ (120572 + 1)[(119905119899

minus 1199050)120572

+ 119905120572

0minus 119905120572

119899

+ (119905119899

minus 1199050)120572

+ (119905119899

minus 1199050)120572

]

+119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

=4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

+119872

Γ (120572 + 1)(119905120572

0minus 119905120572

119899)

lt4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

(44)


0minus 119905120572

119899lt 0




0) This proves that



1003816100381610038161003816119866 (119909 119910) (119905)1003816100381610038161003816

=1003816100381610038161003816F (119909 119910) (119905) sdot G (119909 119910) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905))

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le [1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 0 0)

1003816100381610038161003816 +1003816100381610038161003816119891 (119905 0 0)

1003816100381610038161003816]

times [1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le1

Γ (120572)[1205931

(max (|119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816)) + 1198961]

times [int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

1003816100381610038161003816119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

times [int

119905

0

1205932

(max (|119909 (119904)| 1003816100381610038161003816119910 (119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

+ 1198962

int

119905

0

119889119904

(119905 minus 119904)1minus120572

]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

sdot [1205932

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198962] int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)(1205931

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198961)

sdot (1205932

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198962)

(45)



times 1198611199030

into 1198611199030


10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le 1205931

(1199030) + 1198961

10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le1205932

(1199030) + 1198962

Γ (120572 + 1)

(46)


1199030

times 1198611199030



1199030

times 1198611199030

with (119909 119910)minus(1199090 1199100) = (119909minus119909

0 119910minus1199100) =max119909minus119909

0 119910minus


1003816100381610038161003816F (119909 119910) (119905) minus F (1199090 1199100) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 119909

0(119905) 1199100

(119905))1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (119905) minus 119909

0(119905)

1003816100381610038161003816 1003816100381610038161003816119910 (119905) 119910

0(119905)

1003816100381610038161003816))

le 1205931

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817))

le 1205931

(120576) lt 120576

(47)


1199030

times 1198611199030



times1198611199030

we have1003816100381610038161003816G (119909 119910) (119905) minus G (119909

0 1199100) (119905)

1003816100381610038161003816

=1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

minus int

119905

0

119892 (119904 1199090

(119904) 1199100

(119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 1199090

(119904) 1199100

(119904))1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)int

119905

0

1205932

(max (1003816100381610038161003816119909 (119904) minus 119909

0(119904)

1003816100381610038161003816 1003816100381610038161003816119910 (119904) minus 119910

0(119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)1205932

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817)) int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)1205932

(120576)

lt120576

Γ (120572 + 1)

(48)


0 1199100)1003817100381710038171003817 lt

120576

Γ (120572 + 1)(49)


times 1198611199030



120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (50)


2of 1198611199030


1 1199052


(119909 119910) isin 1198831


1003816100381610038161003816F (119909 119910) (1199051) minus F (119909 119910) (119905

2)1003816100381610038161003816

=1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

1 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

+1003816100381610038161003816119891 (1199051 119909 (1199052) 119910 (119905

2)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (1199051) minus 119909 (119905

2)1003816100381610038161003816

1003816100381610038161003816119910 (1199051) minus 119910 (119905

2)1003816100381610038161003816))

+ 120596 (119891 120576)

le 1205931

(max (120596 (119909 120576) 120596 (119910 120576))) + 120596 (119891 120576)

(51)


120596 (119891 120576) = sup 1003816100381610038161003816119891 (119905 119909 119910) minus 119891 (119904 119909 119910)

1003816100381610038161003816 119905 119904 isin [0 1]


(52)


1times 1198832) 120576)

le 1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576))) + 120596 (119891 120576)

(53)


1205960

(F (1198831


1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576)))

(54)



1205960

(F (1198831

times 1198832)) le 120593

1(max (120596

0(1198831) 1205960

(1198832))) (55)


times 1198832))

Fix 120576 gt 0 1199051 1199052

isin [0 1] with |1199051

minus 1199052| le 120576 and (119909 119910) isin 119883

1times


1lt 1199052

then we have1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

0

119892 (119904 119909 (119904) 119910 (119904))

(1199052


119889119904

minus int

1199051

0

119892 (119904 119909 (119904) 119910 (119904))

(1199051


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)[int

1199051

0

10038161003816100381610038161003816(1199052


minus (1199051

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

le1

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


]

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

(56)


0 1199030] times


0 1199030]


2) minus G (119909 119910) (119905

1)1003816100381610038161003816

le119871

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


] 119889119904

+ int

1199052

1199051

(1199052


119889119904]

le119871

Γ (120572 + 1)[(1199052

minus 1199051)120572

+ 119905120572

1minus 119905120572

2+ (1199052

minus 1199051)120572

]

le2119871

Γ (120572 + 1)(1199052

minus 1199051)120572

le2119871

Γ (120572 + 1)120576120572

(57)



1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)


1205960

(G (1198831

times 1198832)) = 0 (59)


1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962


1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)




and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0



119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)


119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)


1) and (H

2) are







120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0


R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572




concave (because 12057210158401015840


1


isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816


101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)



2are nondecreas-





11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






119864such

that 119883119899+1

sub 119883119899


120583(119883119899) = 0 then

119883infin

= capinfin

119899=1119883119899

= 120601


infin


) le

120583(119883119899) for 119899 = 1 2 we infer that 120583(119883


that 119883infin




120583 (119879 (119883)) le 119896120583 (119883) (18)




120583 (119879 (119883)) lt 120583 (119883) (19)




120583 (119883119884) le 119883 120583 (119884) + 119884 120583 (119883) (20)

for any 119883 119884 isin M119864



119909 = sup |119909 (119905)| 119905 isin [0 1] (21)



119862[01]and



Put

120596 (119883 120576) = sup 120596 (119909 120576) 119909 isin 119883

1205960

(119883) = lim120576rarr0

120596 (119883 120576) (23)


ness in 119862[0 1]



Proof Fix 119883 119884 isin M119862[01]



1003816100381610038161003816 le1003816100381610038161003816119909 (119905) 119910 (119905) minus 119909 (119905) 119910 (119904)

1003816100381610038161003816

+1003816100381610038161003816119909 (119905) 119910 (119904) minus 119909 (119904) 119910 (119904)

1003816100381610038161003816

= |119909 (119905)|1003816100381610038161003816119910 (119905) minus 119910 (119904)

1003816100381610038161003816

+1003816100381610038161003816119910 (119904)

1003816100381610038161003816 |119909 (119905) minus 119909 (119904)|

le 119909 120596 (119910 120576) +1003817100381710038171003817119910

1003817100381710038171003817 120596 (119909 120576)

(24)

This means that

120596 (119909119910 120576) le 119909 120596 (119910 120576) +1003817100381710038171003817119910

1003817100381710038171003817 120596 (119909 120576) (25)

and therefore

120596 (119883119884 120576) le 119883 120596 (119884 120576) + 119884 120596 (119883 120576) (26)


1205960

(119883119884) le 119883 1205960

(119884) + 119884 1205960

(119883) (27)



3 Main Results


A = 120593 R+

997888rarr R+



120593119899

(119905) = 0 for any 119905 gt 0

(28)




0gt 0 and 119905

0le 120593(119905

0)


0 lt 1199050

le 120593 (1199050) le 1205932


119899




120593119899




= 0



120583 (119879 (119883)) le 120593 (120583 (119883)) (30)




Theorem 10 Let 1205831 1205832 120583




that 119865 [0 infin)119899


119899) = 0 if and only if 119909

119894= 0 for 119894 = 1 2 119899

Then

120583 (119883) = 119865 (1205831

(1198831) 1205832

(1198832) 120583

119899(119883119899)) (31)


times 1198642


119894




0 if and only if 119909 = 119910 = 0 120583(119883) = max120583(1198831) 120583(119883

2) defines






120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (32)





2)






120583 (119866 (119883)) = 120583 (119866 (1198831

times 1198832))

= 120583 (119866 (1198831

times 1198832) times 119866 (119883

2times 1198831))

= max 120583 (119866 (1198831

times 1198832)) 120583 (119866 (119883

2times 1198831))

le max 120593 (max (120583 (1198831) 120583 (119883

2)))

120593 (max (120583 (1198832) 120583 (119883

1)))

= 120593 (max (120583 (1198831) 120583 (119883

2)))

= 120593 (120583 (1198831

times 1198832))

(33)



that 119866(1199090 1199100) = (119909

0 1199100) or equivalently 119866(119909

0 1199100) = 1199090and


point


119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1]

(34)






1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le 1205931

(max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

1003816100381610038161003816119892 (119905 1199091 1199102) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le 1205932

(max (10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 10038161003816100381610038161199092 minus 119910

2

1003816100381610038161003816))

(35)



isin Rwhere 120593

1 1205932




2such that

|119891(119905 0 0)| le 1198961and |119892(119905 0 0)| le 119896

2 for any 119905 isin [0 1]



(1205931

(119903) + 1198961) sdot (1205932

(119903) + 1198962) le 119903Γ (120572 + 1)

1205932

(119903) + 1198962

le Γ (120572 + 1)

(36)





119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(37)




119866 (119909 119910) (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(38)


F (119909 119910) (119905) = 119891 (119905 119909 (119905) 119910 (119905))

G (119909 119910) (119905) =1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

(39)


119866 (119909 119910) = F (119909 119910) sdot G (119909 119910) (40)




0isin


119899rarr


119899) rarr G(119909 119910)(119905

0)


gt 1199050 Then

we have

1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904

+1

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))1003816100381610038161003816 119889119904

(41)



119872 = sup 1003816100381610038161003816119892 (119904 119909

1 1199101)1003816100381610038161003816 119904 isin [0 1] 119909

1isin [minus 119909 119909]

1199101

isin [minus1003817100381710038171003817119910

1003817100381710038171003817 1003817100381710038171003817119910

1003817100381710038171003817]

(42)


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+119872

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

(43)


As 0 lt 120572 lt 1 and 119905119899


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)[int

1199050

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+ int

119905119899

1199050

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

=119872

Γ (120572)[ int

1199050

0

[(1199050


minus (119905119899


] 119889119904

+ int

119905119899

1199050

119889119904

(119905119899


+ int

119905119899

1199050

119889119904

(119904 minus 1199050)1minus120572

]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

le119872

Γ (120572 + 1)[(119905119899

minus 1199050)120572

+ 119905120572

0minus 119905120572

119899

+ (119905119899

minus 1199050)120572

+ (119905119899

minus 1199050)120572

]

+119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

=4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

+119872

Γ (120572 + 1)(119905120572

0minus 119905120572

119899)

lt4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

(44)


0minus 119905120572

119899lt 0




0) This proves that



1003816100381610038161003816119866 (119909 119910) (119905)1003816100381610038161003816

=1003816100381610038161003816F (119909 119910) (119905) sdot G (119909 119910) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905))

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le [1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 0 0)

1003816100381610038161003816 +1003816100381610038161003816119891 (119905 0 0)

1003816100381610038161003816]

times [1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le1

Γ (120572)[1205931

(max (|119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816)) + 1198961]

times [int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

1003816100381610038161003816119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

times [int

119905

0

1205932

(max (|119909 (119904)| 1003816100381610038161003816119910 (119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

+ 1198962

int

119905

0

119889119904

(119905 minus 119904)1minus120572

]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

sdot [1205932

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198962] int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)(1205931

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198961)

sdot (1205932

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198962)

(45)



times 1198611199030

into 1198611199030


10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le 1205931

(1199030) + 1198961

10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le1205932

(1199030) + 1198962

Γ (120572 + 1)

(46)


1199030

times 1198611199030



1199030

times 1198611199030

with (119909 119910)minus(1199090 1199100) = (119909minus119909

0 119910minus1199100) =max119909minus119909

0 119910minus


1003816100381610038161003816F (119909 119910) (119905) minus F (1199090 1199100) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 119909

0(119905) 1199100

(119905))1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (119905) minus 119909

0(119905)

1003816100381610038161003816 1003816100381610038161003816119910 (119905) 119910

0(119905)

1003816100381610038161003816))

le 1205931

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817))

le 1205931

(120576) lt 120576

(47)


1199030

times 1198611199030



times1198611199030

we have1003816100381610038161003816G (119909 119910) (119905) minus G (119909

0 1199100) (119905)

1003816100381610038161003816

=1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

minus int

119905

0

119892 (119904 1199090

(119904) 1199100

(119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 1199090

(119904) 1199100

(119904))1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)int

119905

0

1205932

(max (1003816100381610038161003816119909 (119904) minus 119909

0(119904)

1003816100381610038161003816 1003816100381610038161003816119910 (119904) minus 119910

0(119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)1205932

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817)) int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)1205932

(120576)

lt120576

Γ (120572 + 1)

(48)


0 1199100)1003817100381710038171003817 lt

120576

Γ (120572 + 1)(49)


times 1198611199030



120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (50)


2of 1198611199030


1 1199052


(119909 119910) isin 1198831


1003816100381610038161003816F (119909 119910) (1199051) minus F (119909 119910) (119905

2)1003816100381610038161003816

=1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

1 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

+1003816100381610038161003816119891 (1199051 119909 (1199052) 119910 (119905

2)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (1199051) minus 119909 (119905

2)1003816100381610038161003816

1003816100381610038161003816119910 (1199051) minus 119910 (119905

2)1003816100381610038161003816))

+ 120596 (119891 120576)

le 1205931

(max (120596 (119909 120576) 120596 (119910 120576))) + 120596 (119891 120576)

(51)


120596 (119891 120576) = sup 1003816100381610038161003816119891 (119905 119909 119910) minus 119891 (119904 119909 119910)

1003816100381610038161003816 119905 119904 isin [0 1]


(52)


1times 1198832) 120576)

le 1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576))) + 120596 (119891 120576)

(53)


1205960

(F (1198831


1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576)))

(54)



1205960

(F (1198831

times 1198832)) le 120593

1(max (120596

0(1198831) 1205960

(1198832))) (55)


times 1198832))

Fix 120576 gt 0 1199051 1199052

isin [0 1] with |1199051

minus 1199052| le 120576 and (119909 119910) isin 119883

1times


1lt 1199052

then we have1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

0

119892 (119904 119909 (119904) 119910 (119904))

(1199052


119889119904

minus int

1199051

0

119892 (119904 119909 (119904) 119910 (119904))

(1199051


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)[int

1199051

0

10038161003816100381610038161003816(1199052


minus (1199051

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

le1

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


]

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

(56)


0 1199030] times


0 1199030]


2) minus G (119909 119910) (119905

1)1003816100381610038161003816

le119871

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


] 119889119904

+ int

1199052

1199051

(1199052


119889119904]

le119871

Γ (120572 + 1)[(1199052

minus 1199051)120572

+ 119905120572

1minus 119905120572

2+ (1199052

minus 1199051)120572

]

le2119871

Γ (120572 + 1)(1199052

minus 1199051)120572

le2119871

Γ (120572 + 1)120576120572

(57)



1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)


1205960

(G (1198831

times 1198832)) = 0 (59)


1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962


1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)




and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0



119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)


119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)


1) and (H

2) are







120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0


R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572




concave (because 12057210158401015840


1


isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816


101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)



2are nondecreas-





11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





0gt 0 and 119905

0le 120593(119905

0)


0 lt 1199050

le 120593 (1199050) le 1205932


119899




120593119899




= 0



120583 (119879 (119883)) le 120593 (120583 (119883)) (30)




Theorem 10 Let 1205831 1205832 120583




that 119865 [0 infin)119899


119899) = 0 if and only if 119909

119894= 0 for 119894 = 1 2 119899

Then

120583 (119883) = 119865 (1205831

(1198831) 1205832

(1198832) 120583

119899(119883119899)) (31)


times 1198642


119894




0 if and only if 119909 = 119910 = 0 120583(119883) = max120583(1198831) 120583(119883

2) defines






120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (32)





2)






120583 (119866 (119883)) = 120583 (119866 (1198831

times 1198832))

= 120583 (119866 (1198831

times 1198832) times 119866 (119883

2times 1198831))

= max 120583 (119866 (1198831

times 1198832)) 120583 (119866 (119883

2times 1198831))

le max 120593 (max (120583 (1198831) 120583 (119883

2)))

120593 (max (120583 (1198832) 120583 (119883

1)))

= 120593 (max (120583 (1198831) 120583 (119883

2)))

= 120593 (120583 (1198831

times 1198832))

(33)



that 119866(1199090 1199100) = (119909

0 1199100) or equivalently 119866(119909

0 1199100) = 1199090and


point


119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 119905 isin [0 1]

(34)






1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le 1205931

(max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

1003816100381610038161003816119892 (119905 1199091 1199102) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le 1205932

(max (10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 10038161003816100381610038161199092 minus 119910

2

1003816100381610038161003816))

(35)



isin Rwhere 120593

1 1205932




2such that

|119891(119905 0 0)| le 1198961and |119892(119905 0 0)| le 119896

2 for any 119905 isin [0 1]



(1205931

(119903) + 1198961) sdot (1205932

(119903) + 1198962) le 119903Γ (120572 + 1)

1205932

(119903) + 1198962

le Γ (120572 + 1)

(36)





119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(37)




119866 (119909 119910) (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(38)


F (119909 119910) (119905) = 119891 (119905 119909 (119905) 119910 (119905))

G (119909 119910) (119905) =1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

(39)


119866 (119909 119910) = F (119909 119910) sdot G (119909 119910) (40)




0isin


119899rarr


119899) rarr G(119909 119910)(119905

0)


gt 1199050 Then

we have

1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904

+1

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))1003816100381610038161003816 119889119904

(41)



119872 = sup 1003816100381610038161003816119892 (119904 119909

1 1199101)1003816100381610038161003816 119904 isin [0 1] 119909

1isin [minus 119909 119909]

1199101

isin [minus1003817100381710038171003817119910

1003817100381710038171003817 1003817100381710038171003817119910

1003817100381710038171003817]

(42)


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+119872

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

(43)


As 0 lt 120572 lt 1 and 119905119899


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)[int

1199050

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+ int

119905119899

1199050

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

=119872

Γ (120572)[ int

1199050

0

[(1199050


minus (119905119899


] 119889119904

+ int

119905119899

1199050

119889119904

(119905119899


+ int

119905119899

1199050

119889119904

(119904 minus 1199050)1minus120572

]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

le119872

Γ (120572 + 1)[(119905119899

minus 1199050)120572

+ 119905120572

0minus 119905120572

119899

+ (119905119899

minus 1199050)120572

+ (119905119899

minus 1199050)120572

]

+119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

=4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

+119872

Γ (120572 + 1)(119905120572

0minus 119905120572

119899)

lt4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

(44)


0minus 119905120572

119899lt 0




0) This proves that



1003816100381610038161003816119866 (119909 119910) (119905)1003816100381610038161003816

=1003816100381610038161003816F (119909 119910) (119905) sdot G (119909 119910) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905))

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le [1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 0 0)

1003816100381610038161003816 +1003816100381610038161003816119891 (119905 0 0)

1003816100381610038161003816]

times [1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le1

Γ (120572)[1205931

(max (|119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816)) + 1198961]

times [int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

1003816100381610038161003816119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

times [int

119905

0

1205932

(max (|119909 (119904)| 1003816100381610038161003816119910 (119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

+ 1198962

int

119905

0

119889119904

(119905 minus 119904)1minus120572

]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

sdot [1205932

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198962] int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)(1205931

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198961)

sdot (1205932

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198962)

(45)



times 1198611199030

into 1198611199030


10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le 1205931

(1199030) + 1198961

10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le1205932

(1199030) + 1198962

Γ (120572 + 1)

(46)


1199030

times 1198611199030



1199030

times 1198611199030

with (119909 119910)minus(1199090 1199100) = (119909minus119909

0 119910minus1199100) =max119909minus119909

0 119910minus


1003816100381610038161003816F (119909 119910) (119905) minus F (1199090 1199100) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 119909

0(119905) 1199100

(119905))1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (119905) minus 119909

0(119905)

1003816100381610038161003816 1003816100381610038161003816119910 (119905) 119910

0(119905)

1003816100381610038161003816))

le 1205931

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817))

le 1205931

(120576) lt 120576

(47)


1199030

times 1198611199030



times1198611199030

we have1003816100381610038161003816G (119909 119910) (119905) minus G (119909

0 1199100) (119905)

1003816100381610038161003816

=1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

minus int

119905

0

119892 (119904 1199090

(119904) 1199100

(119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 1199090

(119904) 1199100

(119904))1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)int

119905

0

1205932

(max (1003816100381610038161003816119909 (119904) minus 119909

0(119904)

1003816100381610038161003816 1003816100381610038161003816119910 (119904) minus 119910

0(119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)1205932

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817)) int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)1205932

(120576)

lt120576

Γ (120572 + 1)

(48)


0 1199100)1003817100381710038171003817 lt

120576

Γ (120572 + 1)(49)


times 1198611199030



120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (50)


2of 1198611199030


1 1199052


(119909 119910) isin 1198831


1003816100381610038161003816F (119909 119910) (1199051) minus F (119909 119910) (119905

2)1003816100381610038161003816

=1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

1 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

+1003816100381610038161003816119891 (1199051 119909 (1199052) 119910 (119905

2)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (1199051) minus 119909 (119905

2)1003816100381610038161003816

1003816100381610038161003816119910 (1199051) minus 119910 (119905

2)1003816100381610038161003816))

+ 120596 (119891 120576)

le 1205931

(max (120596 (119909 120576) 120596 (119910 120576))) + 120596 (119891 120576)

(51)


120596 (119891 120576) = sup 1003816100381610038161003816119891 (119905 119909 119910) minus 119891 (119904 119909 119910)

1003816100381610038161003816 119905 119904 isin [0 1]


(52)


1times 1198832) 120576)

le 1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576))) + 120596 (119891 120576)

(53)


1205960

(F (1198831


1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576)))

(54)



1205960

(F (1198831

times 1198832)) le 120593

1(max (120596

0(1198831) 1205960

(1198832))) (55)


times 1198832))

Fix 120576 gt 0 1199051 1199052

isin [0 1] with |1199051

minus 1199052| le 120576 and (119909 119910) isin 119883

1times


1lt 1199052

then we have1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

0

119892 (119904 119909 (119904) 119910 (119904))

(1199052


119889119904

minus int

1199051

0

119892 (119904 119909 (119904) 119910 (119904))

(1199051


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)[int

1199051

0

10038161003816100381610038161003816(1199052


minus (1199051

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

le1

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


]

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

(56)


0 1199030] times


0 1199030]


2) minus G (119909 119910) (119905

1)1003816100381610038161003816

le119871

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


] 119889119904

+ int

1199052

1199051

(1199052


119889119904]

le119871

Γ (120572 + 1)[(1199052

minus 1199051)120572

+ 119905120572

1minus 119905120572

2+ (1199052

minus 1199051)120572

]

le2119871

Γ (120572 + 1)(1199052

minus 1199051)120572

le2119871

Γ (120572 + 1)120576120572

(57)



1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)


1205960

(G (1198831

times 1198832)) = 0 (59)


1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962


1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)




and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0



119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)


119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)


1) and (H

2) are







120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0


R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572




concave (because 12057210158401015840


1


isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816


101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)



2are nondecreas-





11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





isin Rwhere 120593

1 1205932




2such that

|119891(119905 0 0)| le 1198961and |119892(119905 0 0)| le 119896

2 for any 119905 isin [0 1]



(1205931

(119903) + 1198961) sdot (1205932

(119903) + 1198962) le 119903Γ (120572 + 1)

1205932

(119903) + 1198962

le Γ (120572 + 1)

(36)





119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(37)




119866 (119909 119910) (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119905 isin [0 1]

(38)


F (119909 119910) (119905) = 119891 (119905 119909 (119905) 119910 (119905))

G (119909 119910) (119905) =1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

(39)


119866 (119909 119910) = F (119909 119910) sdot G (119909 119910) (40)




0isin


119899rarr


119899) rarr G(119909 119910)(119905

0)


gt 1199050 Then

we have

1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(119905119899


119889119904

minus int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

minus int

1199050

0

119892 (119904 119909 (119904) 119910 (119904))

(1199050


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904

+1

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))1003816100381610038161003816 119889119904

(41)



119872 = sup 1003816100381610038161003816119892 (119904 119909

1 1199101)1003816100381610038161003816 119904 isin [0 1] 119909

1isin [minus 119909 119909]

1199101

isin [minus1003817100381710038171003817119910

1003817100381710038171003817 1003817100381710038171003817119910

1003817100381710038171003817]

(42)


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)int

119905119899

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+119872

Γ (120572)int

119905119899

1199050

10038161003816100381610038161003816(1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

(43)


As 0 lt 120572 lt 1 and 119905119899


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)[int

1199050

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+ int

119905119899

1199050

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

=119872

Γ (120572)[ int

1199050

0

[(1199050


minus (119905119899


] 119889119904

+ int

119905119899

1199050

119889119904

(119905119899


+ int

119905119899

1199050

119889119904

(119904 minus 1199050)1minus120572

]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

le119872

Γ (120572 + 1)[(119905119899

minus 1199050)120572

+ 119905120572

0minus 119905120572

119899

+ (119905119899

minus 1199050)120572

+ (119905119899

minus 1199050)120572

]

+119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

=4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

+119872

Γ (120572 + 1)(119905120572

0minus 119905120572

119899)

lt4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

(44)


0minus 119905120572

119899lt 0




0) This proves that



1003816100381610038161003816119866 (119909 119910) (119905)1003816100381610038161003816

=1003816100381610038161003816F (119909 119910) (119905) sdot G (119909 119910) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905))

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le [1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 0 0)

1003816100381610038161003816 +1003816100381610038161003816119891 (119905 0 0)

1003816100381610038161003816]

times [1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le1

Γ (120572)[1205931

(max (|119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816)) + 1198961]

times [int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

1003816100381610038161003816119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

times [int

119905

0

1205932

(max (|119909 (119904)| 1003816100381610038161003816119910 (119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

+ 1198962

int

119905

0

119889119904

(119905 minus 119904)1minus120572

]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

sdot [1205932

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198962] int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)(1205931

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198961)

sdot (1205932

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198962)

(45)



times 1198611199030

into 1198611199030


10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le 1205931

(1199030) + 1198961

10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le1205932

(1199030) + 1198962

Γ (120572 + 1)

(46)


1199030

times 1198611199030



1199030

times 1198611199030

with (119909 119910)minus(1199090 1199100) = (119909minus119909

0 119910minus1199100) =max119909minus119909

0 119910minus


1003816100381610038161003816F (119909 119910) (119905) minus F (1199090 1199100) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 119909

0(119905) 1199100

(119905))1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (119905) minus 119909

0(119905)

1003816100381610038161003816 1003816100381610038161003816119910 (119905) 119910

0(119905)

1003816100381610038161003816))

le 1205931

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817))

le 1205931

(120576) lt 120576

(47)


1199030

times 1198611199030



times1198611199030

we have1003816100381610038161003816G (119909 119910) (119905) minus G (119909

0 1199100) (119905)

1003816100381610038161003816

=1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

minus int

119905

0

119892 (119904 1199090

(119904) 1199100

(119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 1199090

(119904) 1199100

(119904))1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)int

119905

0

1205932

(max (1003816100381610038161003816119909 (119904) minus 119909

0(119904)

1003816100381610038161003816 1003816100381610038161003816119910 (119904) minus 119910

0(119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)1205932

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817)) int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)1205932

(120576)

lt120576

Γ (120572 + 1)

(48)


0 1199100)1003817100381710038171003817 lt

120576

Γ (120572 + 1)(49)


times 1198611199030



120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (50)


2of 1198611199030


1 1199052


(119909 119910) isin 1198831


1003816100381610038161003816F (119909 119910) (1199051) minus F (119909 119910) (119905

2)1003816100381610038161003816

=1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

1 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

+1003816100381610038161003816119891 (1199051 119909 (1199052) 119910 (119905

2)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (1199051) minus 119909 (119905

2)1003816100381610038161003816

1003816100381610038161003816119910 (1199051) minus 119910 (119905

2)1003816100381610038161003816))

+ 120596 (119891 120576)

le 1205931

(max (120596 (119909 120576) 120596 (119910 120576))) + 120596 (119891 120576)

(51)


120596 (119891 120576) = sup 1003816100381610038161003816119891 (119905 119909 119910) minus 119891 (119904 119909 119910)

1003816100381610038161003816 119905 119904 isin [0 1]


(52)


1times 1198832) 120576)

le 1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576))) + 120596 (119891 120576)

(53)


1205960

(F (1198831


1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576)))

(54)



1205960

(F (1198831

times 1198832)) le 120593

1(max (120596

0(1198831) 1205960

(1198832))) (55)


times 1198832))

Fix 120576 gt 0 1199051 1199052

isin [0 1] with |1199051

minus 1199052| le 120576 and (119909 119910) isin 119883

1times


1lt 1199052

then we have1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

0

119892 (119904 119909 (119904) 119910 (119904))

(1199052


119889119904

minus int

1199051

0

119892 (119904 119909 (119904) 119910 (119904))

(1199051


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)[int

1199051

0

10038161003816100381610038161003816(1199052


minus (1199051

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

le1

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


]

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

(56)


0 1199030] times


0 1199030]


2) minus G (119909 119910) (119905

1)1003816100381610038161003816

le119871

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


] 119889119904

+ int

1199052

1199051

(1199052


119889119904]

le119871

Γ (120572 + 1)[(1199052

minus 1199051)120572

+ 119905120572

1minus 119905120572

2+ (1199052

minus 1199051)120572

]

le2119871

Γ (120572 + 1)(1199052

minus 1199051)120572

le2119871

Γ (120572 + 1)120576120572

(57)



1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)


1205960

(G (1198831

times 1198832)) = 0 (59)


1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962


1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)




and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0



119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)


119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)


1) and (H

2) are







120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0


R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572




concave (because 12057210158401015840


1


isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816


101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)



2are nondecreas-





11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




As 0 lt 120572 lt 1 and 119905119899


1003816100381610038161003816G (119909 119910) (119905119899) minus G (119909 119910) (119905

0)1003816100381610038161003816

le119872

Γ (120572)[int

1199050

0

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904

+ int

119905119899

1199050

10038161003816100381610038161003816(119905119899


minus (1199050

minus 119904)120572minus110038161003816100381610038161003816

119889119904]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

=119872

Γ (120572)[ int

1199050

0

[(1199050


minus (119905119899


] 119889119904

+ int

119905119899

1199050

119889119904

(119905119899


+ int

119905119899

1199050

119889119904

(119904 minus 1199050)1minus120572

]

+119872

Γ (120572)int

119905119899

1199050

1

(119904 minus 1199050)1minus120572

119889119904

le119872

Γ (120572 + 1)[(119905119899

minus 1199050)120572

+ 119905120572

0minus 119905120572

119899

+ (119905119899

minus 1199050)120572

+ (119905119899

minus 1199050)120572

]

+119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

=4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

+119872

Γ (120572 + 1)(119905120572

0minus 119905120572

119899)

lt4119872

Γ (120572 + 1)(119905119899

minus 1199050)120572

(44)


0minus 119905120572

119899lt 0




0) This proves that



1003816100381610038161003816119866 (119909 119910) (119905)1003816100381610038161003816

=1003816100381610038161003816F (119909 119910) (119905) sdot G (119909 119910) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905))

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

1

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le [1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 0 0)

1003816100381610038161003816 +1003816100381610038161003816119891 (119905 0 0)

1003816100381610038161003816]

times [1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

119892 (119904 0 0)

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le1

Γ (120572)[1205931

(max (|119909 (119905)| 1003816100381610038161003816119910 (119905)

1003816100381610038161003816)) + 1198961]

times [int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

+ int

119905

0

1003816100381610038161003816119892 (119904 0 0)1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

times [int

119905

0

1205932

(max (|119909 (119904)| 1003816100381610038161003816119910 (119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

+ 1198962

int

119905

0

119889119904

(119905 minus 119904)1minus120572

]

le1

Γ (120572)[1205931

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198961]

sdot [1205932

(max (119909 1003817100381710038171003817119910

1003817100381710038171003817)) + 1198962] int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)(1205931

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198961)

sdot (1205932

(1003817100381710038171003817(119909 119910)

1003817100381710038171003817) + 1198962)

(45)



times 1198611199030

into 1198611199030


10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le 1205931

(1199030) + 1198961

10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

le1205932

(1199030) + 1198962

Γ (120572 + 1)

(46)


1199030

times 1198611199030



1199030

times 1198611199030

with (119909 119910)minus(1199090 1199100) = (119909minus119909

0 119910minus1199100) =max119909minus119909

0 119910minus


1003816100381610038161003816F (119909 119910) (119905) minus F (1199090 1199100) (119905)

1003816100381610038161003816

=1003816100381610038161003816119891 (119905 119909 (119905) 119910 (119905)) minus 119891 (119905 119909

0(119905) 1199100

(119905))1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (119905) minus 119909

0(119905)

1003816100381610038161003816 1003816100381610038161003816119910 (119905) 119910

0(119905)

1003816100381610038161003816))

le 1205931

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817))

le 1205931

(120576) lt 120576

(47)


1199030

times 1198611199030



times1198611199030

we have1003816100381610038161003816G (119909 119910) (119905) minus G (119909

0 1199100) (119905)

1003816100381610038161003816

=1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

minus int

119905

0

119892 (119904 1199090

(119904) 1199100

(119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 1199090

(119904) 1199100

(119904))1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)int

119905

0

1205932

(max (1003816100381610038161003816119909 (119904) minus 119909

0(119904)

1003816100381610038161003816 1003816100381610038161003816119910 (119904) minus 119910

0(119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)1205932

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817)) int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)1205932

(120576)

lt120576

Γ (120572 + 1)

(48)


0 1199100)1003817100381710038171003817 lt

120576

Γ (120572 + 1)(49)


times 1198611199030



120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (50)


2of 1198611199030


1 1199052


(119909 119910) isin 1198831


1003816100381610038161003816F (119909 119910) (1199051) minus F (119909 119910) (119905

2)1003816100381610038161003816

=1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

1 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

+1003816100381610038161003816119891 (1199051 119909 (1199052) 119910 (119905

2)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (1199051) minus 119909 (119905

2)1003816100381610038161003816

1003816100381610038161003816119910 (1199051) minus 119910 (119905

2)1003816100381610038161003816))

+ 120596 (119891 120576)

le 1205931

(max (120596 (119909 120576) 120596 (119910 120576))) + 120596 (119891 120576)

(51)


120596 (119891 120576) = sup 1003816100381610038161003816119891 (119905 119909 119910) minus 119891 (119904 119909 119910)

1003816100381610038161003816 119905 119904 isin [0 1]


(52)


1times 1198832) 120576)

le 1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576))) + 120596 (119891 120576)

(53)


1205960

(F (1198831


1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576)))

(54)



1205960

(F (1198831

times 1198832)) le 120593

1(max (120596

0(1198831) 1205960

(1198832))) (55)


times 1198832))

Fix 120576 gt 0 1199051 1199052

isin [0 1] with |1199051

minus 1199052| le 120576 and (119909 119910) isin 119883

1times


1lt 1199052

then we have1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

0

119892 (119904 119909 (119904) 119910 (119904))

(1199052


119889119904

minus int

1199051

0

119892 (119904 119909 (119904) 119910 (119904))

(1199051


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)[int

1199051

0

10038161003816100381610038161003816(1199052


minus (1199051

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

le1

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


]

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

(56)


0 1199030] times


0 1199030]


2) minus G (119909 119910) (119905

1)1003816100381610038161003816

le119871

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


] 119889119904

+ int

1199052

1199051

(1199052


119889119904]

le119871

Γ (120572 + 1)[(1199052

minus 1199051)120572

+ 119905120572

1minus 119905120572

2+ (1199052

minus 1199051)120572

]

le2119871

Γ (120572 + 1)(1199052

minus 1199051)120572

le2119871

Γ (120572 + 1)120576120572

(57)



1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)


1205960

(G (1198831

times 1198832)) = 0 (59)


1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962


1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)




and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0



119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)


119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)


1) and (H

2) are







120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0


R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572




concave (because 12057210158401015840


1


isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816


101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)



2are nondecreas-





11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





times1198611199030

we have1003816100381610038161003816G (119909 119910) (119905) minus G (119909

0 1199100) (119905)

1003816100381610038161003816

=1

Γ (120572)

100381610038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

minus int

119905

0

119892 (119904 1199090

(119904) 1199100

(119904))

(119905 minus 119904)1minus120572

119889119904

100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)int

119905

0

1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904)) minus 119892 (119904 1199090

(119904) 1199100

(119904))1003816100381610038161003816

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)int

119905

0

1205932

(max (1003816100381610038161003816119909 (119904) minus 119909

0(119904)

1003816100381610038161003816 1003816100381610038161003816119910 (119904) minus 119910

0(119904)

1003816100381610038161003816))

(119905 minus 119904)1minus120572

119889119904

le1

Γ (120572)1205932

(max (1003817100381710038171003817119909 minus 119909

0

1003817100381710038171003817 1003817100381710038171003817119910 minus 119910

0

1003817100381710038171003817)) int

119905

0

119889119904

(119905 minus 119904)1minus120572

le1

Γ (120572 + 1)1205932

(120576)

lt120576

Γ (120572 + 1)

(48)


0 1199100)1003817100381710038171003817 lt

120576

Γ (120572 + 1)(49)


times 1198611199030



120583 (119866 (1198831

times 1198832)) le 120593 (max (120583 (119883

1) 120583 (119883

2))) (50)


2of 1198611199030


1 1199052


(119909 119910) isin 1198831


1003816100381610038161003816F (119909 119910) (1199051) minus F (119909 119910) (119905

2)1003816100381610038161003816

=1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le1003816100381610038161003816119891 (1199051 119909 (1199051) 119910 (119905

1)) minus 119891 (119905

1 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

+1003816100381610038161003816119891 (1199051 119909 (1199052) 119910 (119905

2)) minus 119891 (119905

2 119909 (1199052) 119910 (119905

2))

1003816100381610038161003816

le 1205931

(max (1003816100381610038161003816119909 (1199051) minus 119909 (119905

2)1003816100381610038161003816

1003816100381610038161003816119910 (1199051) minus 119910 (119905

2)1003816100381610038161003816))

+ 120596 (119891 120576)

le 1205931

(max (120596 (119909 120576) 120596 (119910 120576))) + 120596 (119891 120576)

(51)


120596 (119891 120576) = sup 1003816100381610038161003816119891 (119905 119909 119910) minus 119891 (119904 119909 119910)

1003816100381610038161003816 119905 119904 isin [0 1]


(52)


1times 1198832) 120576)

le 1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576))) + 120596 (119891 120576)

(53)


1205960

(F (1198831


1205931

(max (120596 (1198831 120576) 120596 (119883

2 120576)))

(54)



1205960

(F (1198831

times 1198832)) le 120593

1(max (120596

0(1198831) 1205960

(1198832))) (55)


times 1198832))

Fix 120576 gt 0 1199051 1199052

isin [0 1] with |1199051

minus 1199052| le 120576 and (119909 119910) isin 119883

1times


1lt 1199052

then we have1003816100381610038161003816G (119909 119910) (119905

2) minus G (119909 119910) (119905

1)1003816100381610038161003816

=1

Γ (120572)

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

0

119892 (119904 119909 (119904) 119910 (119904))

(1199052


119889119904

minus int

1199051

0

119892 (119904 119909 (119904) 119910 (119904))

(1199051


119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)[int

1199051

0

10038161003816100381610038161003816(1199052


minus (1199051

minus 119904)120572minus110038161003816100381610038161003816

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

le1

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


]

times1003816100381610038161003816119892 (119904 119909 (119904) 119909 (120590119904))

1003816100381610038161003816 119889119904

+ int

1199052

1199051

(1199052

minus 119904)120572minus1 1003816100381610038161003816119892 (119904 119909 (119904) 119910 (119904))

1003816100381610038161003816 119889119904]

(56)


0 1199030] times


0 1199030]


2) minus G (119909 119910) (119905

1)1003816100381610038161003816

le119871

Γ (120572)[int

1199051

0

[(1199051


minus (1199052


] 119889119904

+ int

1199052

1199051

(1199052


119889119904]

le119871

Γ (120572 + 1)[(1199052

minus 1199051)120572

+ 119905120572

1minus 119905120572

2+ (1199052

minus 1199051)120572

]

le2119871

Γ (120572 + 1)(1199052

minus 1199051)120572

le2119871

Γ (120572 + 1)120576120572

(57)



1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)


1205960

(G (1198831

times 1198832)) = 0 (59)


1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962


1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)




and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0



119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)


119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)


1) and (H

2) are







120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0


R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572




concave (because 12057210158401015840


1


isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816


101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)



2are nondecreas-





11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





1minus 119905120572

2le 0 Therefore

120596 (G (1198831

times 1198832) 120576) le

2119871

Γ (120572 + 1)120576120572

(58)


1205960

(G (1198831

times 1198832)) = 0 (59)


1205960

(119866 (1198831

times 1198832))

= 1205960

(F (1198831

times 1198832) sdot G (119883

1times 1198832))

le1003817100381710038171003817F (119883

1times 1198832)1003817100381710038171003817 1205960

(G (1198831

times 1198832))

+1003817100381710038171003817G (119883

1times 1198832)1003817100381710038171003817 1205960

(F (1198831

times 1198832))

le10038171003817100381710038171003817F (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(G (1198831

times 1198832))

+10038171003817100381710038171003817G (1198611199030

times 1198611199030

)10038171003817100381710038171003817

1205960

(F (1198831

times 1198832))

le1205932

(1199030) + 1198962

Γ (120572 + 1)1205931

(max (1205960

(1198831) 1205960

(1198832)))

(60)


2(1199030) + 1198962


1205960

(119866 (1198831

times 1198832)) le 120593 (max (120596

0(1198831) 1205960

(1198832))) (61)




and 119892(119905 119909 119910) ge 0 or 119891(119905 119909 119910) lt 0 and 119892(119905 119909 119910) le 0



119909 (119905) =119891 (119905 119909 (119905) 119910 (119905))

Γ (120572)int

119905

0

119892 (119904 119909 (119904) 119910 (119904))

(119905 minus 119904)1minus120572

119889119904

119910 (119905) =119891 (119905 119910 (119905) 119909 (119905))

Γ (120572)int

119905

0

119892 (119904 119910 (119904) 119909 (119904))

(119905 minus 119904)1minus120572

119889119904 0 le 119905 le 1

(62)


119863120572

0+ [

119909 (119905)

119891 (119905 119909 (119905) 119910 (119905))] = 119892 (119905 119909 (119905) 119910 (119905)) + 120578 (119905)

119863120572

0+ [

119910 (119905)

119891 (119905 119910 (119905) 119909 (119905))] = 119892 (119905 119910 (119905) 119909 (119905)) + 120578 (119905)

0 lt 119905 lt 1

(63)


1) and (H

2) are







120593119899

(119905) = 0 for any 119905 ge 0

(ii) 120593(119905) lt 119905 for any 119905 gt 0


R+

rarr R+given by

1205721(119905) = arctan 119905 and 120572




concave (because 12057210158401015840


1


isin R+ we have

100381610038161003816100381610038161205721

(119905) minus 1205721

(1199051015840

)10038161003816100381610038161003816


101584010038161003816100381610038161003816

le arctan 10038161003816100381610038161003816119905 minus 119905101584010038161003816100381610038161003816

(64)



2are nondecreas-





11986312

0+ [119909 (119905) times (

1

4+ (

1

10) arctan |119909 (119905)|

+(1

20) (

1003816100381610038161003816119910 (119905)1003816100381610038161003816

(1 +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)))

minus1

]

=1

7+

1

9119909 (119905) +

1

10119910 (119905)


11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




11986312

0+ [119910 (119905) times (

1

4+ (

1

10) arctan 1003816100381610038161003816119910 (119905)

1003816100381610038161003816

+ (1

20) (

|119909 (119905)|

(1 + |119909 (119905)|)))

minus1

]

=1

7+

1

9119910 (119905) +

1

10119909 (119905)

0 lt 119905 lt 1

119909 (0) = 119910 (0) = 0

(65)




= sup|119891(119905 0 0)| 119905 isin

[0 1] = 14 and 1198962




isin R we have1003816100381610038161003816119891 (119905 119909

1 1199101) minus 119891 (119905 119909

2 1199102)1003816100381610038161003816

le1

10

1003816100381610038161003816arctan10038161003816100381610038161199091

1003816100381610038161003816 minus arctan 100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816

1 +10038161003816100381610038161199101

1003816100381610038161003816

minus

100381610038161003816100381611991021003816100381610038161003816

1 +10038161003816100381610038161199102

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan 1003816100381610038161003816

100381610038161003816100381611990911003816100381610038161003816 minus

100381610038161003816100381611990921003816100381610038161003816

1003816100381610038161003816

+1

20

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381611991011003816100381610038161003816 minus

100381610038161003816100381611991021003816100381610038161003816

(1 +10038161003816100381610038161199101

1003816100381610038161003816) (1 +10038161003816100381610038161199102

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

le1

10arctan (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

20

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816

1 +10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816

=1

101205721

(10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816)

+1

201205722

(10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)

le1

10max (120572

1 1205722) (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816)

+1

10max (120572

1 1205722) (

10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816)

le1

10[2max (120572

1 1205722)

times max (10038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816)]

=1

5max (120572

1 1205722) (max (

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 10038161003816100381610038161199101 minus 119910

2

1003816100381610038161003816))

(66)

Therefore 1205931(119905) = (15)max(120572

1(119905) 1205722(119905)) and 120593

1isin A

On the other hand1003816100381610038161003816119892 (119905 119909

1 1199101) minus 119892 (119905 119909

2 1199102)1003816100381610038161003816

le1

9

10038161003816100381610038161199091 minus 1199092

1003816100381610038161003816 +1

10

10038161003816100381610038161199092 minus 1199102

1003816100381610038161003816

le1

9(10038161003816100381610038161199091 minus 119910

1

1003816100381610038161003816 +10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

le1

9(2max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816))

=2

9max (

10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 10038161003816100381610038161199102 minus 119910

2

1003816100381610038161003816)

(67)


2isin A Therefore




[1

5max(arctan 119903

119903

1 + 119903) +

1

4] [

2

9119903 +

1

7] le 119903Γ (

3

2) (68)



2

91199030

+1

7=

2

9+

1

7le Γ (

3

2) cong 088623 (69)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



Documents

Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions