Research ArticleFixed Point Theorems for Noncyclic Monotone Relativelyρ-Nonexpansive Mappings in Modular Spaces

Nour-eddine El Harmouchi Karim Chaira and El Miloudi Marhrani

Laboratory of Algebra Analysis and Applications Hassan II University of Casablanca Faculty of Sciences Ben MrsquoSikAvenue Driss El Harti B P 7955 Sidi Othmane Casablanca Morocco

Correspondence should be addressed to El Miloudi Marhrani marhranigmailcom

Received 21 September 2019 Accepted 6 May 2020 Published 22 June 2020

Academic Editor Luca Vitagliano

Copyright copy 2020 Nour-Eddine El Harmouchi et al )is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

In this paper we obtain some fixed point results for noncyclic monotone ρ-nonexpansive mappings in uniformly convex modularspaces and uniformly convex in every direction modular spaces As an application we prove the existence of the solution of anintegral equation

1 Introduction

)e notion of modular spaces as a generalization of Banachspaces was firstly introduced by Nakano [1] in connectionwith the theory of ordered spaces )ese spaces were de-veloped and generalized by Orlicz and Musielak [2]

It is well known that fixed point theory is an active fieldof research a powerful tool in solving integral and differ-ential equations Following the publications of Ran andReuring [3] and Neito and Rrodriguez-Lopez [4] fixed pointtheory in partially ordered modular spaces has recentlyreceived a good attention from researchers

Alfuraidan et al [5] gave a modular version of Ran andReuring fixed point theorem and proved the existence offixed point of monotone contraction mappings in modularfunction spaces )ereafter they gave an extension of theirmain results for pointwise monotone contractions In [6]Gordji et al have proved that any quasi-contraction map-pings in partially ordered modular spaces withoutΔ2-condition have a fixed point

In 2016 Bin Dehaish and Khamsi proved in [7] thefollowing theorems

Theorem 1 (see [7]) Let ρ isinR be a (UUC1) and C be anonempty convex ρ-closed ρ-bounded subset of Lρ not re-duced to one point Let T C⟶ C be a monotone

ρ-nonexpansive mapping and ρ-continuous Assume thereexists f0 isin C such that f0 and Tf0 are comparable 1en T

has a fixed point

Theorem 2 (see [7]) Assume that ρ isinR is UUCED anduniformly continuous Assume that Lρ satisfies the property(R) Let C be a nonempty convex ρ-closed ρ-bounded subset ofLρ not reduced to one point Let T C⟶ C be a monotoneρ-nonexpansive mapping and ρ-continuous Assume thereexists f0 isin C such that f0 and Tf0 are comparable 1en T

has a fixed pointThese theorems are a generalization to modular function

spaces of Browder and Gohde fixed point theorem formonotone nonexpansive mappings in Banach spaces

A mapping T A cup B⟶ A cup B is said to be noncyclicprovided that T(A) sube A and T(B) sube B where (A B) is anonempty pair in a modular space When we consider thistype of mappings it is interesting to ask if it is possible tofind a pair (xlowast ylowast) isin A times B such that

Txlowast

xlowast

Tylowast

ylowast

ρ xlowast

minus ylowast

( 1113857 dρ(A B)

(1)

that is (xlowast ylowast) is a best proximity pair

HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2020 Article ID 4271728 8 pageshttpsdoiorg10115520204271728

In this paper we discuss some best proximity pair resultsfor the class of monotone noncyclic relatively ρ-non-expansive mappings in the framework of modular spacesequipped with a partial order defined by a ρ-closed convexcone P that is for any x and y in the modular space Xρ onehas x⪯y if and only if y minus x in P We took this orderingbecause in the case of modular spaces the order intervals arenot convex closed as in modular function spaces (see)eorem 24 in [7]) Our results generalize)eorems 1 and 2and others for monotone ρ-nonexpansive mappings to thecase of noncyclic monotone relatively ρ-nonexpansivemappings in modular spaces

To illustrate the effectiveness of our main results we givean application to an integral equation which involvesnoncyclic monotone ρ-nonexpansive mappings

2 Materials and Methods

)e materials used in this study are obtained from journalarticles and books existing on the Internet )e main toolsused in this manuscript are the uniform convexity and theuniform convexity in every direction of a modular ρ)e aimof this work is to extend some fixed point theorems fornonexansive mappings to best proximity pair for monotonenoncyclic relatively ρ-nonexpansive mappings in modularspaces

)e paper is organized as follows In Section 3 we beginwith recollection of some basic definitions and lemmas withcorresponding references that will be used in the sequel)ereafter we give the main results of the paper with someexamples To validate the utility of our results we give inSection 4 an application to an integral equation

3 Preliminaries

)roughout this work X stands for a linear vector space onthe field R Let us start with some preliminaries andnotations

Definition 1 (see [8]) A function ρ X⟶ [0 +infin] is calleda modular if the following holds

(1) ρ(x) 0 if and only if x 0(2) ρ(minus x) ρ(x)

(3) ρ(αx + (1 minus α)y)le ρ(x) + ρ(y) for any α isin [0 1]

and for any x and y in X

If (3) is replaced by (3prime) ρ(αx + (1 minus α)y)le αρ(x) + (1 minus

α)ρ(y) for any α isin [0 1] and x and y in X then ρ is called aconvex modular

)e modular space is defined asXρ x isin X limλ⟶0ρ(λx) 01113864 1113865 )roughout this paperwe will assume that the modular ρ is convex)e Luxemburgnorm in Xρ is defined as

xρ inf λgt 0 ρx

λ1113874 1113875le 11113882 1113883 (2)

Associated to a modular we introduce some basic no-tions needed throughout this work

Definition 2 (see [8]) Let ρ be a modular defined on a vectorspace X

(1) We say that a sequence (xn)nisinN sub Xρ is ρ-convergentto x isin Xρ if and only if limn⟶infinρ(xn minus x) 0 Notethat the limit is unique

(2) A sequence (xn)n sub Xρ is called ρ-Cauchy if ρ(xn minus

xm)⟶ 0 as n m⟶ +infin(3) We say that Xρ is ρ-complete if and only if any

ρ-Cauchy sequence is ρ-convergent(4) A subset C of Xρ is said ρ-closed if the ρ-limit of a

ρ-convergent sequence of C always belong to C(5) A subset C of Xρ is said ρ-bounded if we have

δρ(C) sup ρ(x minus y) x y isin C1113864 1113865ltinfin

(6) A subset K ofXρ is said ρ-sequentially compact if anysequence (xn)n of C has a subsequence ρ-convergentto a point x isin C

(7) We say that ρ satisfies the Fatou property if ρ(x minus

y)le limn⟶+infinρ(x minus yn) whenever (yn)n ρ-con-verges to y for any x y and yn in Xρ

Let us note that ρ-convergence does not imply ρ-Cauchycondition Also xn⟶

ρx does not imply in general

λxn⟶ρ

λx for every λgt 1

Definition 3 Let ρ be amodular and C be a nonempty subsetof the modular space Xρ A mapping T C⪯C is said to be

(a) Monotone if T(x)⪯T(y) for any x y isin C such thatx⪯y

(b) Monotone ρ-nonexpansive if T is monotone and

ρ(T(x) minus T(y))⪯ ρ(x minus y) (3)

whenever x y isin Xρ and x⪯yRecall that T C⟶ C is said to be ρ-continuous if

(T(xn))n ρ-converges to T(x) whenever (xn)n ρ-convergesto x It is not true that a monotone ρ-nonexpansive mappingis ρ-continuous since this result is not true in general when ρis a norm

Let A and B be nonempty subsets of a modular space XρWe adopt the notation dρ(A B) inf ρ(x minus y)1113864

x isin A y isin BA pair (A B) is said to satisfy a property if both A and B

satisfy that property For instance (A B) is ρ-closed (respconvex ρ-bounded) if and only if A and B are ρ-closed (respconvex ρ-bounded) A pair (A B) is not reduced to onepoint means that A and B are not singletons

Recall the definition of the modular uniform convexity

Definition 4 (see [8]) Let ρ be a modular and rgt 0 and εgt 0Define for i isin 1 2

Di(r ε) (x y) isin Xρ times Xρ ρ(x)le r ρ(y)le r ρx minus y

i1113874 1113875ge rε1113882 1113883

(4)

If Di(r ε)neempty let

2 International Journal of Mathematics and Mathematical Sciences

δi(r ε) inf 1 minus1rρ

x + y

21113874 1113875 (x y) isin Di1113882 1113883 (5)

If Di(r ε) empty we set δi(r ε) 1

(i) We say that ρ is uniformly convex (UCi) if for everyrgt 0 and εgt 0 and we have δi(r ε)gt 0

(ii) We say that ρ is unique uniformly convex (UUCi) iffor all sge 0 and εgt 0 and there exists η(s ε)gt 0 suchthat δi(r ε)gt η(s ε) for rgt s

(iii) We say that ρ is strictly convex (SC) if for every xy isin Xρ such that ρ(x) ρ(y) andρ((x + y)2) ρ(x) + ρ(y)2 and we have x y

)e following proposition characterizes the relationshipbetween the above notions

Proposition 1 (see [8])

(a) (UUCi) implies (UCi) for i 1 2(b) δ1(r ε)le δ2(r ε) for rgt 0 and εgt 0(c) (UC1) implies (UC2) implies (SC)

(d) (UUC1) implies (UUC2)

Definition 5 (see [9]) Let ρ be a modular We say that themodular space Xρ satisfies the property (R) if and only if forevery decreasing sequence (Cn)nisinN of nonempty ρ-closedconvex and ρ-bounded subsets of Xρ has a nonemptyintersection

Lemma 1 (see [8]) Let ρ be a convex modular satisfying theFatou property Assume that Xρ is ρ-complete and ρ is(UUC2) 1en Xρ satisfies the property (R)

Definition 6 (see [8]) Let (xn)n be a sequence in Xρ and K

be a nonempty subset of Xρ )e function τ K⟶ [0infin]

defined by τ(x) lim supn⟶infinρ(xn minus x) is called a ρ-typefunction

Note that the ρ-type function τ is convex since ρ isconvex A sequence (cn)n sub K is called a minimizing se-quence of τ if limn⟶+infinτ(cn) infxisinKτ(x)

)e following result found in [8] plays a crucial role inthe proof of many fixed point results in modular spaces

Lemma 2 (see [8]) Let ρ be a convex modular (UUC1)satisfying the Fatou property and Xρ a ρ-compete modularspace Let C be a nonempty ρ-closed convex subset of XρConsider the ρ-type function τ C⟶ [0 +infin] generated bya sequence (xn)n in Xρ Assume thatτ0 inf τ(x) x isin C ltinfin 1en all the minimizing se-quences of τ are ρ-convergent to the same limit

4 Main Results

A subset P sub Xρ is a pointed ρ-closed convex cone if P is anonempty ρ-closed subset of Xρ satisfying the followingproperties

(i) P + P sub P

(ii) λP sub P for all λ isin R+

(iii) Pcap (minus P) 0

Using P we define an ordering on Xρ byx⪯y if and only if y minus x isin P

Let ρ be a convex modular and T C⟶ C be amonotone mapping where C is a nonempty convex subsetof the modular space Xρ Let x0 isin C and λ isin (0 1) ConsiderKrasnoselskii-Ishikawa iteration sequence (xn)nisinN in C

defined by xn+1 (1 minus λ)xn + λT(xn) for all n isin N Assumethat x0 ⪯T(x0) (resp T(x0)⪯ x0) By the definition of thepartial ordering ⪯ we obtain x0 ⪯x1 ⪯T(x0) (respT(x0)⪯x1 ⪯ x0)

Since T is monotone one has T(x0)⪯T(x1) (respT(x1)⪯T(x0)) By induction we prove that

xn ⪯xn+1 ⪯T xn( 1113857⪯T xn+1( 1113857 respT xn+1( 1113857⪯T xn( 1113857(

⪯xn+1 ⪯xn1113857 for all n isin N(6)

Let us introduce the class of mappings for which theproblem of fixed point will be considered

Definition 7 Let (A B) be a nonempty pair of a modularspace Xρ A mapping T A cup B⟶ A cup B is said to bemonotone noncyclic relatively ρ-nonexpansive if it satisfiesthe following conditions

(i) For all (x y) isin (A cup B)2 x⪯y implies thatTx⪯Ty

(ii) T(A) sube A and T(B) sube B

(iii) For all x isin A and y isin B such that x⪯y we haveρ(Tx minus Ty)le ρ(x minus y)

Definition 8 (see [10]) Let (A B) be a pair of a modularspace Xρ and T A cup B⟶ A cup B be a noncyclic mappingA pair (x y) isin A times B is said to be a best proximity pair forthe noncyclic mapping T if

Tx x

Ty y

ρ(x minus y) dρ(A B)

(7)

Definition 9 A space Xρ is said to satisfy the property (p) ifxn⟶

ρx and yn⟶

ρy with xn ⪯yn for all n isin N then

x⪯yWe use (A⪯0 B⪯0 ) to denote the ordered ρ-proximal pair

obtained from (A B) by

A⪯0 x isin A ρ(x minus y) dρ(A B) andx⪯y for somey isin B1113966 1113967

B⪯0 y isin B ρ(x minus y) dρ(A B) and x⪯y for somex isin A1113966 1113967

(8)

International Journal of Mathematics and Mathematical Sciences 3

Note that if A⪯0 is nonempty if and only if B⪯0 is non-empty Indeed if x0 isin A⪯0 then there exists y isin B such thatx0 ⪯y and ρ(x0 minus y) dρ(A B) )us for this y we takex x0 isin A we have x⪯y and ρ(x minus y) dρ(A B) that isy isin B⪯0 As the same we prove the converse

Proposition 2 Let ρ be a convex modular satisfying theFatou property and Xρ satisfies the property (P) Let (A B) bea nonempty convex pair of Xρ Assume that A⪯0 is nonempty

(i) If (A B) is ρ-sequentially compact then (A⪯0 B⪯0 ) isρ-sequentially compact

(ii) (A⪯0 B⪯0 ) is convex(iii) dρ(A⪯0 B⪯0 ) dρ(A B)

(iv) If T C⟶ C is a noncyclic monotone relativelyρ-nonexpansive mapping then T(A ⪯0 )subeA⪯0 andT(B⪯0 )subeB⪯0 that is A⪯0 and B⪯0 are T-invariant

Proof It is quite easy to verify (iii)

(i) Let (xn)n be a sequence in A⪯0 )en there exists asequence (yn)n in B such that xn ⪯yn and ρ(xn minus

yn) dρ(A B) for all nge 0 Since (A B) is ρ-se-quentially compact there exists subsequences (xnk

)k

and (ynk)k of (xn)n and (yn)n respectively such that

xnk⪯ ynk

for all kge 0 and ρ-converge to x isin A andy isin B respectivelyFrom property (P) one has x⪯ y Since ρ satisfies theFatou property then

dρ(A B)le ρ(x minus y)le lim infk⟶infin

ρ xnkminus ynk

1113872 1113873 dρ(A B)

(9)

)us ρ(x minus y) dρ(A B) )erefore A⪯0 is ρ-se-quentially compact Likewise we prove that B⪯0 isalso ρ-sequentially compact

(ii) Let x1 x2 isin A⪯0 δ isin (0 1) and setz δx1 + (1 minus δ)x2 Since x1 x2 isin A⪯0 then thereexists y1 y2 isin B such that x1 ⪯y1 x2 ⪯y2 andρ(x1 minus y1) ρ(x2 minus y2) dρ(A B) Moreoverδx1 ⪯ δy1 and (1 minus δ)x2 ⪯ (1 minus δ)y2 Hencez δx1 + (1 minus δ)x2 ⪯ zprime δy1 + (1 minus δ)y2 Since ρis convex then

dρ(A B)le ρ z minus zprime( 1113857 ρ δx1 +(1 minus δ)x2( 1113857 minus δx1 +(1 minus δ)x2( 1113857( 1113857

le δρ x1 minus y1( 1113857 +(1 minus δ)ρ x2 minus y2( 1113857

le δdρ(A B) +(1 minus δ)dρ(A B) dρ(A B)

(10)

)en ρ(z minus zprime) dρ(A B) Hence z isin A⪯0 )erefore A⪯0 is convex Using the same argumentwe prove that B⪯0 is convex

(iii) Let x isin A⪯0 then there exists y isin B such that x⪯y

and ρ(x minus y) dρ(A B) Since T is monotone

noncyclic relatively ρ-nonexpansive then Tx⪯Ty

and dρ(A B)le ρ(Tx minus Ty)le ρ(x minus y) dρ(A B))us ρ(Tx minus Ty) dρ(A B) )erefore Tx isin A ⪯0 As the same way we obtain T(B⪯0 )subeB⪯0

Theorem 3 Let ρ be a convex modular (UUC1) satisfyingthe Fatou property and Xρ be a ρ-complete modular space Let(A B) be a nonempty convex and ρ-bounded pair of Xρ suchthat the subset B⪯0 is nonempty ρ-closed LetT AcupB⟶ A cup B be a monotone noncyclic relativelyρ-nonexpansive mapping and ρ-continuous on B Assumethat there exists x0 isin A⪯0 such that x0 ⪯Tx0 1en T has abest proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)n defined by xn+1 λTxn + (1 minus λ)xn for all nge 0and λ isin (0 1) Set Cn y isin B⪯0 xn ⪯y1113864 1113865 for any nge 0 )esequence (Cn)nisinN is a decreasing sequence of nonemptyρ-closed convex ρ-bounded subsets

Since A⪯0 is T-invariant then Tx0 isin A⪯0 Using theconvexity of A⪯0 one has x1 λTx0 + (1 minus λ)x0 isin A⪯0 Assume that xn isin A⪯0 then Txn isin A⪯0 Again the convexityof A⪯0 implies that xn+1 λTxn + (1 minus λ)xn isin A⪯0 )ere-fore xn isin A⪯0 for all nge 0 Hence for all nge 0 there existsy isin B such that xn ⪯y and ρ(xn minus y) dρ(A B) )us thereexists y isin B⪯0 such that xn ⪯y)ereforeCn is nonempty forall nge 0 Moreover since xn ⪯xn+1 then (Cn)nisinN is a de-creasing sequence

Let (cp)pge 0 be a sequence in Cn ρ-converges to b isin B⪯0 We have cp isin Cn then cp minus xn isin P and cp isin B⪯0 for all pge 0Since B⪯0 is ρ-closed one has b isin B⪯0 Moreovercp minus xn⟶

ρb minus xn )us b minus xn isin P since P is ρ-closed

Hence b isin Cn )erefore Cn is ρ-closed for all nge 0As P and B⪯0 are convex it is easy to see that Cn is convex

for all nge 0 Furthermore Cn is ρ-bounded for all nge 0By Lemma 1 Xρ satisfies the property (R) )en Cinfin

cap nge0Cn is nonempty and ρ-closed convexMoreover T(Cinfin) sube Cinfin In fact let z isin Cinfin then xn ⪯ z

for all nge 0 Since T is monotone and xn ⪯Txn one hasxn ⪯Tz for all nge 0 Moreover Tz isin B⪯0 since B⪯0 isT-invariant Hence T(Cinfin) sube Cinfin

Consider the type function τ Cinfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that is τ(z) lim supn⟶+infinρ(xn minus z) for all z isin Cinfin Let (zp)pisinN be a minimizing se-quence of τ By Lemma 2 (zp)p ρ-converges to a pointy isin Cinfin since Cinfin is ρ-closed Otherwise

τ Tzp1113872 1113873 lim supn⟶+infin

ρ xn+1 minus Tzp1113872 1113873

le (1 minus λ) lim supn⟶+infin

ρ xn minus Tzp1113872 1113873 + λ lim supn⟶+infin

ρ xn minus zp1113872 1113873

le (1 minus λ)τ Tzp1113872 1113873 + λτ zp1113872 1113873

(11)

Hence τ(Tzp)le τ(zp) )en (Tzp)pge 0 is also a mini-mizing sequence of τ By Lemma 2 (Tzp)p ρ-converges to yAs T is ρ-continuous on B then (Tzp)p ρ-converges to Ty)erefore Ty y )en there exists y isin B⪯0 such that Ty

4 International Journal of Mathematics and Mathematical Sciences

y and xn ⪯y for all n isin N By the definition of B⪯0 thereexists x isin A such that x⪯y and ρ(x minus y) dρ(A B) Inorder to finish the proof we show that x is a fixed point of T

on A We have

dρ(A B)le ρ(Tx minus y) ρ(Tx minus Ty)

le ρ(x minus y) dρ(A B)(12)

)en ρ(Tx minus y) dρ(A B) Moreover

dρ(A B)le ρTx minus y

2+

x minus y

21113874 1113875

leρ(Tx minus y)

2+ρ(x minus y)

2 dρ(A B)

(13)

)us ρ(((Tx minus y) + (x minus y))2) dρ(A B) Since ρ is(UUC2) it is (SC) Hence Tx minus y x minus y which impliesthat Tx x )erefore (x y) is a best proximity pair of themapping T

To illustrate )eorem 3 we consider the followingexample

Example 1 Let X R2 we define the modularρ X⟶ [0 +infin[ by ρ(x) |x1|

2 + |x2|2 for all

x (x1 x2) isin R2 )e modular ρ is convex satisfying theFatou property and (UUC1) and Xρ is a ρ-completemodular space Consider the ρ-closed convex coneP (x1 x2) isin Xρ (x1 x2)ge (0 0)1113966 1113967 Let A [(minus 1 minus 2)

(minus 1 2)] and B [(1 minus 4) (1 0)] )e pair (A B) is ρ-closedconvex and ρ-bounded in Xρ with dρ(A B) 4 MoreoverA0⪯ [(minus 1 minus 2) (minus 1 0)] and B⪯0 [(1 minus 2) (1 0)] are

ρ-closed Furthermore we have

(i) For all x isin [(minus 1 minus 2) (minus 1 0)] there existsy isin [(1 minus 2) (1 0)] such that x⪯ y

(ii) For the other cases the elements of A and B are notcomparable

Consider the mapping T A cup B⟶ A cup B defined by

T x1 x2( 1113857 (minus 1 0) if x1 x2( 1113857 isin [(minus 1 minus 2) (minus 1 0)]

x1 x2( 1113857 if x1 x2( 1113857 isin [(minus 1 0) (minus 1 2)]1113896

T y1 y2( 1113857 (1 0) if y1 y2( 1113857 isin B

(14)

)e mapping T is monotone noncyclic relativelyρ-nonexpansive ρ-continuous on A cup B and forx0 (minus 1 minus 2) isin Ale0 we have x0 ⪯ Tx0 )en T has a bestproximity pair ((1 0) (minus 1 0)) wheredρ(A B) ρ((1 0) minus (minus 1 0)) 4

)e following corollary is an immediate consequence of)eorem 3 and it suffices to take A B therefore A A⪯0and B B⪯0 It will be considered as a generalization ofBrowder and Gohde fixed point theorem for monotoneρ-nonexpansive mappings in modular spaces (see[7 11 12]) )is corollary result has already been mentionedby Bin Dehaish and Khamsi (see )eorem 1) [7] in theframework of modular function spaces

Corollary 4 Let ρ be a convex modular (UUC1) satisfyingthe Fatou property and Xρ be a ρ-complete modular space LetC be a nonempty ρ-closed convex ρ-bounded subset of Xρ andT C⟶ C be a monotone ρ-nonexpansive mapping andρ-continuous If there exists x0 isin C such that x0 ⪯T(x0) thenT has a fixed point

Particular case of normed vector spaces

Remark 1(i) If (X middot) is a normed vector space then the norm

middot is continuous and satisfies the Fatou propertyMoreover if (X middot) is a uniformly convex Banachspace then (X middot) is reflexive Hence it satisfies theproperty (R)

(ii) If (X middot) is reflexive and (A B) is a nonemptyclosed and bounded pair of X then (A⪯0 B⪯0 ) is alsoclosed convex pair of X

First case if (A⪯0 B⪯0 ) is empty then there is nothing toprove

Second case assume that (A⪯0 B⪯0 ) is nonempty Let(xn)n be a sequence in A⪯0 such that xn ρ-converges to x isin A)ere exists a sequence (yn)n in B such that xn ⪯yn andxn minus yn d(A B) for all nge 0 Since X is reflexive and(yn)n is bounded then there exists a subsequence (yφ(n))n inB which converges weakly to y isin B Since (xφ(n))n convergesweakly to x then

x minus yle lim infn⟶infin

xφ(n) minus yφ(n)

(15)

)us x minus y d(A B) and x⪯y Hence x isin A ⪯0 )erefore A⪯0 is closed Using the same argument we provethat B⪯0 is also closed

In the context of uniformly convex normed vectorspaces )eorem 3 is given as follows

Theorem 5 Let (A B) be a nonempty convex and boundedpair of a partially ordered uniformly convex Banach space X

such that B⪯0 nonempty and closed Let T AcupB⟶ AcupB

be a noncyclic monotone relatively nonexpansive mappingAssume that there exists x0 isin A⪯0 such that x0 ⪯Tx0 1en T

has a best proximity pairIn order to weaken the assumptions of Theorem 3 we

introduce the notions of uniform continuity and uniformconvexity in every direction

Definition 10 A modular ρ is said to be uniformly con-tinuous if for any εgt 0 and Rgt 0 there exists ηgt 0 such that|ρ(y) minus ρ(x + y)|le ε whenever ρ(x)le η and ρ(y)leR

Definition 11 Let ρ be a modular We say that ρ is uniformlyconvex in every direction (UCED) if for any rgt 0 and a nonnull z isin Xρ we have

δ(r z) inf 1 minus1rρ x +

z

21113874 1113875 ρ(x)le r ρ(x + z)le r1113882 1113883gt 0

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International Journal of Mathematics and Mathematical Sciences 5

We say that ρ is unique uniform convexity in everydirection (UUCED) if there exists η(s z)gt 0 for sge 0 and z

nonnull in Xρ such that

δ(r z)gt η(s z) for rgt s (17)

)e following proposition characterizes relationshipsbetween uniform convexity uniform convexity in everydirection and strict convexity of a modular

Proposition 3(a) (UCi) (resp (UUCi)) implies (UCED) (resp

(UUCED)) for i 1 2(b) (UUCED) implies (UCED)

(c) (UCED) implies (SC)

Proof It is quite easy to show (a) and (b) To prove (c) let xy isin Xρ such that xney If ρ(x)ne ρ(y) there is nothing toprove Otherwise we assume that ρ(x) ρ(y) rgt 0 andwe consider z y minus x Hence ρ(x + z) ρ(y) r Since ρis (UCED) then δ(r z)gt 0 which implies

1 minus1rρ x +

z

21113874 1113875ge δ(r z)gt 0 (18)

)us

ρx + y

21113874 1113875 ρ x +

y minus x

21113874 1113875le (1 minus δ(r z))rlt r (19)

)at is ρ((x + y)2)lt r (ρ(x) + ρ(y))2As an example of a modular which is (UUCED) but it is

not (UUC1) consider the function ρ defined on X RN by

ρ(x) ρ xn( 1113857( 1113857 1113944infin

n1xn

11138681113868111386811138681113868111386811138681113868n+1

(20)

Abdou and Khamsi [8] proved that this convex modularis (UUC2) which imply that ρ is (UUCED) but it is not(UUC1)

)e following lemma plays an important role in theproof of the next fixed point theorem To prove it we use thesame ideas as those used to prove Lemma 35 in [7]

Lemma 3 Let ρ be a convex modular uniformly continuousand (UUCED) Assume that Xρ satisfies the property (R) LetC be a nonempty ρ-closed convex and ρ-bounded subset of Xρand K be a nonempty ρ-closed convex subset of C Let (xk)kisinNbe a sequence in C and consider the ρ-type functionτ K⟶ [0 +infin] defined by

τ(y) lim supk⟶+infin

ρ y minus xk( 1113857 (21)

1en τ has a unique minimum point in K

Theorem 6 Let ρ be a convex modular (UUCED) and isuniformly continuous Assume that Xρ satisfies the property(R) Let (A B) be a nonempty convex and ρ-bounded pair inXρ such that B⪯0 is nonempty and ρ-closed LetT AcupB⟶ AcupB be a monotone noncyclic relativelyρ-nonexpansive mapping Assume that there exists x0 isin A⪯0such that x0 ⪯Tx0 1en T has a best proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)nge 0 given by xn+1 (1 minus λ)xn + λTxn whereα isin (0 1) For any nge 0 set Bn y isin B⪯0 xn ⪯y1113864 1113865 Similarto )eorem 3 one has Binfin cap nge0Bn is nonempty ρ-closedconvex and T-invariant

Consider the type function τ Binfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that isτ(z) lim supn⟶+infinρ(xn minus z) for all z isin Binfin By Lemma 3we know that τ has a unique minimum point y isin Binfin SinceT is monotone noncyclic relatively ρ-nonexpansive we haveτ(T(y)) lim sup

n

ρ xn+1 minus T(y)( 1113857

le lim supn

(1 minus λ)ρ xn minus T(y)( 1113857 + λρ T xn( 1113857 minus T(y)( 1113857( 1113857

le (1 minus λ) lim supn

ρ xn minus T(y)( 1113857

+ λ lim supn

ρ T xn( 1113857 minus T(y)( 1113857

le (1 minus λ)τ(T(y)) + λτ(y)

(22)which implies that τ(T(y))le τ(y) since λ isin (0 1))ereforeT(y) is also a minimum point of τ in Binfin )e uniqueness ofthe minimum point of τ implies that T(y) y

Since y isin Binfin then y isin B⪯0 )us there exists x isin A suchthat x⪯y and ρ(x minus y) dρ(A B) Using the same idea asin the proof of )eorem 3 we show that x is a fixed point ofT on A )erefore Tx x Ty y andρ(x minus y) dρ(A B) that is (x y) is a best proximity pair

In the case A B therefore A A⪯0 and B B⪯0 weobtain the following corollary)e result of this corollary hasalready been mentioned by Khamsi and Bin Dehaish (see)eorem 2) [7] in modular function spaces

Corollary 7 Let ρ be a convex modular (UUCED) anduniformly continuous Assume that Xρ satisfies the property(R) Let C be a nonempty ρ-closed convex and ρ-boundedsubset of Xρ Let T C⟶ C be a monotone ρ-nonexpansivemapping If there exists x0 isin C such that x0 ⪯T(x0) then T

has a fixed point

5 Application

In this section we present an application of our main resultsto prove the existence of the solution for nonlinear integralequations Let X L2([0 1]R) be the space of measurableand square-integrable functions on [0 1] Recall that X is aρ-complete modular space denoted Xρ whereρ(f) 1113938

[01]|f(t)|2dt is a modular Next we consider the

following integral equation

x(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(23)

where

sign(x(t)) 1 if x(t)ge 0

minus 1 if x(t)lt 01113896 (24)

6 International Journal of Mathematics and Mathematical Sciences

g isin Xρ such that g(t)ge 0 for all t isin [0 1] and F [0 1] times

[0 1] times R⟶ R is measurable in both variables t and s forevery x such that

11139461

0F(t s x(s))dsltinfin forallt isin [0 1] (25)

Recall that x⪯y if and only if y minus x isin P whereP x isin Xρ x(t)ge 0 ae1113966 1113967 Consider the ρ-closed convexsubsets A and B of Xρ given by A x isin Xρ c⪯x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 where c η ϕ and ψ are in XρSinceρ(x minus y)lemax ρ(η minus c) + ρ(ψ minus ϕ) ρ(ϕ minus η) + ρ(ψ minus c)1113864 1113865

ltinfin for all x y isin AcupB then (A B) is ρ-bounded More-over A ⪯0 η1113864 1113865 and B⪯0 ϕ1113864 1113865 are ρ-closed

Theorem 8 Let the mapping J be defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1] (26)

Assume that the following conditions are fulfilled

(c1) η(t)lt 0lt ϕ(t) for all t isin [0 1] and ρ(η minus ϕ)gt 0

(c2)c(t)le minus g(t) + 1113938

10 F(t s c(s))ds

η(t)ge minus g(t) + 111393810 F(t s η(s))ds

⎧⎨

⎩ and

ϕ(t)leg(t) + 111393810 F(t s ϕ(s))ds

ψ(t)geg(t) + 111393810 F(t sψ(s))ds

⎧⎨

⎩

(c3) F fulfill the following monotonicity condition0leF(t s x(s)) minus F(t s y(s)) for all (x y) isin Xρ suchthat y⪯ x

(c4) ρ(J(x) minus J(y))le ρ(x minus y) whenever x y isin AcupB

such that x⪯y

1en there exists xlowast isin A and ylowast isin B such that xlowast ⪯ylowastJ(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B)

Proof ρ is a (UUC1) convex modular then ρ is (UUCED)Moreover Xρ satisfies the property (R) since Xρ isρ-complete modular space and ρ is (UUC2) (from (d) ofProposition 1) and satisfies the Fatou property

ρ is uniformly continuous Indeed let εgt 0 and rgt 0 andwe prove that there exists αgt 0 such that

|ρ(y) minus ρ(x + y)|le ε (27)

where ρ(x)le α and ρ(y)le r For all t isin [0 1] we have||y(t)|2 minus |x(t) + y(t)|2|le |x(t)|2 + 2|y(t)||x(t)| )en

|ρ(y) minus ρ(x + y)|le1113946[01]

|x(t)|2dt + 21113946

[01]|y(t)||x(t)|dt

le ρ(x) + 2

ρ(x)

1113969

ρ(y)

1113969

le

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)

(28)

If ρ(x)le 1 then

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)le

ρ(x)

1113969

(1 + 2r

radic) (29)

Sinceρ(x)

1113968(1 + 2

r

radic)le ε this is equivalent to

ρ(x)1113968le ε(2

r

radic+ 1))erefore ρ(x)le (ε(2

r

radic+ 1))2We

take α min 1 (ε(2r

radic+ 1))21113966 1113967 Hence ρ is uniformly

continuous)e condition (c2) implies that J(A)subeA and J(B)subeB

Indeed let x isin A

J(x)(t) minus c(t) minus g(t) + 11139461

0F(t s x(s))ds minus c(t)

ge 11139461

0(F(t s x(s)) minus F(t s c(s)))dsge 0

(30)

)en c⪯ J(x) In addition

η(t) minus J(x)(t) η(t) minus minus g(t) + 1113946SF(t s x(s))ds1113874 1113875

ge1113946S(F(t s η(s)) minus F(t s x(s)))ds ge 0

(31)

)en J(x)⪯ η Hence J(A)subeA Using the same ar-guments we show that J(B)subeB From condition (c1) onehas dρ(A B)gt 0 By condition (c3) and since 0⪯g on AcupBwe conclude that J is monotone on AcupB

Consider x0 c Condition (c2) implies that x0 ⪯ J(x0)Hence by )eorem 6 there exists xlowast isin A and ylowast isin B suchthat xlowast ⪯ylowast J(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B) that is a best proximity pair

Example 2 Take F(t s x(s)) (ts3)x(s) for all(t s) isin [0 1]2 and x isin Xρ

Consider the functions c η ϕ ψ and g defined on Xρ byc(t) minus 2t η(t) minus t ϕ(t) t ψ(t) 2t and g(t) (89)tfor all t isin [0 1] Denote the sets A x isin Xρ c⪯ x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 Let the mappingJ AcupB⟶ AcupB defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(32)

We will check that the mapping J satisfies conditions(c1) minus (c4)

(c1) For every t isin [0 1] the inequality η(t)lt 0ltϕ(t)

holds and dρ(A B) ρ(ϕ minus η) 111393810 (2t)2dt 43

(c2) It easy to see that

c(t)le minus g(t) + 11139461

0F(t s c(s))ds

η(t)ge minus g(t) + 11139461

0F(t s η(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ϕ(t)leg(t) + 11139461

0F(t s ϕ(s))ds

ψ(t)geg(t) + 11139461

0F(t sψ(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

then J(A) sub A and J(B) sub B

International Journal of Mathematics and Mathematical Sciences 7

(c3) We have 0leF(t s y(s)) minus F(t s

x(s)) (ts3)(y(s) minus x(s)) for all (t s) isin [0 1]2 and(x y) isin Xρ times Xρ such that xley(c4) Let x isin A and y isin B such that y⪯x we have

ρ(J(y) minus J(x)) 11139461

011139461

0(F(t s y(s)) minus F(t s y(s)))ds + 2g(t)1113888 1113889

2

dt

11139461

0(2g(t))

2 1 +

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

dt

(34)

Let u0 isin ]0 1] for any u isin R such that uge uo we haveuo(1 + u)le u(1 + uo) )us

1113946z

uo

uo(1 + u)dule 1113946z

uo

u 1 + uo( 1113857du

uo (1 + z)2

minus 1 + uo( 11138572

1113872 1113873le 1 + uo( 1113857 z2

minus u2o1113872 1113873

(1 + z)2 le 1 + uo( 1113857 1 + uo +

z2 minus u2o

uo

1113888 1113889

(1 + z)2 le 1 + uo( 1113857 1 +

1uo

z2

1113888 1113889

(35)

for all zge u0 Since 111393810(ts3)(y(s) minus x(s))ds

2g(t)ge 18 for all t isin [0 1] then

ρ(J(y) minus J(x))le 11139461

0(2g(t))

2 1 +18

1113874 1113875

1 +118

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠dt

le98

11139461

0(2g(t))

2 1 +932

11139461

0s(y(s) minus x(s))ds1113888 1113889

2⎛⎝ ⎞⎠dt

le3227

1 +932

11139461

0s2ds1113888 1113889 1113946

1

0(y(s) minus x(s))

2ds1113888 11138891113888 1113889

le3227

1 +332

ρ(y(s) minus x(s))1113874 1113875

le89dρ(A B) +

19ρ(y(s) minus x(s))

le ρ(y(s) minus x(s))

(36)

where x isin A and y isin B such that y⪯x which implies that J

is a monotone relatively ρ-nonexpansive mapping )usJ(η) η J(ϕ) ϕ and ρ(ϕ minus β) dρ(A B) that is (η ϕ) isa best proximity pair of J

6 Conclusions

)roughout the paper we have discussed some best prox-imity pair results for the class of monotone noncyclic rel-atively ρ-nonexpansive mappings in the framework ofmodular spaces equipped with a partial order defined by aρ-closed convex cone Our results are extensions of severalresults as in relevant items from the reference section of thispaper as well as in the literature in general In order to showthe utility of our main results we prove the existence of thesolution of an integral equation which involves monotonenoncyclic relatively ρ-nonexpansive mappinngs

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] H NakanoModulared Semi-Ordered Linear Spaces MaruzenCo Tokyo Japan 1st edition 1950

[2] W Orlicz and J Musielak Orlicz Spaces and Modular SpacesLecture Notes in Mathematics Springer-Verlag Berlin Ger-many 1983

[3] A C M Ran andM C B Reurings ldquoA fixed point theorem inpartially ordered sets and some applications to matrixequationsrdquo Proceedings of the AmericanMathematical Societyvol 132 no 5 pp 1435ndash1443 2003

[4] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[5] M R Alfuraidan M Bachar and M A Khamsi ldquoOnmonotone contraction mappings in modular function spacesrdquoFixed Point 1eory and Applications vol 2015 no 1 2015

[6] M E Gordji F Sajadian Y J Cho andM Ramezani ldquoA fixedpoint theorem for quasi-contraction mappings in partiallyorder modular spaces with applicationrdquo UPB Scientific Bul-letin Series A Applied Mathematics and Physics vol 76pp 135ndash146 2014

[7] B A Bin Dehaish and M A Khamsi ldquoBrowder and Gohdefixed point theorem for monotone nonexpansive mappingsrdquoFixed Point 1eory and Applications vol 2016 no 1 2016

[8] A A N Abdou and M A Khamsi ldquoFixed point theorems inmodular vector spacesrdquo1e Journal of Nonlinear Sciences andApplications vol 10 no 8 pp 4046ndash4057 2017

[9] P Kumam ldquoFixed point theorems for nonexpansive map-pings in modular spacesrdquo Archivum Mathematicum vol 40pp 345ndash353 2004

[10] K Chaira and S Lazaiz ldquoBest proximity pair and fixed pointresults for noncyclic mappings in modular spacesrdquo ArabJournal of Mathematical Sciences vol 24 no 2 p 147 2018

[11] B A B Dehaish and M A Khamsi ldquoOn monotone mappingsin modular function spacesrdquo Journal of Nonlinear Sciencesand Applications vol 9 no 8 pp 5219ndash5228 2016

[12] F E Browder ldquoNonexpansive nonlinear operators in aBanach spacerdquo Proceedings of the National Academy of Sci-ences vol 54 no 4 pp 1041ndash1044 1965

8 International Journal of Mathematics and Mathematical Sciences

δi(r ε) inf 1 minus1rρ

x + y

21113874 1113875 (x y) isin Di1113882 1113883 (5)

If Di(r ε) empty we set δi(r ε) 1

(i) We say that ρ is uniformly convex (UCi) if for everyrgt 0 and εgt 0 and we have δi(r ε)gt 0

(ii) We say that ρ is unique uniformly convex (UUCi) iffor all sge 0 and εgt 0 and there exists η(s ε)gt 0 suchthat δi(r ε)gt η(s ε) for rgt s

(iii) We say that ρ is strictly convex (SC) if for every xy isin Xρ such that ρ(x) ρ(y) andρ((x + y)2) ρ(x) + ρ(y)2 and we have x y

)e following proposition characterizes the relationshipbetween the above notions

Proposition 1 (see [8])

(a) (UUCi) implies (UCi) for i 1 2(b) δ1(r ε)le δ2(r ε) for rgt 0 and εgt 0(c) (UC1) implies (UC2) implies (SC)

(d) (UUC1) implies (UUC2)

Definition 5 (see [9]) Let ρ be a modular We say that themodular space Xρ satisfies the property (R) if and only if forevery decreasing sequence (Cn)nisinN of nonempty ρ-closedconvex and ρ-bounded subsets of Xρ has a nonemptyintersection

Lemma 1 (see [8]) Let ρ be a convex modular satisfying theFatou property Assume that Xρ is ρ-complete and ρ is(UUC2) 1en Xρ satisfies the property (R)

Definition 6 (see [8]) Let (xn)n be a sequence in Xρ and K

be a nonempty subset of Xρ )e function τ K⟶ [0infin]

defined by τ(x) lim supn⟶infinρ(xn minus x) is called a ρ-typefunction

Note that the ρ-type function τ is convex since ρ isconvex A sequence (cn)n sub K is called a minimizing se-quence of τ if limn⟶+infinτ(cn) infxisinKτ(x)

)e following result found in [8] plays a crucial role inthe proof of many fixed point results in modular spaces

Lemma 2 (see [8]) Let ρ be a convex modular (UUC1)satisfying the Fatou property and Xρ a ρ-compete modularspace Let C be a nonempty ρ-closed convex subset of XρConsider the ρ-type function τ C⟶ [0 +infin] generated bya sequence (xn)n in Xρ Assume thatτ0 inf τ(x) x isin C ltinfin 1en all the minimizing se-quences of τ are ρ-convergent to the same limit

4 Main Results

A subset P sub Xρ is a pointed ρ-closed convex cone if P is anonempty ρ-closed subset of Xρ satisfying the followingproperties

(i) P + P sub P

(ii) λP sub P for all λ isin R+

(iii) Pcap (minus P) 0

Using P we define an ordering on Xρ byx⪯y if and only if y minus x isin P

Let ρ be a convex modular and T C⟶ C be amonotone mapping where C is a nonempty convex subsetof the modular space Xρ Let x0 isin C and λ isin (0 1) ConsiderKrasnoselskii-Ishikawa iteration sequence (xn)nisinN in C

defined by xn+1 (1 minus λ)xn + λT(xn) for all n isin N Assumethat x0 ⪯T(x0) (resp T(x0)⪯ x0) By the definition of thepartial ordering ⪯ we obtain x0 ⪯x1 ⪯T(x0) (respT(x0)⪯x1 ⪯ x0)

Since T is monotone one has T(x0)⪯T(x1) (respT(x1)⪯T(x0)) By induction we prove that

xn ⪯xn+1 ⪯T xn( 1113857⪯T xn+1( 1113857 respT xn+1( 1113857⪯T xn( 1113857(

⪯xn+1 ⪯xn1113857 for all n isin N(6)

Let us introduce the class of mappings for which theproblem of fixed point will be considered

Definition 7 Let (A B) be a nonempty pair of a modularspace Xρ A mapping T A cup B⟶ A cup B is said to bemonotone noncyclic relatively ρ-nonexpansive if it satisfiesthe following conditions

(i) For all (x y) isin (A cup B)2 x⪯y implies thatTx⪯Ty

(ii) T(A) sube A and T(B) sube B

(iii) For all x isin A and y isin B such that x⪯y we haveρ(Tx minus Ty)le ρ(x minus y)

Definition 8 (see [10]) Let (A B) be a pair of a modularspace Xρ and T A cup B⟶ A cup B be a noncyclic mappingA pair (x y) isin A times B is said to be a best proximity pair forthe noncyclic mapping T if

Tx x

Ty y

ρ(x minus y) dρ(A B)

(7)

Definition 9 A space Xρ is said to satisfy the property (p) ifxn⟶

ρx and yn⟶

ρy with xn ⪯yn for all n isin N then

x⪯yWe use (A⪯0 B⪯0 ) to denote the ordered ρ-proximal pair

obtained from (A B) by

A⪯0 x isin A ρ(x minus y) dρ(A B) andx⪯y for somey isin B1113966 1113967

B⪯0 y isin B ρ(x minus y) dρ(A B) and x⪯y for somex isin A1113966 1113967

(8)

International Journal of Mathematics and Mathematical Sciences 3

Note that if A⪯0 is nonempty if and only if B⪯0 is non-empty Indeed if x0 isin A⪯0 then there exists y isin B such thatx0 ⪯y and ρ(x0 minus y) dρ(A B) )us for this y we takex x0 isin A we have x⪯y and ρ(x minus y) dρ(A B) that isy isin B⪯0 As the same we prove the converse

Proposition 2 Let ρ be a convex modular satisfying theFatou property and Xρ satisfies the property (P) Let (A B) bea nonempty convex pair of Xρ Assume that A⪯0 is nonempty

(i) If (A B) is ρ-sequentially compact then (A⪯0 B⪯0 ) isρ-sequentially compact

(ii) (A⪯0 B⪯0 ) is convex(iii) dρ(A⪯0 B⪯0 ) dρ(A B)

(iv) If T C⟶ C is a noncyclic monotone relativelyρ-nonexpansive mapping then T(A ⪯0 )subeA⪯0 andT(B⪯0 )subeB⪯0 that is A⪯0 and B⪯0 are T-invariant

Proof It is quite easy to verify (iii)

(i) Let (xn)n be a sequence in A⪯0 )en there exists asequence (yn)n in B such that xn ⪯yn and ρ(xn minus

yn) dρ(A B) for all nge 0 Since (A B) is ρ-se-quentially compact there exists subsequences (xnk

)k

and (ynk)k of (xn)n and (yn)n respectively such that

xnk⪯ ynk

for all kge 0 and ρ-converge to x isin A andy isin B respectivelyFrom property (P) one has x⪯ y Since ρ satisfies theFatou property then

dρ(A B)le ρ(x minus y)le lim infk⟶infin

ρ xnkminus ynk

1113872 1113873 dρ(A B)

(9)

)us ρ(x minus y) dρ(A B) )erefore A⪯0 is ρ-se-quentially compact Likewise we prove that B⪯0 isalso ρ-sequentially compact

(ii) Let x1 x2 isin A⪯0 δ isin (0 1) and setz δx1 + (1 minus δ)x2 Since x1 x2 isin A⪯0 then thereexists y1 y2 isin B such that x1 ⪯y1 x2 ⪯y2 andρ(x1 minus y1) ρ(x2 minus y2) dρ(A B) Moreoverδx1 ⪯ δy1 and (1 minus δ)x2 ⪯ (1 minus δ)y2 Hencez δx1 + (1 minus δ)x2 ⪯ zprime δy1 + (1 minus δ)y2 Since ρis convex then

dρ(A B)le ρ z minus zprime( 1113857 ρ δx1 +(1 minus δ)x2( 1113857 minus δx1 +(1 minus δ)x2( 1113857( 1113857

le δρ x1 minus y1( 1113857 +(1 minus δ)ρ x2 minus y2( 1113857

le δdρ(A B) +(1 minus δ)dρ(A B) dρ(A B)

(10)

)en ρ(z minus zprime) dρ(A B) Hence z isin A⪯0 )erefore A⪯0 is convex Using the same argumentwe prove that B⪯0 is convex

(iii) Let x isin A⪯0 then there exists y isin B such that x⪯y

and ρ(x minus y) dρ(A B) Since T is monotone

noncyclic relatively ρ-nonexpansive then Tx⪯Ty

and dρ(A B)le ρ(Tx minus Ty)le ρ(x minus y) dρ(A B))us ρ(Tx minus Ty) dρ(A B) )erefore Tx isin A ⪯0 As the same way we obtain T(B⪯0 )subeB⪯0

Theorem 3 Let ρ be a convex modular (UUC1) satisfyingthe Fatou property and Xρ be a ρ-complete modular space Let(A B) be a nonempty convex and ρ-bounded pair of Xρ suchthat the subset B⪯0 is nonempty ρ-closed LetT AcupB⟶ A cup B be a monotone noncyclic relativelyρ-nonexpansive mapping and ρ-continuous on B Assumethat there exists x0 isin A⪯0 such that x0 ⪯Tx0 1en T has abest proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)n defined by xn+1 λTxn + (1 minus λ)xn for all nge 0and λ isin (0 1) Set Cn y isin B⪯0 xn ⪯y1113864 1113865 for any nge 0 )esequence (Cn)nisinN is a decreasing sequence of nonemptyρ-closed convex ρ-bounded subsets

Since A⪯0 is T-invariant then Tx0 isin A⪯0 Using theconvexity of A⪯0 one has x1 λTx0 + (1 minus λ)x0 isin A⪯0 Assume that xn isin A⪯0 then Txn isin A⪯0 Again the convexityof A⪯0 implies that xn+1 λTxn + (1 minus λ)xn isin A⪯0 )ere-fore xn isin A⪯0 for all nge 0 Hence for all nge 0 there existsy isin B such that xn ⪯y and ρ(xn minus y) dρ(A B) )us thereexists y isin B⪯0 such that xn ⪯y)ereforeCn is nonempty forall nge 0 Moreover since xn ⪯xn+1 then (Cn)nisinN is a de-creasing sequence

Let (cp)pge 0 be a sequence in Cn ρ-converges to b isin B⪯0 We have cp isin Cn then cp minus xn isin P and cp isin B⪯0 for all pge 0Since B⪯0 is ρ-closed one has b isin B⪯0 Moreovercp minus xn⟶

ρb minus xn )us b minus xn isin P since P is ρ-closed

Hence b isin Cn )erefore Cn is ρ-closed for all nge 0As P and B⪯0 are convex it is easy to see that Cn is convex

for all nge 0 Furthermore Cn is ρ-bounded for all nge 0By Lemma 1 Xρ satisfies the property (R) )en Cinfin

cap nge0Cn is nonempty and ρ-closed convexMoreover T(Cinfin) sube Cinfin In fact let z isin Cinfin then xn ⪯ z

for all nge 0 Since T is monotone and xn ⪯Txn one hasxn ⪯Tz for all nge 0 Moreover Tz isin B⪯0 since B⪯0 isT-invariant Hence T(Cinfin) sube Cinfin

Consider the type function τ Cinfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that is τ(z) lim supn⟶+infinρ(xn minus z) for all z isin Cinfin Let (zp)pisinN be a minimizing se-quence of τ By Lemma 2 (zp)p ρ-converges to a pointy isin Cinfin since Cinfin is ρ-closed Otherwise

τ Tzp1113872 1113873 lim supn⟶+infin

ρ xn+1 minus Tzp1113872 1113873

le (1 minus λ) lim supn⟶+infin

ρ xn minus Tzp1113872 1113873 + λ lim supn⟶+infin

ρ xn minus zp1113872 1113873

le (1 minus λ)τ Tzp1113872 1113873 + λτ zp1113872 1113873

(11)

Hence τ(Tzp)le τ(zp) )en (Tzp)pge 0 is also a mini-mizing sequence of τ By Lemma 2 (Tzp)p ρ-converges to yAs T is ρ-continuous on B then (Tzp)p ρ-converges to Ty)erefore Ty y )en there exists y isin B⪯0 such that Ty

4 International Journal of Mathematics and Mathematical Sciences

y and xn ⪯y for all n isin N By the definition of B⪯0 thereexists x isin A such that x⪯y and ρ(x minus y) dρ(A B) Inorder to finish the proof we show that x is a fixed point of T

on A We have

dρ(A B)le ρ(Tx minus y) ρ(Tx minus Ty)

le ρ(x minus y) dρ(A B)(12)

)en ρ(Tx minus y) dρ(A B) Moreover

dρ(A B)le ρTx minus y

2+

x minus y

21113874 1113875

leρ(Tx minus y)

2+ρ(x minus y)

2 dρ(A B)

(13)

)us ρ(((Tx minus y) + (x minus y))2) dρ(A B) Since ρ is(UUC2) it is (SC) Hence Tx minus y x minus y which impliesthat Tx x )erefore (x y) is a best proximity pair of themapping T

To illustrate )eorem 3 we consider the followingexample

Example 1 Let X R2 we define the modularρ X⟶ [0 +infin[ by ρ(x) |x1|

2 + |x2|2 for all

x (x1 x2) isin R2 )e modular ρ is convex satisfying theFatou property and (UUC1) and Xρ is a ρ-completemodular space Consider the ρ-closed convex coneP (x1 x2) isin Xρ (x1 x2)ge (0 0)1113966 1113967 Let A [(minus 1 minus 2)

(minus 1 2)] and B [(1 minus 4) (1 0)] )e pair (A B) is ρ-closedconvex and ρ-bounded in Xρ with dρ(A B) 4 MoreoverA0⪯ [(minus 1 minus 2) (minus 1 0)] and B⪯0 [(1 minus 2) (1 0)] are

ρ-closed Furthermore we have

(i) For all x isin [(minus 1 minus 2) (minus 1 0)] there existsy isin [(1 minus 2) (1 0)] such that x⪯ y

(ii) For the other cases the elements of A and B are notcomparable

Consider the mapping T A cup B⟶ A cup B defined by

T x1 x2( 1113857 (minus 1 0) if x1 x2( 1113857 isin [(minus 1 minus 2) (minus 1 0)]

x1 x2( 1113857 if x1 x2( 1113857 isin [(minus 1 0) (minus 1 2)]1113896

T y1 y2( 1113857 (1 0) if y1 y2( 1113857 isin B

(14)

)e mapping T is monotone noncyclic relativelyρ-nonexpansive ρ-continuous on A cup B and forx0 (minus 1 minus 2) isin Ale0 we have x0 ⪯ Tx0 )en T has a bestproximity pair ((1 0) (minus 1 0)) wheredρ(A B) ρ((1 0) minus (minus 1 0)) 4

)e following corollary is an immediate consequence of)eorem 3 and it suffices to take A B therefore A A⪯0and B B⪯0 It will be considered as a generalization ofBrowder and Gohde fixed point theorem for monotoneρ-nonexpansive mappings in modular spaces (see[7 11 12]) )is corollary result has already been mentionedby Bin Dehaish and Khamsi (see )eorem 1) [7] in theframework of modular function spaces

Corollary 4 Let ρ be a convex modular (UUC1) satisfyingthe Fatou property and Xρ be a ρ-complete modular space LetC be a nonempty ρ-closed convex ρ-bounded subset of Xρ andT C⟶ C be a monotone ρ-nonexpansive mapping andρ-continuous If there exists x0 isin C such that x0 ⪯T(x0) thenT has a fixed point

Particular case of normed vector spaces

Remark 1(i) If (X middot) is a normed vector space then the norm

middot is continuous and satisfies the Fatou propertyMoreover if (X middot) is a uniformly convex Banachspace then (X middot) is reflexive Hence it satisfies theproperty (R)

(ii) If (X middot) is reflexive and (A B) is a nonemptyclosed and bounded pair of X then (A⪯0 B⪯0 ) is alsoclosed convex pair of X

First case if (A⪯0 B⪯0 ) is empty then there is nothing toprove

Second case assume that (A⪯0 B⪯0 ) is nonempty Let(xn)n be a sequence in A⪯0 such that xn ρ-converges to x isin A)ere exists a sequence (yn)n in B such that xn ⪯yn andxn minus yn d(A B) for all nge 0 Since X is reflexive and(yn)n is bounded then there exists a subsequence (yφ(n))n inB which converges weakly to y isin B Since (xφ(n))n convergesweakly to x then

x minus yle lim infn⟶infin

xφ(n) minus yφ(n)

(15)

)us x minus y d(A B) and x⪯y Hence x isin A ⪯0 )erefore A⪯0 is closed Using the same argument we provethat B⪯0 is also closed

In the context of uniformly convex normed vectorspaces )eorem 3 is given as follows

Theorem 5 Let (A B) be a nonempty convex and boundedpair of a partially ordered uniformly convex Banach space X

such that B⪯0 nonempty and closed Let T AcupB⟶ AcupB

be a noncyclic monotone relatively nonexpansive mappingAssume that there exists x0 isin A⪯0 such that x0 ⪯Tx0 1en T

has a best proximity pairIn order to weaken the assumptions of Theorem 3 we

introduce the notions of uniform continuity and uniformconvexity in every direction

Definition 10 A modular ρ is said to be uniformly con-tinuous if for any εgt 0 and Rgt 0 there exists ηgt 0 such that|ρ(y) minus ρ(x + y)|le ε whenever ρ(x)le η and ρ(y)leR

Definition 11 Let ρ be a modular We say that ρ is uniformlyconvex in every direction (UCED) if for any rgt 0 and a nonnull z isin Xρ we have

δ(r z) inf 1 minus1rρ x +

z

21113874 1113875 ρ(x)le r ρ(x + z)le r1113882 1113883gt 0

(16)

International Journal of Mathematics and Mathematical Sciences 5

We say that ρ is unique uniform convexity in everydirection (UUCED) if there exists η(s z)gt 0 for sge 0 and z

nonnull in Xρ such that

δ(r z)gt η(s z) for rgt s (17)

)e following proposition characterizes relationshipsbetween uniform convexity uniform convexity in everydirection and strict convexity of a modular

Proposition 3(a) (UCi) (resp (UUCi)) implies (UCED) (resp

(UUCED)) for i 1 2(b) (UUCED) implies (UCED)

(c) (UCED) implies (SC)

Proof It is quite easy to show (a) and (b) To prove (c) let xy isin Xρ such that xney If ρ(x)ne ρ(y) there is nothing toprove Otherwise we assume that ρ(x) ρ(y) rgt 0 andwe consider z y minus x Hence ρ(x + z) ρ(y) r Since ρis (UCED) then δ(r z)gt 0 which implies

1 minus1rρ x +

z

21113874 1113875ge δ(r z)gt 0 (18)

)us

ρx + y

21113874 1113875 ρ x +

y minus x

21113874 1113875le (1 minus δ(r z))rlt r (19)

)at is ρ((x + y)2)lt r (ρ(x) + ρ(y))2As an example of a modular which is (UUCED) but it is

not (UUC1) consider the function ρ defined on X RN by

ρ(x) ρ xn( 1113857( 1113857 1113944infin

n1xn

11138681113868111386811138681113868111386811138681113868n+1

(20)

Abdou and Khamsi [8] proved that this convex modularis (UUC2) which imply that ρ is (UUCED) but it is not(UUC1)

)e following lemma plays an important role in theproof of the next fixed point theorem To prove it we use thesame ideas as those used to prove Lemma 35 in [7]

Lemma 3 Let ρ be a convex modular uniformly continuousand (UUCED) Assume that Xρ satisfies the property (R) LetC be a nonempty ρ-closed convex and ρ-bounded subset of Xρand K be a nonempty ρ-closed convex subset of C Let (xk)kisinNbe a sequence in C and consider the ρ-type functionτ K⟶ [0 +infin] defined by

τ(y) lim supk⟶+infin

ρ y minus xk( 1113857 (21)

1en τ has a unique minimum point in K

Theorem 6 Let ρ be a convex modular (UUCED) and isuniformly continuous Assume that Xρ satisfies the property(R) Let (A B) be a nonempty convex and ρ-bounded pair inXρ such that B⪯0 is nonempty and ρ-closed LetT AcupB⟶ AcupB be a monotone noncyclic relativelyρ-nonexpansive mapping Assume that there exists x0 isin A⪯0such that x0 ⪯Tx0 1en T has a best proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)nge 0 given by xn+1 (1 minus λ)xn + λTxn whereα isin (0 1) For any nge 0 set Bn y isin B⪯0 xn ⪯y1113864 1113865 Similarto )eorem 3 one has Binfin cap nge0Bn is nonempty ρ-closedconvex and T-invariant

Consider the type function τ Binfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that isτ(z) lim supn⟶+infinρ(xn minus z) for all z isin Binfin By Lemma 3we know that τ has a unique minimum point y isin Binfin SinceT is monotone noncyclic relatively ρ-nonexpansive we haveτ(T(y)) lim sup

n

ρ xn+1 minus T(y)( 1113857

le lim supn

(1 minus λ)ρ xn minus T(y)( 1113857 + λρ T xn( 1113857 minus T(y)( 1113857( 1113857

le (1 minus λ) lim supn

ρ xn minus T(y)( 1113857

+ λ lim supn

ρ T xn( 1113857 minus T(y)( 1113857

le (1 minus λ)τ(T(y)) + λτ(y)

(22)which implies that τ(T(y))le τ(y) since λ isin (0 1))ereforeT(y) is also a minimum point of τ in Binfin )e uniqueness ofthe minimum point of τ implies that T(y) y

Since y isin Binfin then y isin B⪯0 )us there exists x isin A suchthat x⪯y and ρ(x minus y) dρ(A B) Using the same idea asin the proof of )eorem 3 we show that x is a fixed point ofT on A )erefore Tx x Ty y andρ(x minus y) dρ(A B) that is (x y) is a best proximity pair

In the case A B therefore A A⪯0 and B B⪯0 weobtain the following corollary)e result of this corollary hasalready been mentioned by Khamsi and Bin Dehaish (see)eorem 2) [7] in modular function spaces

Corollary 7 Let ρ be a convex modular (UUCED) anduniformly continuous Assume that Xρ satisfies the property(R) Let C be a nonempty ρ-closed convex and ρ-boundedsubset of Xρ Let T C⟶ C be a monotone ρ-nonexpansivemapping If there exists x0 isin C such that x0 ⪯T(x0) then T

has a fixed point

5 Application

In this section we present an application of our main resultsto prove the existence of the solution for nonlinear integralequations Let X L2([0 1]R) be the space of measurableand square-integrable functions on [0 1] Recall that X is aρ-complete modular space denoted Xρ whereρ(f) 1113938

[01]|f(t)|2dt is a modular Next we consider the

following integral equation

x(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(23)

where

sign(x(t)) 1 if x(t)ge 0

minus 1 if x(t)lt 01113896 (24)

6 International Journal of Mathematics and Mathematical Sciences

g isin Xρ such that g(t)ge 0 for all t isin [0 1] and F [0 1] times

[0 1] times R⟶ R is measurable in both variables t and s forevery x such that

11139461

0F(t s x(s))dsltinfin forallt isin [0 1] (25)

Recall that x⪯y if and only if y minus x isin P whereP x isin Xρ x(t)ge 0 ae1113966 1113967 Consider the ρ-closed convexsubsets A and B of Xρ given by A x isin Xρ c⪯x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 where c η ϕ and ψ are in XρSinceρ(x minus y)lemax ρ(η minus c) + ρ(ψ minus ϕ) ρ(ϕ minus η) + ρ(ψ minus c)1113864 1113865

ltinfin for all x y isin AcupB then (A B) is ρ-bounded More-over A ⪯0 η1113864 1113865 and B⪯0 ϕ1113864 1113865 are ρ-closed

Theorem 8 Let the mapping J be defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1] (26)

Assume that the following conditions are fulfilled

(c1) η(t)lt 0lt ϕ(t) for all t isin [0 1] and ρ(η minus ϕ)gt 0

(c2)c(t)le minus g(t) + 1113938

10 F(t s c(s))ds

η(t)ge minus g(t) + 111393810 F(t s η(s))ds

⎧⎨

⎩ and

ϕ(t)leg(t) + 111393810 F(t s ϕ(s))ds

ψ(t)geg(t) + 111393810 F(t sψ(s))ds

⎧⎨

⎩

(c3) F fulfill the following monotonicity condition0leF(t s x(s)) minus F(t s y(s)) for all (x y) isin Xρ suchthat y⪯ x

(c4) ρ(J(x) minus J(y))le ρ(x minus y) whenever x y isin AcupB

such that x⪯y

1en there exists xlowast isin A and ylowast isin B such that xlowast ⪯ylowastJ(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B)

Proof ρ is a (UUC1) convex modular then ρ is (UUCED)Moreover Xρ satisfies the property (R) since Xρ isρ-complete modular space and ρ is (UUC2) (from (d) ofProposition 1) and satisfies the Fatou property

ρ is uniformly continuous Indeed let εgt 0 and rgt 0 andwe prove that there exists αgt 0 such that

|ρ(y) minus ρ(x + y)|le ε (27)

where ρ(x)le α and ρ(y)le r For all t isin [0 1] we have||y(t)|2 minus |x(t) + y(t)|2|le |x(t)|2 + 2|y(t)||x(t)| )en

|ρ(y) minus ρ(x + y)|le1113946[01]

|x(t)|2dt + 21113946

[01]|y(t)||x(t)|dt

le ρ(x) + 2

ρ(x)

1113969

ρ(y)

1113969

le

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)

(28)

If ρ(x)le 1 then

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)le

ρ(x)

1113969

(1 + 2r

radic) (29)

Sinceρ(x)

1113968(1 + 2

r

radic)le ε this is equivalent to

ρ(x)1113968le ε(2

r

radic+ 1))erefore ρ(x)le (ε(2

r

radic+ 1))2We

take α min 1 (ε(2r

radic+ 1))21113966 1113967 Hence ρ is uniformly

continuous)e condition (c2) implies that J(A)subeA and J(B)subeB

Indeed let x isin A

J(x)(t) minus c(t) minus g(t) + 11139461

0F(t s x(s))ds minus c(t)

ge 11139461

0(F(t s x(s)) minus F(t s c(s)))dsge 0

(30)

)en c⪯ J(x) In addition

η(t) minus J(x)(t) η(t) minus minus g(t) + 1113946SF(t s x(s))ds1113874 1113875

ge1113946S(F(t s η(s)) minus F(t s x(s)))ds ge 0

(31)

)en J(x)⪯ η Hence J(A)subeA Using the same ar-guments we show that J(B)subeB From condition (c1) onehas dρ(A B)gt 0 By condition (c3) and since 0⪯g on AcupBwe conclude that J is monotone on AcupB

Consider x0 c Condition (c2) implies that x0 ⪯ J(x0)Hence by )eorem 6 there exists xlowast isin A and ylowast isin B suchthat xlowast ⪯ylowast J(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B) that is a best proximity pair

Example 2 Take F(t s x(s)) (ts3)x(s) for all(t s) isin [0 1]2 and x isin Xρ

Consider the functions c η ϕ ψ and g defined on Xρ byc(t) minus 2t η(t) minus t ϕ(t) t ψ(t) 2t and g(t) (89)tfor all t isin [0 1] Denote the sets A x isin Xρ c⪯ x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 Let the mappingJ AcupB⟶ AcupB defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(32)

We will check that the mapping J satisfies conditions(c1) minus (c4)

(c1) For every t isin [0 1] the inequality η(t)lt 0ltϕ(t)

holds and dρ(A B) ρ(ϕ minus η) 111393810 (2t)2dt 43

(c2) It easy to see that

c(t)le minus g(t) + 11139461

0F(t s c(s))ds

η(t)ge minus g(t) + 11139461

0F(t s η(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ϕ(t)leg(t) + 11139461

0F(t s ϕ(s))ds

ψ(t)geg(t) + 11139461

0F(t sψ(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

then J(A) sub A and J(B) sub B

International Journal of Mathematics and Mathematical Sciences 7

(c3) We have 0leF(t s y(s)) minus F(t s

x(s)) (ts3)(y(s) minus x(s)) for all (t s) isin [0 1]2 and(x y) isin Xρ times Xρ such that xley(c4) Let x isin A and y isin B such that y⪯x we have

ρ(J(y) minus J(x)) 11139461

011139461

0(F(t s y(s)) minus F(t s y(s)))ds + 2g(t)1113888 1113889

2

dt

11139461

0(2g(t))

2 1 +

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

dt

(34)

Let u0 isin ]0 1] for any u isin R such that uge uo we haveuo(1 + u)le u(1 + uo) )us

1113946z

uo

uo(1 + u)dule 1113946z

uo

u 1 + uo( 1113857du

uo (1 + z)2

minus 1 + uo( 11138572

1113872 1113873le 1 + uo( 1113857 z2

minus u2o1113872 1113873

(1 + z)2 le 1 + uo( 1113857 1 + uo +

z2 minus u2o

uo

1113888 1113889

(1 + z)2 le 1 + uo( 1113857 1 +

1uo

z2

1113888 1113889

(35)

for all zge u0 Since 111393810(ts3)(y(s) minus x(s))ds

2g(t)ge 18 for all t isin [0 1] then

ρ(J(y) minus J(x))le 11139461

0(2g(t))

2 1 +18

1113874 1113875

1 +118

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠dt

le98

11139461

0(2g(t))

2 1 +932

11139461

0s(y(s) minus x(s))ds1113888 1113889

2⎛⎝ ⎞⎠dt

le3227

1 +932

11139461

0s2ds1113888 1113889 1113946

1

0(y(s) minus x(s))

2ds1113888 11138891113888 1113889

le3227

1 +332

ρ(y(s) minus x(s))1113874 1113875

le89dρ(A B) +

19ρ(y(s) minus x(s))

le ρ(y(s) minus x(s))

(36)

where x isin A and y isin B such that y⪯x which implies that J

is a monotone relatively ρ-nonexpansive mapping )usJ(η) η J(ϕ) ϕ and ρ(ϕ minus β) dρ(A B) that is (η ϕ) isa best proximity pair of J

6 Conclusions

)roughout the paper we have discussed some best prox-imity pair results for the class of monotone noncyclic rel-atively ρ-nonexpansive mappings in the framework ofmodular spaces equipped with a partial order defined by aρ-closed convex cone Our results are extensions of severalresults as in relevant items from the reference section of thispaper as well as in the literature in general In order to showthe utility of our main results we prove the existence of thesolution of an integral equation which involves monotonenoncyclic relatively ρ-nonexpansive mappinngs

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] H NakanoModulared Semi-Ordered Linear Spaces MaruzenCo Tokyo Japan 1st edition 1950

[2] W Orlicz and J Musielak Orlicz Spaces and Modular SpacesLecture Notes in Mathematics Springer-Verlag Berlin Ger-many 1983

[3] A C M Ran andM C B Reurings ldquoA fixed point theorem inpartially ordered sets and some applications to matrixequationsrdquo Proceedings of the AmericanMathematical Societyvol 132 no 5 pp 1435ndash1443 2003

[4] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[5] M R Alfuraidan M Bachar and M A Khamsi ldquoOnmonotone contraction mappings in modular function spacesrdquoFixed Point 1eory and Applications vol 2015 no 1 2015

[6] M E Gordji F Sajadian Y J Cho andM Ramezani ldquoA fixedpoint theorem for quasi-contraction mappings in partiallyorder modular spaces with applicationrdquo UPB Scientific Bul-letin Series A Applied Mathematics and Physics vol 76pp 135ndash146 2014

[7] B A Bin Dehaish and M A Khamsi ldquoBrowder and Gohdefixed point theorem for monotone nonexpansive mappingsrdquoFixed Point 1eory and Applications vol 2016 no 1 2016

[8] A A N Abdou and M A Khamsi ldquoFixed point theorems inmodular vector spacesrdquo1e Journal of Nonlinear Sciences andApplications vol 10 no 8 pp 4046ndash4057 2017

[9] P Kumam ldquoFixed point theorems for nonexpansive map-pings in modular spacesrdquo Archivum Mathematicum vol 40pp 345ndash353 2004

[10] K Chaira and S Lazaiz ldquoBest proximity pair and fixed pointresults for noncyclic mappings in modular spacesrdquo ArabJournal of Mathematical Sciences vol 24 no 2 p 147 2018

[11] B A B Dehaish and M A Khamsi ldquoOn monotone mappingsin modular function spacesrdquo Journal of Nonlinear Sciencesand Applications vol 9 no 8 pp 5219ndash5228 2016

[12] F E Browder ldquoNonexpansive nonlinear operators in aBanach spacerdquo Proceedings of the National Academy of Sci-ences vol 54 no 4 pp 1041ndash1044 1965

8 International Journal of Mathematics and Mathematical Sciences

Note that if A⪯0 is nonempty if and only if B⪯0 is non-empty Indeed if x0 isin A⪯0 then there exists y isin B such thatx0 ⪯y and ρ(x0 minus y) dρ(A B) )us for this y we takex x0 isin A we have x⪯y and ρ(x minus y) dρ(A B) that isy isin B⪯0 As the same we prove the converse

Proposition 2 Let ρ be a convex modular satisfying theFatou property and Xρ satisfies the property (P) Let (A B) bea nonempty convex pair of Xρ Assume that A⪯0 is nonempty

(i) If (A B) is ρ-sequentially compact then (A⪯0 B⪯0 ) isρ-sequentially compact

(ii) (A⪯0 B⪯0 ) is convex(iii) dρ(A⪯0 B⪯0 ) dρ(A B)

(iv) If T C⟶ C is a noncyclic monotone relativelyρ-nonexpansive mapping then T(A ⪯0 )subeA⪯0 andT(B⪯0 )subeB⪯0 that is A⪯0 and B⪯0 are T-invariant

Proof It is quite easy to verify (iii)

(i) Let (xn)n be a sequence in A⪯0 )en there exists asequence (yn)n in B such that xn ⪯yn and ρ(xn minus

yn) dρ(A B) for all nge 0 Since (A B) is ρ-se-quentially compact there exists subsequences (xnk

)k

and (ynk)k of (xn)n and (yn)n respectively such that

xnk⪯ ynk

for all kge 0 and ρ-converge to x isin A andy isin B respectivelyFrom property (P) one has x⪯ y Since ρ satisfies theFatou property then

dρ(A B)le ρ(x minus y)le lim infk⟶infin

ρ xnkminus ynk

1113872 1113873 dρ(A B)

(9)

)us ρ(x minus y) dρ(A B) )erefore A⪯0 is ρ-se-quentially compact Likewise we prove that B⪯0 isalso ρ-sequentially compact

(ii) Let x1 x2 isin A⪯0 δ isin (0 1) and setz δx1 + (1 minus δ)x2 Since x1 x2 isin A⪯0 then thereexists y1 y2 isin B such that x1 ⪯y1 x2 ⪯y2 andρ(x1 minus y1) ρ(x2 minus y2) dρ(A B) Moreoverδx1 ⪯ δy1 and (1 minus δ)x2 ⪯ (1 minus δ)y2 Hencez δx1 + (1 minus δ)x2 ⪯ zprime δy1 + (1 minus δ)y2 Since ρis convex then

dρ(A B)le ρ z minus zprime( 1113857 ρ δx1 +(1 minus δ)x2( 1113857 minus δx1 +(1 minus δ)x2( 1113857( 1113857

le δρ x1 minus y1( 1113857 +(1 minus δ)ρ x2 minus y2( 1113857

le δdρ(A B) +(1 minus δ)dρ(A B) dρ(A B)

(10)

)en ρ(z minus zprime) dρ(A B) Hence z isin A⪯0 )erefore A⪯0 is convex Using the same argumentwe prove that B⪯0 is convex

(iii) Let x isin A⪯0 then there exists y isin B such that x⪯y

and ρ(x minus y) dρ(A B) Since T is monotone

noncyclic relatively ρ-nonexpansive then Tx⪯Ty

and dρ(A B)le ρ(Tx minus Ty)le ρ(x minus y) dρ(A B))us ρ(Tx minus Ty) dρ(A B) )erefore Tx isin A ⪯0 As the same way we obtain T(B⪯0 )subeB⪯0

Theorem 3 Let ρ be a convex modular (UUC1) satisfyingthe Fatou property and Xρ be a ρ-complete modular space Let(A B) be a nonempty convex and ρ-bounded pair of Xρ suchthat the subset B⪯0 is nonempty ρ-closed LetT AcupB⟶ A cup B be a monotone noncyclic relativelyρ-nonexpansive mapping and ρ-continuous on B Assumethat there exists x0 isin A⪯0 such that x0 ⪯Tx0 1en T has abest proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)n defined by xn+1 λTxn + (1 minus λ)xn for all nge 0and λ isin (0 1) Set Cn y isin B⪯0 xn ⪯y1113864 1113865 for any nge 0 )esequence (Cn)nisinN is a decreasing sequence of nonemptyρ-closed convex ρ-bounded subsets

Since A⪯0 is T-invariant then Tx0 isin A⪯0 Using theconvexity of A⪯0 one has x1 λTx0 + (1 minus λ)x0 isin A⪯0 Assume that xn isin A⪯0 then Txn isin A⪯0 Again the convexityof A⪯0 implies that xn+1 λTxn + (1 minus λ)xn isin A⪯0 )ere-fore xn isin A⪯0 for all nge 0 Hence for all nge 0 there existsy isin B such that xn ⪯y and ρ(xn minus y) dρ(A B) )us thereexists y isin B⪯0 such that xn ⪯y)ereforeCn is nonempty forall nge 0 Moreover since xn ⪯xn+1 then (Cn)nisinN is a de-creasing sequence

Let (cp)pge 0 be a sequence in Cn ρ-converges to b isin B⪯0 We have cp isin Cn then cp minus xn isin P and cp isin B⪯0 for all pge 0Since B⪯0 is ρ-closed one has b isin B⪯0 Moreovercp minus xn⟶

ρb minus xn )us b minus xn isin P since P is ρ-closed

Hence b isin Cn )erefore Cn is ρ-closed for all nge 0As P and B⪯0 are convex it is easy to see that Cn is convex

for all nge 0 Furthermore Cn is ρ-bounded for all nge 0By Lemma 1 Xρ satisfies the property (R) )en Cinfin

cap nge0Cn is nonempty and ρ-closed convexMoreover T(Cinfin) sube Cinfin In fact let z isin Cinfin then xn ⪯ z

for all nge 0 Since T is monotone and xn ⪯Txn one hasxn ⪯Tz for all nge 0 Moreover Tz isin B⪯0 since B⪯0 isT-invariant Hence T(Cinfin) sube Cinfin

Consider the type function τ Cinfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that is τ(z) lim supn⟶+infinρ(xn minus z) for all z isin Cinfin Let (zp)pisinN be a minimizing se-quence of τ By Lemma 2 (zp)p ρ-converges to a pointy isin Cinfin since Cinfin is ρ-closed Otherwise

τ Tzp1113872 1113873 lim supn⟶+infin

ρ xn+1 minus Tzp1113872 1113873

le (1 minus λ) lim supn⟶+infin

ρ xn minus Tzp1113872 1113873 + λ lim supn⟶+infin

ρ xn minus zp1113872 1113873

le (1 minus λ)τ Tzp1113872 1113873 + λτ zp1113872 1113873

(11)

Hence τ(Tzp)le τ(zp) )en (Tzp)pge 0 is also a mini-mizing sequence of τ By Lemma 2 (Tzp)p ρ-converges to yAs T is ρ-continuous on B then (Tzp)p ρ-converges to Ty)erefore Ty y )en there exists y isin B⪯0 such that Ty

4 International Journal of Mathematics and Mathematical Sciences

y and xn ⪯y for all n isin N By the definition of B⪯0 thereexists x isin A such that x⪯y and ρ(x minus y) dρ(A B) Inorder to finish the proof we show that x is a fixed point of T

on A We have

dρ(A B)le ρ(Tx minus y) ρ(Tx minus Ty)

le ρ(x minus y) dρ(A B)(12)

)en ρ(Tx minus y) dρ(A B) Moreover

dρ(A B)le ρTx minus y

2+

x minus y

21113874 1113875

leρ(Tx minus y)

2+ρ(x minus y)

2 dρ(A B)

(13)

)us ρ(((Tx minus y) + (x minus y))2) dρ(A B) Since ρ is(UUC2) it is (SC) Hence Tx minus y x minus y which impliesthat Tx x )erefore (x y) is a best proximity pair of themapping T

To illustrate )eorem 3 we consider the followingexample

Example 1 Let X R2 we define the modularρ X⟶ [0 +infin[ by ρ(x) |x1|

2 + |x2|2 for all

x (x1 x2) isin R2 )e modular ρ is convex satisfying theFatou property and (UUC1) and Xρ is a ρ-completemodular space Consider the ρ-closed convex coneP (x1 x2) isin Xρ (x1 x2)ge (0 0)1113966 1113967 Let A [(minus 1 minus 2)

(minus 1 2)] and B [(1 minus 4) (1 0)] )e pair (A B) is ρ-closedconvex and ρ-bounded in Xρ with dρ(A B) 4 MoreoverA0⪯ [(minus 1 minus 2) (minus 1 0)] and B⪯0 [(1 minus 2) (1 0)] are

ρ-closed Furthermore we have

(i) For all x isin [(minus 1 minus 2) (minus 1 0)] there existsy isin [(1 minus 2) (1 0)] such that x⪯ y

(ii) For the other cases the elements of A and B are notcomparable

Consider the mapping T A cup B⟶ A cup B defined by

T x1 x2( 1113857 (minus 1 0) if x1 x2( 1113857 isin [(minus 1 minus 2) (minus 1 0)]

x1 x2( 1113857 if x1 x2( 1113857 isin [(minus 1 0) (minus 1 2)]1113896

T y1 y2( 1113857 (1 0) if y1 y2( 1113857 isin B

(14)

)e mapping T is monotone noncyclic relativelyρ-nonexpansive ρ-continuous on A cup B and forx0 (minus 1 minus 2) isin Ale0 we have x0 ⪯ Tx0 )en T has a bestproximity pair ((1 0) (minus 1 0)) wheredρ(A B) ρ((1 0) minus (minus 1 0)) 4

)e following corollary is an immediate consequence of)eorem 3 and it suffices to take A B therefore A A⪯0and B B⪯0 It will be considered as a generalization ofBrowder and Gohde fixed point theorem for monotoneρ-nonexpansive mappings in modular spaces (see[7 11 12]) )is corollary result has already been mentionedby Bin Dehaish and Khamsi (see )eorem 1) [7] in theframework of modular function spaces

Corollary 4 Let ρ be a convex modular (UUC1) satisfyingthe Fatou property and Xρ be a ρ-complete modular space LetC be a nonempty ρ-closed convex ρ-bounded subset of Xρ andT C⟶ C be a monotone ρ-nonexpansive mapping andρ-continuous If there exists x0 isin C such that x0 ⪯T(x0) thenT has a fixed point

Particular case of normed vector spaces

Remark 1(i) If (X middot) is a normed vector space then the norm

middot is continuous and satisfies the Fatou propertyMoreover if (X middot) is a uniformly convex Banachspace then (X middot) is reflexive Hence it satisfies theproperty (R)

(ii) If (X middot) is reflexive and (A B) is a nonemptyclosed and bounded pair of X then (A⪯0 B⪯0 ) is alsoclosed convex pair of X

First case if (A⪯0 B⪯0 ) is empty then there is nothing toprove

Second case assume that (A⪯0 B⪯0 ) is nonempty Let(xn)n be a sequence in A⪯0 such that xn ρ-converges to x isin A)ere exists a sequence (yn)n in B such that xn ⪯yn andxn minus yn d(A B) for all nge 0 Since X is reflexive and(yn)n is bounded then there exists a subsequence (yφ(n))n inB which converges weakly to y isin B Since (xφ(n))n convergesweakly to x then

x minus yle lim infn⟶infin

xφ(n) minus yφ(n)

(15)

)us x minus y d(A B) and x⪯y Hence x isin A ⪯0 )erefore A⪯0 is closed Using the same argument we provethat B⪯0 is also closed

In the context of uniformly convex normed vectorspaces )eorem 3 is given as follows

Theorem 5 Let (A B) be a nonempty convex and boundedpair of a partially ordered uniformly convex Banach space X

such that B⪯0 nonempty and closed Let T AcupB⟶ AcupB

be a noncyclic monotone relatively nonexpansive mappingAssume that there exists x0 isin A⪯0 such that x0 ⪯Tx0 1en T

has a best proximity pairIn order to weaken the assumptions of Theorem 3 we

introduce the notions of uniform continuity and uniformconvexity in every direction

Definition 10 A modular ρ is said to be uniformly con-tinuous if for any εgt 0 and Rgt 0 there exists ηgt 0 such that|ρ(y) minus ρ(x + y)|le ε whenever ρ(x)le η and ρ(y)leR

Definition 11 Let ρ be a modular We say that ρ is uniformlyconvex in every direction (UCED) if for any rgt 0 and a nonnull z isin Xρ we have

δ(r z) inf 1 minus1rρ x +

z

21113874 1113875 ρ(x)le r ρ(x + z)le r1113882 1113883gt 0

(16)

International Journal of Mathematics and Mathematical Sciences 5

We say that ρ is unique uniform convexity in everydirection (UUCED) if there exists η(s z)gt 0 for sge 0 and z

nonnull in Xρ such that

δ(r z)gt η(s z) for rgt s (17)

)e following proposition characterizes relationshipsbetween uniform convexity uniform convexity in everydirection and strict convexity of a modular

Proposition 3(a) (UCi) (resp (UUCi)) implies (UCED) (resp

(UUCED)) for i 1 2(b) (UUCED) implies (UCED)

(c) (UCED) implies (SC)

Proof It is quite easy to show (a) and (b) To prove (c) let xy isin Xρ such that xney If ρ(x)ne ρ(y) there is nothing toprove Otherwise we assume that ρ(x) ρ(y) rgt 0 andwe consider z y minus x Hence ρ(x + z) ρ(y) r Since ρis (UCED) then δ(r z)gt 0 which implies

1 minus1rρ x +

z

21113874 1113875ge δ(r z)gt 0 (18)

)us

ρx + y

21113874 1113875 ρ x +

y minus x

21113874 1113875le (1 minus δ(r z))rlt r (19)

)at is ρ((x + y)2)lt r (ρ(x) + ρ(y))2As an example of a modular which is (UUCED) but it is

not (UUC1) consider the function ρ defined on X RN by

ρ(x) ρ xn( 1113857( 1113857 1113944infin

n1xn

11138681113868111386811138681113868111386811138681113868n+1

(20)

Abdou and Khamsi [8] proved that this convex modularis (UUC2) which imply that ρ is (UUCED) but it is not(UUC1)

)e following lemma plays an important role in theproof of the next fixed point theorem To prove it we use thesame ideas as those used to prove Lemma 35 in [7]

Lemma 3 Let ρ be a convex modular uniformly continuousand (UUCED) Assume that Xρ satisfies the property (R) LetC be a nonempty ρ-closed convex and ρ-bounded subset of Xρand K be a nonempty ρ-closed convex subset of C Let (xk)kisinNbe a sequence in C and consider the ρ-type functionτ K⟶ [0 +infin] defined by

τ(y) lim supk⟶+infin

ρ y minus xk( 1113857 (21)

1en τ has a unique minimum point in K

Theorem 6 Let ρ be a convex modular (UUCED) and isuniformly continuous Assume that Xρ satisfies the property(R) Let (A B) be a nonempty convex and ρ-bounded pair inXρ such that B⪯0 is nonempty and ρ-closed LetT AcupB⟶ AcupB be a monotone noncyclic relativelyρ-nonexpansive mapping Assume that there exists x0 isin A⪯0such that x0 ⪯Tx0 1en T has a best proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)nge 0 given by xn+1 (1 minus λ)xn + λTxn whereα isin (0 1) For any nge 0 set Bn y isin B⪯0 xn ⪯y1113864 1113865 Similarto )eorem 3 one has Binfin cap nge0Bn is nonempty ρ-closedconvex and T-invariant

Consider the type function τ Binfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that isτ(z) lim supn⟶+infinρ(xn minus z) for all z isin Binfin By Lemma 3we know that τ has a unique minimum point y isin Binfin SinceT is monotone noncyclic relatively ρ-nonexpansive we haveτ(T(y)) lim sup

n

ρ xn+1 minus T(y)( 1113857

le lim supn

(1 minus λ)ρ xn minus T(y)( 1113857 + λρ T xn( 1113857 minus T(y)( 1113857( 1113857

le (1 minus λ) lim supn

ρ xn minus T(y)( 1113857

+ λ lim supn

ρ T xn( 1113857 minus T(y)( 1113857

le (1 minus λ)τ(T(y)) + λτ(y)

(22)which implies that τ(T(y))le τ(y) since λ isin (0 1))ereforeT(y) is also a minimum point of τ in Binfin )e uniqueness ofthe minimum point of τ implies that T(y) y

Since y isin Binfin then y isin B⪯0 )us there exists x isin A suchthat x⪯y and ρ(x minus y) dρ(A B) Using the same idea asin the proof of )eorem 3 we show that x is a fixed point ofT on A )erefore Tx x Ty y andρ(x minus y) dρ(A B) that is (x y) is a best proximity pair

In the case A B therefore A A⪯0 and B B⪯0 weobtain the following corollary)e result of this corollary hasalready been mentioned by Khamsi and Bin Dehaish (see)eorem 2) [7] in modular function spaces

Corollary 7 Let ρ be a convex modular (UUCED) anduniformly continuous Assume that Xρ satisfies the property(R) Let C be a nonempty ρ-closed convex and ρ-boundedsubset of Xρ Let T C⟶ C be a monotone ρ-nonexpansivemapping If there exists x0 isin C such that x0 ⪯T(x0) then T

has a fixed point

5 Application

In this section we present an application of our main resultsto prove the existence of the solution for nonlinear integralequations Let X L2([0 1]R) be the space of measurableand square-integrable functions on [0 1] Recall that X is aρ-complete modular space denoted Xρ whereρ(f) 1113938

[01]|f(t)|2dt is a modular Next we consider the

following integral equation

x(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(23)

where

sign(x(t)) 1 if x(t)ge 0

minus 1 if x(t)lt 01113896 (24)

6 International Journal of Mathematics and Mathematical Sciences

g isin Xρ such that g(t)ge 0 for all t isin [0 1] and F [0 1] times

[0 1] times R⟶ R is measurable in both variables t and s forevery x such that

11139461

0F(t s x(s))dsltinfin forallt isin [0 1] (25)

Recall that x⪯y if and only if y minus x isin P whereP x isin Xρ x(t)ge 0 ae1113966 1113967 Consider the ρ-closed convexsubsets A and B of Xρ given by A x isin Xρ c⪯x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 where c η ϕ and ψ are in XρSinceρ(x minus y)lemax ρ(η minus c) + ρ(ψ minus ϕ) ρ(ϕ minus η) + ρ(ψ minus c)1113864 1113865

ltinfin for all x y isin AcupB then (A B) is ρ-bounded More-over A ⪯0 η1113864 1113865 and B⪯0 ϕ1113864 1113865 are ρ-closed

Theorem 8 Let the mapping J be defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1] (26)

Assume that the following conditions are fulfilled

(c1) η(t)lt 0lt ϕ(t) for all t isin [0 1] and ρ(η minus ϕ)gt 0

(c2)c(t)le minus g(t) + 1113938

10 F(t s c(s))ds

η(t)ge minus g(t) + 111393810 F(t s η(s))ds

⎧⎨

⎩ and

ϕ(t)leg(t) + 111393810 F(t s ϕ(s))ds

ψ(t)geg(t) + 111393810 F(t sψ(s))ds

⎧⎨

⎩

(c3) F fulfill the following monotonicity condition0leF(t s x(s)) minus F(t s y(s)) for all (x y) isin Xρ suchthat y⪯ x

(c4) ρ(J(x) minus J(y))le ρ(x minus y) whenever x y isin AcupB

such that x⪯y

1en there exists xlowast isin A and ylowast isin B such that xlowast ⪯ylowastJ(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B)

Proof ρ is a (UUC1) convex modular then ρ is (UUCED)Moreover Xρ satisfies the property (R) since Xρ isρ-complete modular space and ρ is (UUC2) (from (d) ofProposition 1) and satisfies the Fatou property

ρ is uniformly continuous Indeed let εgt 0 and rgt 0 andwe prove that there exists αgt 0 such that

|ρ(y) minus ρ(x + y)|le ε (27)

where ρ(x)le α and ρ(y)le r For all t isin [0 1] we have||y(t)|2 minus |x(t) + y(t)|2|le |x(t)|2 + 2|y(t)||x(t)| )en

|ρ(y) minus ρ(x + y)|le1113946[01]

|x(t)|2dt + 21113946

[01]|y(t)||x(t)|dt

le ρ(x) + 2

ρ(x)

1113969

ρ(y)

1113969

le

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)

(28)

If ρ(x)le 1 then

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)le

ρ(x)

1113969

(1 + 2r

radic) (29)

Sinceρ(x)

1113968(1 + 2

r

radic)le ε this is equivalent to

ρ(x)1113968le ε(2

r

radic+ 1))erefore ρ(x)le (ε(2

r

radic+ 1))2We

take α min 1 (ε(2r

radic+ 1))21113966 1113967 Hence ρ is uniformly

continuous)e condition (c2) implies that J(A)subeA and J(B)subeB

Indeed let x isin A

J(x)(t) minus c(t) minus g(t) + 11139461

0F(t s x(s))ds minus c(t)

ge 11139461

0(F(t s x(s)) minus F(t s c(s)))dsge 0

(30)

)en c⪯ J(x) In addition

η(t) minus J(x)(t) η(t) minus minus g(t) + 1113946SF(t s x(s))ds1113874 1113875

ge1113946S(F(t s η(s)) minus F(t s x(s)))ds ge 0

(31)

)en J(x)⪯ η Hence J(A)subeA Using the same ar-guments we show that J(B)subeB From condition (c1) onehas dρ(A B)gt 0 By condition (c3) and since 0⪯g on AcupBwe conclude that J is monotone on AcupB

Consider x0 c Condition (c2) implies that x0 ⪯ J(x0)Hence by )eorem 6 there exists xlowast isin A and ylowast isin B suchthat xlowast ⪯ylowast J(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B) that is a best proximity pair

Example 2 Take F(t s x(s)) (ts3)x(s) for all(t s) isin [0 1]2 and x isin Xρ

Consider the functions c η ϕ ψ and g defined on Xρ byc(t) minus 2t η(t) minus t ϕ(t) t ψ(t) 2t and g(t) (89)tfor all t isin [0 1] Denote the sets A x isin Xρ c⪯ x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 Let the mappingJ AcupB⟶ AcupB defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(32)

We will check that the mapping J satisfies conditions(c1) minus (c4)

(c1) For every t isin [0 1] the inequality η(t)lt 0ltϕ(t)

holds and dρ(A B) ρ(ϕ minus η) 111393810 (2t)2dt 43

(c2) It easy to see that

c(t)le minus g(t) + 11139461

0F(t s c(s))ds

η(t)ge minus g(t) + 11139461

0F(t s η(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ϕ(t)leg(t) + 11139461

0F(t s ϕ(s))ds

ψ(t)geg(t) + 11139461

0F(t sψ(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

then J(A) sub A and J(B) sub B

International Journal of Mathematics and Mathematical Sciences 7

(c3) We have 0leF(t s y(s)) minus F(t s

x(s)) (ts3)(y(s) minus x(s)) for all (t s) isin [0 1]2 and(x y) isin Xρ times Xρ such that xley(c4) Let x isin A and y isin B such that y⪯x we have

ρ(J(y) minus J(x)) 11139461

011139461

0(F(t s y(s)) minus F(t s y(s)))ds + 2g(t)1113888 1113889

2

dt

11139461

0(2g(t))

2 1 +

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

dt

(34)

Let u0 isin ]0 1] for any u isin R such that uge uo we haveuo(1 + u)le u(1 + uo) )us

1113946z

uo

uo(1 + u)dule 1113946z

uo

u 1 + uo( 1113857du

uo (1 + z)2

minus 1 + uo( 11138572

1113872 1113873le 1 + uo( 1113857 z2

minus u2o1113872 1113873

(1 + z)2 le 1 + uo( 1113857 1 + uo +

z2 minus u2o

uo

1113888 1113889

(1 + z)2 le 1 + uo( 1113857 1 +

1uo

z2

1113888 1113889

(35)

for all zge u0 Since 111393810(ts3)(y(s) minus x(s))ds

2g(t)ge 18 for all t isin [0 1] then

ρ(J(y) minus J(x))le 11139461

0(2g(t))

2 1 +18

1113874 1113875

1 +118

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠dt

le98

11139461

0(2g(t))

2 1 +932

11139461

0s(y(s) minus x(s))ds1113888 1113889

2⎛⎝ ⎞⎠dt

le3227

1 +932

11139461

0s2ds1113888 1113889 1113946

1

0(y(s) minus x(s))

2ds1113888 11138891113888 1113889

le3227

1 +332

ρ(y(s) minus x(s))1113874 1113875

le89dρ(A B) +

19ρ(y(s) minus x(s))

le ρ(y(s) minus x(s))

(36)

where x isin A and y isin B such that y⪯x which implies that J

is a monotone relatively ρ-nonexpansive mapping )usJ(η) η J(ϕ) ϕ and ρ(ϕ minus β) dρ(A B) that is (η ϕ) isa best proximity pair of J

6 Conclusions

)roughout the paper we have discussed some best prox-imity pair results for the class of monotone noncyclic rel-atively ρ-nonexpansive mappings in the framework ofmodular spaces equipped with a partial order defined by aρ-closed convex cone Our results are extensions of severalresults as in relevant items from the reference section of thispaper as well as in the literature in general In order to showthe utility of our main results we prove the existence of thesolution of an integral equation which involves monotonenoncyclic relatively ρ-nonexpansive mappinngs

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] H NakanoModulared Semi-Ordered Linear Spaces MaruzenCo Tokyo Japan 1st edition 1950

[2] W Orlicz and J Musielak Orlicz Spaces and Modular SpacesLecture Notes in Mathematics Springer-Verlag Berlin Ger-many 1983

[3] A C M Ran andM C B Reurings ldquoA fixed point theorem inpartially ordered sets and some applications to matrixequationsrdquo Proceedings of the AmericanMathematical Societyvol 132 no 5 pp 1435ndash1443 2003

[4] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[5] M R Alfuraidan M Bachar and M A Khamsi ldquoOnmonotone contraction mappings in modular function spacesrdquoFixed Point 1eory and Applications vol 2015 no 1 2015

[6] M E Gordji F Sajadian Y J Cho andM Ramezani ldquoA fixedpoint theorem for quasi-contraction mappings in partiallyorder modular spaces with applicationrdquo UPB Scientific Bul-letin Series A Applied Mathematics and Physics vol 76pp 135ndash146 2014

[7] B A Bin Dehaish and M A Khamsi ldquoBrowder and Gohdefixed point theorem for monotone nonexpansive mappingsrdquoFixed Point 1eory and Applications vol 2016 no 1 2016

[8] A A N Abdou and M A Khamsi ldquoFixed point theorems inmodular vector spacesrdquo1e Journal of Nonlinear Sciences andApplications vol 10 no 8 pp 4046ndash4057 2017

[9] P Kumam ldquoFixed point theorems for nonexpansive map-pings in modular spacesrdquo Archivum Mathematicum vol 40pp 345ndash353 2004

[10] K Chaira and S Lazaiz ldquoBest proximity pair and fixed pointresults for noncyclic mappings in modular spacesrdquo ArabJournal of Mathematical Sciences vol 24 no 2 p 147 2018

[11] B A B Dehaish and M A Khamsi ldquoOn monotone mappingsin modular function spacesrdquo Journal of Nonlinear Sciencesand Applications vol 9 no 8 pp 5219ndash5228 2016

[12] F E Browder ldquoNonexpansive nonlinear operators in aBanach spacerdquo Proceedings of the National Academy of Sci-ences vol 54 no 4 pp 1041ndash1044 1965

8 International Journal of Mathematics and Mathematical Sciences

y and xn ⪯y for all n isin N By the definition of B⪯0 thereexists x isin A such that x⪯y and ρ(x minus y) dρ(A B) Inorder to finish the proof we show that x is a fixed point of T

on A We have

dρ(A B)le ρ(Tx minus y) ρ(Tx minus Ty)

le ρ(x minus y) dρ(A B)(12)

)en ρ(Tx minus y) dρ(A B) Moreover

dρ(A B)le ρTx minus y

2+

x minus y

21113874 1113875

leρ(Tx minus y)

2+ρ(x minus y)

2 dρ(A B)

(13)

)us ρ(((Tx minus y) + (x minus y))2) dρ(A B) Since ρ is(UUC2) it is (SC) Hence Tx minus y x minus y which impliesthat Tx x )erefore (x y) is a best proximity pair of themapping T

To illustrate )eorem 3 we consider the followingexample

Example 1 Let X R2 we define the modularρ X⟶ [0 +infin[ by ρ(x) |x1|

2 + |x2|2 for all

x (x1 x2) isin R2 )e modular ρ is convex satisfying theFatou property and (UUC1) and Xρ is a ρ-completemodular space Consider the ρ-closed convex coneP (x1 x2) isin Xρ (x1 x2)ge (0 0)1113966 1113967 Let A [(minus 1 minus 2)

(minus 1 2)] and B [(1 minus 4) (1 0)] )e pair (A B) is ρ-closedconvex and ρ-bounded in Xρ with dρ(A B) 4 MoreoverA0⪯ [(minus 1 minus 2) (minus 1 0)] and B⪯0 [(1 minus 2) (1 0)] are

ρ-closed Furthermore we have

(i) For all x isin [(minus 1 minus 2) (minus 1 0)] there existsy isin [(1 minus 2) (1 0)] such that x⪯ y

(ii) For the other cases the elements of A and B are notcomparable

Consider the mapping T A cup B⟶ A cup B defined by

T x1 x2( 1113857 (minus 1 0) if x1 x2( 1113857 isin [(minus 1 minus 2) (minus 1 0)]

x1 x2( 1113857 if x1 x2( 1113857 isin [(minus 1 0) (minus 1 2)]1113896

T y1 y2( 1113857 (1 0) if y1 y2( 1113857 isin B

(14)

)e mapping T is monotone noncyclic relativelyρ-nonexpansive ρ-continuous on A cup B and forx0 (minus 1 minus 2) isin Ale0 we have x0 ⪯ Tx0 )en T has a bestproximity pair ((1 0) (minus 1 0)) wheredρ(A B) ρ((1 0) minus (minus 1 0)) 4

)e following corollary is an immediate consequence of)eorem 3 and it suffices to take A B therefore A A⪯0and B B⪯0 It will be considered as a generalization ofBrowder and Gohde fixed point theorem for monotoneρ-nonexpansive mappings in modular spaces (see[7 11 12]) )is corollary result has already been mentionedby Bin Dehaish and Khamsi (see )eorem 1) [7] in theframework of modular function spaces

Corollary 4 Let ρ be a convex modular (UUC1) satisfyingthe Fatou property and Xρ be a ρ-complete modular space LetC be a nonempty ρ-closed convex ρ-bounded subset of Xρ andT C⟶ C be a monotone ρ-nonexpansive mapping andρ-continuous If there exists x0 isin C such that x0 ⪯T(x0) thenT has a fixed point

Particular case of normed vector spaces

Remark 1(i) If (X middot) is a normed vector space then the norm

middot is continuous and satisfies the Fatou propertyMoreover if (X middot) is a uniformly convex Banachspace then (X middot) is reflexive Hence it satisfies theproperty (R)

(ii) If (X middot) is reflexive and (A B) is a nonemptyclosed and bounded pair of X then (A⪯0 B⪯0 ) is alsoclosed convex pair of X

First case if (A⪯0 B⪯0 ) is empty then there is nothing toprove

Second case assume that (A⪯0 B⪯0 ) is nonempty Let(xn)n be a sequence in A⪯0 such that xn ρ-converges to x isin A)ere exists a sequence (yn)n in B such that xn ⪯yn andxn minus yn d(A B) for all nge 0 Since X is reflexive and(yn)n is bounded then there exists a subsequence (yφ(n))n inB which converges weakly to y isin B Since (xφ(n))n convergesweakly to x then

x minus yle lim infn⟶infin

xφ(n) minus yφ(n)

(15)

)us x minus y d(A B) and x⪯y Hence x isin A ⪯0 )erefore A⪯0 is closed Using the same argument we provethat B⪯0 is also closed

In the context of uniformly convex normed vectorspaces )eorem 3 is given as follows

Theorem 5 Let (A B) be a nonempty convex and boundedpair of a partially ordered uniformly convex Banach space X

such that B⪯0 nonempty and closed Let T AcupB⟶ AcupB

be a noncyclic monotone relatively nonexpansive mappingAssume that there exists x0 isin A⪯0 such that x0 ⪯Tx0 1en T

has a best proximity pairIn order to weaken the assumptions of Theorem 3 we

introduce the notions of uniform continuity and uniformconvexity in every direction

Definition 10 A modular ρ is said to be uniformly con-tinuous if for any εgt 0 and Rgt 0 there exists ηgt 0 such that|ρ(y) minus ρ(x + y)|le ε whenever ρ(x)le η and ρ(y)leR

Definition 11 Let ρ be a modular We say that ρ is uniformlyconvex in every direction (UCED) if for any rgt 0 and a nonnull z isin Xρ we have

δ(r z) inf 1 minus1rρ x +

z

21113874 1113875 ρ(x)le r ρ(x + z)le r1113882 1113883gt 0

(16)

International Journal of Mathematics and Mathematical Sciences 5

We say that ρ is unique uniform convexity in everydirection (UUCED) if there exists η(s z)gt 0 for sge 0 and z

nonnull in Xρ such that

δ(r z)gt η(s z) for rgt s (17)

)e following proposition characterizes relationshipsbetween uniform convexity uniform convexity in everydirection and strict convexity of a modular

Proposition 3(a) (UCi) (resp (UUCi)) implies (UCED) (resp

(UUCED)) for i 1 2(b) (UUCED) implies (UCED)

(c) (UCED) implies (SC)

Proof It is quite easy to show (a) and (b) To prove (c) let xy isin Xρ such that xney If ρ(x)ne ρ(y) there is nothing toprove Otherwise we assume that ρ(x) ρ(y) rgt 0 andwe consider z y minus x Hence ρ(x + z) ρ(y) r Since ρis (UCED) then δ(r z)gt 0 which implies

1 minus1rρ x +

z

21113874 1113875ge δ(r z)gt 0 (18)

)us

ρx + y

21113874 1113875 ρ x +

y minus x

21113874 1113875le (1 minus δ(r z))rlt r (19)

)at is ρ((x + y)2)lt r (ρ(x) + ρ(y))2As an example of a modular which is (UUCED) but it is

not (UUC1) consider the function ρ defined on X RN by

ρ(x) ρ xn( 1113857( 1113857 1113944infin

n1xn

11138681113868111386811138681113868111386811138681113868n+1

(20)

Abdou and Khamsi [8] proved that this convex modularis (UUC2) which imply that ρ is (UUCED) but it is not(UUC1)

)e following lemma plays an important role in theproof of the next fixed point theorem To prove it we use thesame ideas as those used to prove Lemma 35 in [7]

Lemma 3 Let ρ be a convex modular uniformly continuousand (UUCED) Assume that Xρ satisfies the property (R) LetC be a nonempty ρ-closed convex and ρ-bounded subset of Xρand K be a nonempty ρ-closed convex subset of C Let (xk)kisinNbe a sequence in C and consider the ρ-type functionτ K⟶ [0 +infin] defined by

τ(y) lim supk⟶+infin

ρ y minus xk( 1113857 (21)

1en τ has a unique minimum point in K

Theorem 6 Let ρ be a convex modular (UUCED) and isuniformly continuous Assume that Xρ satisfies the property(R) Let (A B) be a nonempty convex and ρ-bounded pair inXρ such that B⪯0 is nonempty and ρ-closed LetT AcupB⟶ AcupB be a monotone noncyclic relativelyρ-nonexpansive mapping Assume that there exists x0 isin A⪯0such that x0 ⪯Tx0 1en T has a best proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)nge 0 given by xn+1 (1 minus λ)xn + λTxn whereα isin (0 1) For any nge 0 set Bn y isin B⪯0 xn ⪯y1113864 1113865 Similarto )eorem 3 one has Binfin cap nge0Bn is nonempty ρ-closedconvex and T-invariant

Consider the type function τ Binfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that isτ(z) lim supn⟶+infinρ(xn minus z) for all z isin Binfin By Lemma 3we know that τ has a unique minimum point y isin Binfin SinceT is monotone noncyclic relatively ρ-nonexpansive we haveτ(T(y)) lim sup

n

ρ xn+1 minus T(y)( 1113857

le lim supn

(1 minus λ)ρ xn minus T(y)( 1113857 + λρ T xn( 1113857 minus T(y)( 1113857( 1113857

le (1 minus λ) lim supn

ρ xn minus T(y)( 1113857

+ λ lim supn

ρ T xn( 1113857 minus T(y)( 1113857

le (1 minus λ)τ(T(y)) + λτ(y)

(22)which implies that τ(T(y))le τ(y) since λ isin (0 1))ereforeT(y) is also a minimum point of τ in Binfin )e uniqueness ofthe minimum point of τ implies that T(y) y

Since y isin Binfin then y isin B⪯0 )us there exists x isin A suchthat x⪯y and ρ(x minus y) dρ(A B) Using the same idea asin the proof of )eorem 3 we show that x is a fixed point ofT on A )erefore Tx x Ty y andρ(x minus y) dρ(A B) that is (x y) is a best proximity pair

In the case A B therefore A A⪯0 and B B⪯0 weobtain the following corollary)e result of this corollary hasalready been mentioned by Khamsi and Bin Dehaish (see)eorem 2) [7] in modular function spaces

Corollary 7 Let ρ be a convex modular (UUCED) anduniformly continuous Assume that Xρ satisfies the property(R) Let C be a nonempty ρ-closed convex and ρ-boundedsubset of Xρ Let T C⟶ C be a monotone ρ-nonexpansivemapping If there exists x0 isin C such that x0 ⪯T(x0) then T

has a fixed point

5 Application

In this section we present an application of our main resultsto prove the existence of the solution for nonlinear integralequations Let X L2([0 1]R) be the space of measurableand square-integrable functions on [0 1] Recall that X is aρ-complete modular space denoted Xρ whereρ(f) 1113938

[01]|f(t)|2dt is a modular Next we consider the

following integral equation

x(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(23)

where

sign(x(t)) 1 if x(t)ge 0

minus 1 if x(t)lt 01113896 (24)

6 International Journal of Mathematics and Mathematical Sciences

g isin Xρ such that g(t)ge 0 for all t isin [0 1] and F [0 1] times

[0 1] times R⟶ R is measurable in both variables t and s forevery x such that

11139461

0F(t s x(s))dsltinfin forallt isin [0 1] (25)

Recall that x⪯y if and only if y minus x isin P whereP x isin Xρ x(t)ge 0 ae1113966 1113967 Consider the ρ-closed convexsubsets A and B of Xρ given by A x isin Xρ c⪯x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 where c η ϕ and ψ are in XρSinceρ(x minus y)lemax ρ(η minus c) + ρ(ψ minus ϕ) ρ(ϕ minus η) + ρ(ψ minus c)1113864 1113865

ltinfin for all x y isin AcupB then (A B) is ρ-bounded More-over A ⪯0 η1113864 1113865 and B⪯0 ϕ1113864 1113865 are ρ-closed

Theorem 8 Let the mapping J be defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1] (26)

Assume that the following conditions are fulfilled

(c1) η(t)lt 0lt ϕ(t) for all t isin [0 1] and ρ(η minus ϕ)gt 0

(c2)c(t)le minus g(t) + 1113938

10 F(t s c(s))ds

η(t)ge minus g(t) + 111393810 F(t s η(s))ds

⎧⎨

⎩ and

ϕ(t)leg(t) + 111393810 F(t s ϕ(s))ds

ψ(t)geg(t) + 111393810 F(t sψ(s))ds

⎧⎨

⎩

(c3) F fulfill the following monotonicity condition0leF(t s x(s)) minus F(t s y(s)) for all (x y) isin Xρ suchthat y⪯ x

(c4) ρ(J(x) minus J(y))le ρ(x minus y) whenever x y isin AcupB

such that x⪯y

1en there exists xlowast isin A and ylowast isin B such that xlowast ⪯ylowastJ(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B)

Proof ρ is a (UUC1) convex modular then ρ is (UUCED)Moreover Xρ satisfies the property (R) since Xρ isρ-complete modular space and ρ is (UUC2) (from (d) ofProposition 1) and satisfies the Fatou property

ρ is uniformly continuous Indeed let εgt 0 and rgt 0 andwe prove that there exists αgt 0 such that

|ρ(y) minus ρ(x + y)|le ε (27)

where ρ(x)le α and ρ(y)le r For all t isin [0 1] we have||y(t)|2 minus |x(t) + y(t)|2|le |x(t)|2 + 2|y(t)||x(t)| )en

|ρ(y) minus ρ(x + y)|le1113946[01]

|x(t)|2dt + 21113946

[01]|y(t)||x(t)|dt

le ρ(x) + 2

ρ(x)

1113969

ρ(y)

1113969

le

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)

(28)

If ρ(x)le 1 then

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)le

ρ(x)

1113969

(1 + 2r

radic) (29)

Sinceρ(x)

1113968(1 + 2

r

radic)le ε this is equivalent to

ρ(x)1113968le ε(2

r

radic+ 1))erefore ρ(x)le (ε(2

r

radic+ 1))2We

take α min 1 (ε(2r

radic+ 1))21113966 1113967 Hence ρ is uniformly

continuous)e condition (c2) implies that J(A)subeA and J(B)subeB

Indeed let x isin A

J(x)(t) minus c(t) minus g(t) + 11139461

0F(t s x(s))ds minus c(t)

ge 11139461

0(F(t s x(s)) minus F(t s c(s)))dsge 0

(30)

)en c⪯ J(x) In addition

η(t) minus J(x)(t) η(t) minus minus g(t) + 1113946SF(t s x(s))ds1113874 1113875

ge1113946S(F(t s η(s)) minus F(t s x(s)))ds ge 0

(31)

)en J(x)⪯ η Hence J(A)subeA Using the same ar-guments we show that J(B)subeB From condition (c1) onehas dρ(A B)gt 0 By condition (c3) and since 0⪯g on AcupBwe conclude that J is monotone on AcupB

Consider x0 c Condition (c2) implies that x0 ⪯ J(x0)Hence by )eorem 6 there exists xlowast isin A and ylowast isin B suchthat xlowast ⪯ylowast J(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B) that is a best proximity pair

Example 2 Take F(t s x(s)) (ts3)x(s) for all(t s) isin [0 1]2 and x isin Xρ

Consider the functions c η ϕ ψ and g defined on Xρ byc(t) minus 2t η(t) minus t ϕ(t) t ψ(t) 2t and g(t) (89)tfor all t isin [0 1] Denote the sets A x isin Xρ c⪯ x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 Let the mappingJ AcupB⟶ AcupB defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(32)

We will check that the mapping J satisfies conditions(c1) minus (c4)

(c1) For every t isin [0 1] the inequality η(t)lt 0ltϕ(t)

holds and dρ(A B) ρ(ϕ minus η) 111393810 (2t)2dt 43

(c2) It easy to see that

c(t)le minus g(t) + 11139461

0F(t s c(s))ds

η(t)ge minus g(t) + 11139461

0F(t s η(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ϕ(t)leg(t) + 11139461

0F(t s ϕ(s))ds

ψ(t)geg(t) + 11139461

0F(t sψ(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

then J(A) sub A and J(B) sub B

International Journal of Mathematics and Mathematical Sciences 7

(c3) We have 0leF(t s y(s)) minus F(t s

x(s)) (ts3)(y(s) minus x(s)) for all (t s) isin [0 1]2 and(x y) isin Xρ times Xρ such that xley(c4) Let x isin A and y isin B such that y⪯x we have

ρ(J(y) minus J(x)) 11139461

011139461

0(F(t s y(s)) minus F(t s y(s)))ds + 2g(t)1113888 1113889

2

dt

11139461

0(2g(t))

2 1 +

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

dt

(34)

Let u0 isin ]0 1] for any u isin R such that uge uo we haveuo(1 + u)le u(1 + uo) )us

1113946z

uo

uo(1 + u)dule 1113946z

uo

u 1 + uo( 1113857du

uo (1 + z)2

minus 1 + uo( 11138572

1113872 1113873le 1 + uo( 1113857 z2

minus u2o1113872 1113873

(1 + z)2 le 1 + uo( 1113857 1 + uo +

z2 minus u2o

uo

1113888 1113889

(1 + z)2 le 1 + uo( 1113857 1 +

1uo

z2

1113888 1113889

(35)

for all zge u0 Since 111393810(ts3)(y(s) minus x(s))ds

2g(t)ge 18 for all t isin [0 1] then

ρ(J(y) minus J(x))le 11139461

0(2g(t))

2 1 +18

1113874 1113875

1 +118

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠dt

le98

11139461

0(2g(t))

2 1 +932

11139461

0s(y(s) minus x(s))ds1113888 1113889

2⎛⎝ ⎞⎠dt

le3227

1 +932

11139461

0s2ds1113888 1113889 1113946

1

0(y(s) minus x(s))

2ds1113888 11138891113888 1113889

le3227

1 +332

ρ(y(s) minus x(s))1113874 1113875

le89dρ(A B) +

19ρ(y(s) minus x(s))

le ρ(y(s) minus x(s))

(36)

where x isin A and y isin B such that y⪯x which implies that J

is a monotone relatively ρ-nonexpansive mapping )usJ(η) η J(ϕ) ϕ and ρ(ϕ minus β) dρ(A B) that is (η ϕ) isa best proximity pair of J

6 Conclusions

)roughout the paper we have discussed some best prox-imity pair results for the class of monotone noncyclic rel-atively ρ-nonexpansive mappings in the framework ofmodular spaces equipped with a partial order defined by aρ-closed convex cone Our results are extensions of severalresults as in relevant items from the reference section of thispaper as well as in the literature in general In order to showthe utility of our main results we prove the existence of thesolution of an integral equation which involves monotonenoncyclic relatively ρ-nonexpansive mappinngs

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] H NakanoModulared Semi-Ordered Linear Spaces MaruzenCo Tokyo Japan 1st edition 1950

[2] W Orlicz and J Musielak Orlicz Spaces and Modular SpacesLecture Notes in Mathematics Springer-Verlag Berlin Ger-many 1983

[3] A C M Ran andM C B Reurings ldquoA fixed point theorem inpartially ordered sets and some applications to matrixequationsrdquo Proceedings of the AmericanMathematical Societyvol 132 no 5 pp 1435ndash1443 2003

[4] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[5] M R Alfuraidan M Bachar and M A Khamsi ldquoOnmonotone contraction mappings in modular function spacesrdquoFixed Point 1eory and Applications vol 2015 no 1 2015

[6] M E Gordji F Sajadian Y J Cho andM Ramezani ldquoA fixedpoint theorem for quasi-contraction mappings in partiallyorder modular spaces with applicationrdquo UPB Scientific Bul-letin Series A Applied Mathematics and Physics vol 76pp 135ndash146 2014

[7] B A Bin Dehaish and M A Khamsi ldquoBrowder and Gohdefixed point theorem for monotone nonexpansive mappingsrdquoFixed Point 1eory and Applications vol 2016 no 1 2016

[8] A A N Abdou and M A Khamsi ldquoFixed point theorems inmodular vector spacesrdquo1e Journal of Nonlinear Sciences andApplications vol 10 no 8 pp 4046ndash4057 2017

[9] P Kumam ldquoFixed point theorems for nonexpansive map-pings in modular spacesrdquo Archivum Mathematicum vol 40pp 345ndash353 2004

[10] K Chaira and S Lazaiz ldquoBest proximity pair and fixed pointresults for noncyclic mappings in modular spacesrdquo ArabJournal of Mathematical Sciences vol 24 no 2 p 147 2018

[11] B A B Dehaish and M A Khamsi ldquoOn monotone mappingsin modular function spacesrdquo Journal of Nonlinear Sciencesand Applications vol 9 no 8 pp 5219ndash5228 2016

[12] F E Browder ldquoNonexpansive nonlinear operators in aBanach spacerdquo Proceedings of the National Academy of Sci-ences vol 54 no 4 pp 1041ndash1044 1965

8 International Journal of Mathematics and Mathematical Sciences

We say that ρ is unique uniform convexity in everydirection (UUCED) if there exists η(s z)gt 0 for sge 0 and z

nonnull in Xρ such that

δ(r z)gt η(s z) for rgt s (17)

)e following proposition characterizes relationshipsbetween uniform convexity uniform convexity in everydirection and strict convexity of a modular

Proposition 3(a) (UCi) (resp (UUCi)) implies (UCED) (resp

(UUCED)) for i 1 2(b) (UUCED) implies (UCED)

(c) (UCED) implies (SC)

Proof It is quite easy to show (a) and (b) To prove (c) let xy isin Xρ such that xney If ρ(x)ne ρ(y) there is nothing toprove Otherwise we assume that ρ(x) ρ(y) rgt 0 andwe consider z y minus x Hence ρ(x + z) ρ(y) r Since ρis (UCED) then δ(r z)gt 0 which implies

1 minus1rρ x +

z

21113874 1113875ge δ(r z)gt 0 (18)

)us

ρx + y

21113874 1113875 ρ x +

y minus x

21113874 1113875le (1 minus δ(r z))rlt r (19)

)at is ρ((x + y)2)lt r (ρ(x) + ρ(y))2As an example of a modular which is (UUCED) but it is

not (UUC1) consider the function ρ defined on X RN by

ρ(x) ρ xn( 1113857( 1113857 1113944infin

n1xn

11138681113868111386811138681113868111386811138681113868n+1

(20)

Abdou and Khamsi [8] proved that this convex modularis (UUC2) which imply that ρ is (UUCED) but it is not(UUC1)

)e following lemma plays an important role in theproof of the next fixed point theorem To prove it we use thesame ideas as those used to prove Lemma 35 in [7]

Lemma 3 Let ρ be a convex modular uniformly continuousand (UUCED) Assume that Xρ satisfies the property (R) LetC be a nonempty ρ-closed convex and ρ-bounded subset of Xρand K be a nonempty ρ-closed convex subset of C Let (xk)kisinNbe a sequence in C and consider the ρ-type functionτ K⟶ [0 +infin] defined by

τ(y) lim supk⟶+infin

ρ y minus xk( 1113857 (21)

1en τ has a unique minimum point in K

Theorem 6 Let ρ be a convex modular (UUCED) and isuniformly continuous Assume that Xρ satisfies the property(R) Let (A B) be a nonempty convex and ρ-bounded pair inXρ such that B⪯0 is nonempty and ρ-closed LetT AcupB⟶ AcupB be a monotone noncyclic relativelyρ-nonexpansive mapping Assume that there exists x0 isin A⪯0such that x0 ⪯Tx0 1en T has a best proximity pair

Proof Let x0 isin A⪯0 such that x0 ⪯Tx0 Consider the se-quence (xn)nge 0 given by xn+1 (1 minus λ)xn + λTxn whereα isin (0 1) For any nge 0 set Bn y isin B⪯0 xn ⪯y1113864 1113865 Similarto )eorem 3 one has Binfin cap nge0Bn is nonempty ρ-closedconvex and T-invariant

Consider the type function τ Binfin⟶ [0 +infin] gener-ated by the sequence (xn)nisinN that isτ(z) lim supn⟶+infinρ(xn minus z) for all z isin Binfin By Lemma 3we know that τ has a unique minimum point y isin Binfin SinceT is monotone noncyclic relatively ρ-nonexpansive we haveτ(T(y)) lim sup

n

ρ xn+1 minus T(y)( 1113857

le lim supn

(1 minus λ)ρ xn minus T(y)( 1113857 + λρ T xn( 1113857 minus T(y)( 1113857( 1113857

le (1 minus λ) lim supn

ρ xn minus T(y)( 1113857

+ λ lim supn

ρ T xn( 1113857 minus T(y)( 1113857

le (1 minus λ)τ(T(y)) + λτ(y)

(22)which implies that τ(T(y))le τ(y) since λ isin (0 1))ereforeT(y) is also a minimum point of τ in Binfin )e uniqueness ofthe minimum point of τ implies that T(y) y

Since y isin Binfin then y isin B⪯0 )us there exists x isin A suchthat x⪯y and ρ(x minus y) dρ(A B) Using the same idea asin the proof of )eorem 3 we show that x is a fixed point ofT on A )erefore Tx x Ty y andρ(x minus y) dρ(A B) that is (x y) is a best proximity pair

In the case A B therefore A A⪯0 and B B⪯0 weobtain the following corollary)e result of this corollary hasalready been mentioned by Khamsi and Bin Dehaish (see)eorem 2) [7] in modular function spaces

Corollary 7 Let ρ be a convex modular (UUCED) anduniformly continuous Assume that Xρ satisfies the property(R) Let C be a nonempty ρ-closed convex and ρ-boundedsubset of Xρ Let T C⟶ C be a monotone ρ-nonexpansivemapping If there exists x0 isin C such that x0 ⪯T(x0) then T

has a fixed point

5 Application

In this section we present an application of our main resultsto prove the existence of the solution for nonlinear integralequations Let X L2([0 1]R) be the space of measurableand square-integrable functions on [0 1] Recall that X is aρ-complete modular space denoted Xρ whereρ(f) 1113938

[01]|f(t)|2dt is a modular Next we consider the

following integral equation

x(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(23)

where

sign(x(t)) 1 if x(t)ge 0

minus 1 if x(t)lt 01113896 (24)

6 International Journal of Mathematics and Mathematical Sciences

g isin Xρ such that g(t)ge 0 for all t isin [0 1] and F [0 1] times

[0 1] times R⟶ R is measurable in both variables t and s forevery x such that

11139461

0F(t s x(s))dsltinfin forallt isin [0 1] (25)

Recall that x⪯y if and only if y minus x isin P whereP x isin Xρ x(t)ge 0 ae1113966 1113967 Consider the ρ-closed convexsubsets A and B of Xρ given by A x isin Xρ c⪯x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 where c η ϕ and ψ are in XρSinceρ(x minus y)lemax ρ(η minus c) + ρ(ψ minus ϕ) ρ(ϕ minus η) + ρ(ψ minus c)1113864 1113865

ltinfin for all x y isin AcupB then (A B) is ρ-bounded More-over A ⪯0 η1113864 1113865 and B⪯0 ϕ1113864 1113865 are ρ-closed

Theorem 8 Let the mapping J be defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1] (26)

Assume that the following conditions are fulfilled

(c1) η(t)lt 0lt ϕ(t) for all t isin [0 1] and ρ(η minus ϕ)gt 0

(c2)c(t)le minus g(t) + 1113938

10 F(t s c(s))ds

η(t)ge minus g(t) + 111393810 F(t s η(s))ds

⎧⎨

⎩ and

ϕ(t)leg(t) + 111393810 F(t s ϕ(s))ds

ψ(t)geg(t) + 111393810 F(t sψ(s))ds

⎧⎨

⎩

(c3) F fulfill the following monotonicity condition0leF(t s x(s)) minus F(t s y(s)) for all (x y) isin Xρ suchthat y⪯ x

(c4) ρ(J(x) minus J(y))le ρ(x minus y) whenever x y isin AcupB

such that x⪯y

1en there exists xlowast isin A and ylowast isin B such that xlowast ⪯ylowastJ(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B)

Proof ρ is a (UUC1) convex modular then ρ is (UUCED)Moreover Xρ satisfies the property (R) since Xρ isρ-complete modular space and ρ is (UUC2) (from (d) ofProposition 1) and satisfies the Fatou property

ρ is uniformly continuous Indeed let εgt 0 and rgt 0 andwe prove that there exists αgt 0 such that

|ρ(y) minus ρ(x + y)|le ε (27)

where ρ(x)le α and ρ(y)le r For all t isin [0 1] we have||y(t)|2 minus |x(t) + y(t)|2|le |x(t)|2 + 2|y(t)||x(t)| )en

|ρ(y) minus ρ(x + y)|le1113946[01]

|x(t)|2dt + 21113946

[01]|y(t)||x(t)|dt

le ρ(x) + 2

ρ(x)

1113969

ρ(y)

1113969

le

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)

(28)

If ρ(x)le 1 then

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)le

ρ(x)

1113969

(1 + 2r

radic) (29)

Sinceρ(x)

1113968(1 + 2

r

radic)le ε this is equivalent to

ρ(x)1113968le ε(2

r

radic+ 1))erefore ρ(x)le (ε(2

r

radic+ 1))2We

take α min 1 (ε(2r

radic+ 1))21113966 1113967 Hence ρ is uniformly

continuous)e condition (c2) implies that J(A)subeA and J(B)subeB

Indeed let x isin A

J(x)(t) minus c(t) minus g(t) + 11139461

0F(t s x(s))ds minus c(t)

ge 11139461

0(F(t s x(s)) minus F(t s c(s)))dsge 0

(30)

)en c⪯ J(x) In addition

η(t) minus J(x)(t) η(t) minus minus g(t) + 1113946SF(t s x(s))ds1113874 1113875

ge1113946S(F(t s η(s)) minus F(t s x(s)))ds ge 0

(31)

)en J(x)⪯ η Hence J(A)subeA Using the same ar-guments we show that J(B)subeB From condition (c1) onehas dρ(A B)gt 0 By condition (c3) and since 0⪯g on AcupBwe conclude that J is monotone on AcupB

Consider x0 c Condition (c2) implies that x0 ⪯ J(x0)Hence by )eorem 6 there exists xlowast isin A and ylowast isin B suchthat xlowast ⪯ylowast J(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B) that is a best proximity pair

Example 2 Take F(t s x(s)) (ts3)x(s) for all(t s) isin [0 1]2 and x isin Xρ

Consider the functions c η ϕ ψ and g defined on Xρ byc(t) minus 2t η(t) minus t ϕ(t) t ψ(t) 2t and g(t) (89)tfor all t isin [0 1] Denote the sets A x isin Xρ c⪯ x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 Let the mappingJ AcupB⟶ AcupB defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(32)

We will check that the mapping J satisfies conditions(c1) minus (c4)

(c1) For every t isin [0 1] the inequality η(t)lt 0ltϕ(t)

holds and dρ(A B) ρ(ϕ minus η) 111393810 (2t)2dt 43

(c2) It easy to see that

c(t)le minus g(t) + 11139461

0F(t s c(s))ds

η(t)ge minus g(t) + 11139461

0F(t s η(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ϕ(t)leg(t) + 11139461

0F(t s ϕ(s))ds

ψ(t)geg(t) + 11139461

0F(t sψ(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

then J(A) sub A and J(B) sub B

International Journal of Mathematics and Mathematical Sciences 7

(c3) We have 0leF(t s y(s)) minus F(t s

x(s)) (ts3)(y(s) minus x(s)) for all (t s) isin [0 1]2 and(x y) isin Xρ times Xρ such that xley(c4) Let x isin A and y isin B such that y⪯x we have

ρ(J(y) minus J(x)) 11139461

011139461

0(F(t s y(s)) minus F(t s y(s)))ds + 2g(t)1113888 1113889

2

dt

11139461

0(2g(t))

2 1 +

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

dt

(34)

Let u0 isin ]0 1] for any u isin R such that uge uo we haveuo(1 + u)le u(1 + uo) )us

1113946z

uo

uo(1 + u)dule 1113946z

uo

u 1 + uo( 1113857du

uo (1 + z)2

minus 1 + uo( 11138572

1113872 1113873le 1 + uo( 1113857 z2

minus u2o1113872 1113873

(1 + z)2 le 1 + uo( 1113857 1 + uo +

z2 minus u2o

uo

1113888 1113889

(1 + z)2 le 1 + uo( 1113857 1 +

1uo

z2

1113888 1113889

(35)

for all zge u0 Since 111393810(ts3)(y(s) minus x(s))ds

2g(t)ge 18 for all t isin [0 1] then

ρ(J(y) minus J(x))le 11139461

0(2g(t))

2 1 +18

1113874 1113875

1 +118

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠dt

le98

11139461

0(2g(t))

2 1 +932

11139461

0s(y(s) minus x(s))ds1113888 1113889

2⎛⎝ ⎞⎠dt

le3227

1 +932

11139461

0s2ds1113888 1113889 1113946

1

0(y(s) minus x(s))

2ds1113888 11138891113888 1113889

le3227

1 +332

ρ(y(s) minus x(s))1113874 1113875

le89dρ(A B) +

19ρ(y(s) minus x(s))

le ρ(y(s) minus x(s))

(36)

where x isin A and y isin B such that y⪯x which implies that J

is a monotone relatively ρ-nonexpansive mapping )usJ(η) η J(ϕ) ϕ and ρ(ϕ minus β) dρ(A B) that is (η ϕ) isa best proximity pair of J

6 Conclusions

)roughout the paper we have discussed some best prox-imity pair results for the class of monotone noncyclic rel-atively ρ-nonexpansive mappings in the framework ofmodular spaces equipped with a partial order defined by aρ-closed convex cone Our results are extensions of severalresults as in relevant items from the reference section of thispaper as well as in the literature in general In order to showthe utility of our main results we prove the existence of thesolution of an integral equation which involves monotonenoncyclic relatively ρ-nonexpansive mappinngs

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] H NakanoModulared Semi-Ordered Linear Spaces MaruzenCo Tokyo Japan 1st edition 1950

[2] W Orlicz and J Musielak Orlicz Spaces and Modular SpacesLecture Notes in Mathematics Springer-Verlag Berlin Ger-many 1983

[3] A C M Ran andM C B Reurings ldquoA fixed point theorem inpartially ordered sets and some applications to matrixequationsrdquo Proceedings of the AmericanMathematical Societyvol 132 no 5 pp 1435ndash1443 2003

[4] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[5] M R Alfuraidan M Bachar and M A Khamsi ldquoOnmonotone contraction mappings in modular function spacesrdquoFixed Point 1eory and Applications vol 2015 no 1 2015

[6] M E Gordji F Sajadian Y J Cho andM Ramezani ldquoA fixedpoint theorem for quasi-contraction mappings in partiallyorder modular spaces with applicationrdquo UPB Scientific Bul-letin Series A Applied Mathematics and Physics vol 76pp 135ndash146 2014

[7] B A Bin Dehaish and M A Khamsi ldquoBrowder and Gohdefixed point theorem for monotone nonexpansive mappingsrdquoFixed Point 1eory and Applications vol 2016 no 1 2016

[8] A A N Abdou and M A Khamsi ldquoFixed point theorems inmodular vector spacesrdquo1e Journal of Nonlinear Sciences andApplications vol 10 no 8 pp 4046ndash4057 2017

[9] P Kumam ldquoFixed point theorems for nonexpansive map-pings in modular spacesrdquo Archivum Mathematicum vol 40pp 345ndash353 2004

[10] K Chaira and S Lazaiz ldquoBest proximity pair and fixed pointresults for noncyclic mappings in modular spacesrdquo ArabJournal of Mathematical Sciences vol 24 no 2 p 147 2018

[11] B A B Dehaish and M A Khamsi ldquoOn monotone mappingsin modular function spacesrdquo Journal of Nonlinear Sciencesand Applications vol 9 no 8 pp 5219ndash5228 2016

[12] F E Browder ldquoNonexpansive nonlinear operators in aBanach spacerdquo Proceedings of the National Academy of Sci-ences vol 54 no 4 pp 1041ndash1044 1965

8 International Journal of Mathematics and Mathematical Sciences

g isin Xρ such that g(t)ge 0 for all t isin [0 1] and F [0 1] times

[0 1] times R⟶ R is measurable in both variables t and s forevery x such that

11139461

0F(t s x(s))dsltinfin forallt isin [0 1] (25)

Recall that x⪯y if and only if y minus x isin P whereP x isin Xρ x(t)ge 0 ae1113966 1113967 Consider the ρ-closed convexsubsets A and B of Xρ given by A x isin Xρ c⪯x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 where c η ϕ and ψ are in XρSinceρ(x minus y)lemax ρ(η minus c) + ρ(ψ minus ϕ) ρ(ϕ minus η) + ρ(ψ minus c)1113864 1113865

ltinfin for all x y isin AcupB then (A B) is ρ-bounded More-over A ⪯0 η1113864 1113865 and B⪯0 ϕ1113864 1113865 are ρ-closed

Theorem 8 Let the mapping J be defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1] (26)

Assume that the following conditions are fulfilled

(c1) η(t)lt 0lt ϕ(t) for all t isin [0 1] and ρ(η minus ϕ)gt 0

(c2)c(t)le minus g(t) + 1113938

10 F(t s c(s))ds

η(t)ge minus g(t) + 111393810 F(t s η(s))ds

⎧⎨

⎩ and

ϕ(t)leg(t) + 111393810 F(t s ϕ(s))ds

ψ(t)geg(t) + 111393810 F(t sψ(s))ds

⎧⎨

⎩

(c3) F fulfill the following monotonicity condition0leF(t s x(s)) minus F(t s y(s)) for all (x y) isin Xρ suchthat y⪯ x

(c4) ρ(J(x) minus J(y))le ρ(x minus y) whenever x y isin AcupB

such that x⪯y

1en there exists xlowast isin A and ylowast isin B such that xlowast ⪯ylowastJ(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B)

Proof ρ is a (UUC1) convex modular then ρ is (UUCED)Moreover Xρ satisfies the property (R) since Xρ isρ-complete modular space and ρ is (UUC2) (from (d) ofProposition 1) and satisfies the Fatou property

ρ is uniformly continuous Indeed let εgt 0 and rgt 0 andwe prove that there exists αgt 0 such that

|ρ(y) minus ρ(x + y)|le ε (27)

where ρ(x)le α and ρ(y)le r For all t isin [0 1] we have||y(t)|2 minus |x(t) + y(t)|2|le |x(t)|2 + 2|y(t)||x(t)| )en

|ρ(y) minus ρ(x + y)|le1113946[01]

|x(t)|2dt + 21113946

[01]|y(t)||x(t)|dt

le ρ(x) + 2

ρ(x)

1113969

ρ(y)

1113969

le

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)

(28)

If ρ(x)le 1 then

ρ(x)

1113969

(

ρ(x)

1113969

+ 2

ρ(y)

1113969

)le

ρ(x)

1113969

(1 + 2r

radic) (29)

Sinceρ(x)

1113968(1 + 2

r

radic)le ε this is equivalent to

ρ(x)1113968le ε(2

r

radic+ 1))erefore ρ(x)le (ε(2

r

radic+ 1))2We

take α min 1 (ε(2r

radic+ 1))21113966 1113967 Hence ρ is uniformly

continuous)e condition (c2) implies that J(A)subeA and J(B)subeB

Indeed let x isin A

J(x)(t) minus c(t) minus g(t) + 11139461

0F(t s x(s))ds minus c(t)

ge 11139461

0(F(t s x(s)) minus F(t s c(s)))dsge 0

(30)

)en c⪯ J(x) In addition

η(t) minus J(x)(t) η(t) minus minus g(t) + 1113946SF(t s x(s))ds1113874 1113875

ge1113946S(F(t s η(s)) minus F(t s x(s)))ds ge 0

(31)

)en J(x)⪯ η Hence J(A)subeA Using the same ar-guments we show that J(B)subeB From condition (c1) onehas dρ(A B)gt 0 By condition (c3) and since 0⪯g on AcupBwe conclude that J is monotone on AcupB

Consider x0 c Condition (c2) implies that x0 ⪯ J(x0)Hence by )eorem 6 there exists xlowast isin A and ylowast isin B suchthat xlowast ⪯ylowast J(xlowast) xlowast J(ylowast) ylowast and ρ(xlowast minus ylowast) dρ(A B) that is a best proximity pair

Example 2 Take F(t s x(s)) (ts3)x(s) for all(t s) isin [0 1]2 and x isin Xρ

Consider the functions c η ϕ ψ and g defined on Xρ byc(t) minus 2t η(t) minus t ϕ(t) t ψ(t) 2t and g(t) (89)tfor all t isin [0 1] Denote the sets A x isin Xρ c⪯ x⪯ η1113966 1113967 andB y isin Xρ ϕ⪯y⪯ψ1113966 1113967 Let the mappingJ AcupB⟶ AcupB defined by

J(x)(t) sign(x(t))g(t) + 11139461

0F(t s x(s))ds t isin [0 1]

(32)

We will check that the mapping J satisfies conditions(c1) minus (c4)

(c1) For every t isin [0 1] the inequality η(t)lt 0ltϕ(t)

holds and dρ(A B) ρ(ϕ minus η) 111393810 (2t)2dt 43

(c2) It easy to see that

c(t)le minus g(t) + 11139461

0F(t s c(s))ds

η(t)ge minus g(t) + 11139461

0F(t s η(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ϕ(t)leg(t) + 11139461

0F(t s ϕ(s))ds

ψ(t)geg(t) + 11139461

0F(t sψ(s))ds

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

then J(A) sub A and J(B) sub B

International Journal of Mathematics and Mathematical Sciences 7

(c3) We have 0leF(t s y(s)) minus F(t s

x(s)) (ts3)(y(s) minus x(s)) for all (t s) isin [0 1]2 and(x y) isin Xρ times Xρ such that xley(c4) Let x isin A and y isin B such that y⪯x we have

ρ(J(y) minus J(x)) 11139461

011139461

0(F(t s y(s)) minus F(t s y(s)))ds + 2g(t)1113888 1113889

2

dt

11139461

0(2g(t))

2 1 +

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

dt

(34)

Let u0 isin ]0 1] for any u isin R such that uge uo we haveuo(1 + u)le u(1 + uo) )us

1113946z

uo

uo(1 + u)dule 1113946z

uo

u 1 + uo( 1113857du

uo (1 + z)2

minus 1 + uo( 11138572

1113872 1113873le 1 + uo( 1113857 z2

minus u2o1113872 1113873

(1 + z)2 le 1 + uo( 1113857 1 + uo +

z2 minus u2o

uo

1113888 1113889

(1 + z)2 le 1 + uo( 1113857 1 +

1uo

z2

1113888 1113889

(35)

for all zge u0 Since 111393810(ts3)(y(s) minus x(s))ds

2g(t)ge 18 for all t isin [0 1] then

ρ(J(y) minus J(x))le 11139461

0(2g(t))

2 1 +18

1113874 1113875

1 +118

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠dt

le98

11139461

0(2g(t))

2 1 +932

11139461

0s(y(s) minus x(s))ds1113888 1113889

2⎛⎝ ⎞⎠dt

le3227

1 +932

11139461

0s2ds1113888 1113889 1113946

1

0(y(s) minus x(s))

2ds1113888 11138891113888 1113889

le3227

1 +332

ρ(y(s) minus x(s))1113874 1113875

le89dρ(A B) +

19ρ(y(s) minus x(s))

le ρ(y(s) minus x(s))

(36)

where x isin A and y isin B such that y⪯x which implies that J

is a monotone relatively ρ-nonexpansive mapping )usJ(η) η J(ϕ) ϕ and ρ(ϕ minus β) dρ(A B) that is (η ϕ) isa best proximity pair of J

6 Conclusions

)roughout the paper we have discussed some best prox-imity pair results for the class of monotone noncyclic rel-atively ρ-nonexpansive mappings in the framework ofmodular spaces equipped with a partial order defined by aρ-closed convex cone Our results are extensions of severalresults as in relevant items from the reference section of thispaper as well as in the literature in general In order to showthe utility of our main results we prove the existence of thesolution of an integral equation which involves monotonenoncyclic relatively ρ-nonexpansive mappinngs

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] H NakanoModulared Semi-Ordered Linear Spaces MaruzenCo Tokyo Japan 1st edition 1950

[2] W Orlicz and J Musielak Orlicz Spaces and Modular SpacesLecture Notes in Mathematics Springer-Verlag Berlin Ger-many 1983

[3] A C M Ran andM C B Reurings ldquoA fixed point theorem inpartially ordered sets and some applications to matrixequationsrdquo Proceedings of the AmericanMathematical Societyvol 132 no 5 pp 1435ndash1443 2003

[4] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[5] M R Alfuraidan M Bachar and M A Khamsi ldquoOnmonotone contraction mappings in modular function spacesrdquoFixed Point 1eory and Applications vol 2015 no 1 2015

[6] M E Gordji F Sajadian Y J Cho andM Ramezani ldquoA fixedpoint theorem for quasi-contraction mappings in partiallyorder modular spaces with applicationrdquo UPB Scientific Bul-letin Series A Applied Mathematics and Physics vol 76pp 135ndash146 2014

[7] B A Bin Dehaish and M A Khamsi ldquoBrowder and Gohdefixed point theorem for monotone nonexpansive mappingsrdquoFixed Point 1eory and Applications vol 2016 no 1 2016

[8] A A N Abdou and M A Khamsi ldquoFixed point theorems inmodular vector spacesrdquo1e Journal of Nonlinear Sciences andApplications vol 10 no 8 pp 4046ndash4057 2017

[9] P Kumam ldquoFixed point theorems for nonexpansive map-pings in modular spacesrdquo Archivum Mathematicum vol 40pp 345ndash353 2004

[10] K Chaira and S Lazaiz ldquoBest proximity pair and fixed pointresults for noncyclic mappings in modular spacesrdquo ArabJournal of Mathematical Sciences vol 24 no 2 p 147 2018

[11] B A B Dehaish and M A Khamsi ldquoOn monotone mappingsin modular function spacesrdquo Journal of Nonlinear Sciencesand Applications vol 9 no 8 pp 5219ndash5228 2016

[12] F E Browder ldquoNonexpansive nonlinear operators in aBanach spacerdquo Proceedings of the National Academy of Sci-ences vol 54 no 4 pp 1041ndash1044 1965

8 International Journal of Mathematics and Mathematical Sciences

(c3) We have 0leF(t s y(s)) minus F(t s

x(s)) (ts3)(y(s) minus x(s)) for all (t s) isin [0 1]2 and(x y) isin Xρ times Xρ such that xley(c4) Let x isin A and y isin B such that y⪯x we have

ρ(J(y) minus J(x)) 11139461

011139461

0(F(t s y(s)) minus F(t s y(s)))ds + 2g(t)1113888 1113889

2

dt

11139461

0(2g(t))

2 1 +

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

dt

(34)

Let u0 isin ]0 1] for any u isin R such that uge uo we haveuo(1 + u)le u(1 + uo) )us

1113946z

uo

uo(1 + u)dule 1113946z

uo

u 1 + uo( 1113857du

uo (1 + z)2

minus 1 + uo( 11138572

1113872 1113873le 1 + uo( 1113857 z2

minus u2o1113872 1113873

(1 + z)2 le 1 + uo( 1113857 1 + uo +

z2 minus u2o

uo

1113888 1113889

(1 + z)2 le 1 + uo( 1113857 1 +

1uo

z2

1113888 1113889

(35)

for all zge u0 Since 111393810(ts3)(y(s) minus x(s))ds

2g(t)ge 18 for all t isin [0 1] then

ρ(J(y) minus J(x))le 11139461

0(2g(t))

2 1 +18

1113874 1113875

1 +118

11139461

0(ts3)(y(s) minus x(s))ds

2g(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠dt

le98

11139461

0(2g(t))

2 1 +932

11139461

0s(y(s) minus x(s))ds1113888 1113889

2⎛⎝ ⎞⎠dt

le3227

1 +932

11139461

0s2ds1113888 1113889 1113946

1

0(y(s) minus x(s))

2ds1113888 11138891113888 1113889

le3227

1 +332

ρ(y(s) minus x(s))1113874 1113875

le89dρ(A B) +

19ρ(y(s) minus x(s))

le ρ(y(s) minus x(s))

(36)

where x isin A and y isin B such that y⪯x which implies that J

is a monotone relatively ρ-nonexpansive mapping )usJ(η) η J(ϕ) ϕ and ρ(ϕ minus β) dρ(A B) that is (η ϕ) isa best proximity pair of J

6 Conclusions

)roughout the paper we have discussed some best prox-imity pair results for the class of monotone noncyclic rel-atively ρ-nonexpansive mappings in the framework ofmodular spaces equipped with a partial order defined by aρ-closed convex cone Our results are extensions of severalresults as in relevant items from the reference section of thispaper as well as in the literature in general In order to showthe utility of our main results we prove the existence of thesolution of an integral equation which involves monotonenoncyclic relatively ρ-nonexpansive mappinngs

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] H NakanoModulared Semi-Ordered Linear Spaces MaruzenCo Tokyo Japan 1st edition 1950

[2] W Orlicz and J Musielak Orlicz Spaces and Modular SpacesLecture Notes in Mathematics Springer-Verlag Berlin Ger-many 1983

[3] A C M Ran andM C B Reurings ldquoA fixed point theorem inpartially ordered sets and some applications to matrixequationsrdquo Proceedings of the AmericanMathematical Societyvol 132 no 5 pp 1435ndash1443 2003

[4] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[5] M R Alfuraidan M Bachar and M A Khamsi ldquoOnmonotone contraction mappings in modular function spacesrdquoFixed Point 1eory and Applications vol 2015 no 1 2015

[6] M E Gordji F Sajadian Y J Cho andM Ramezani ldquoA fixedpoint theorem for quasi-contraction mappings in partiallyorder modular spaces with applicationrdquo UPB Scientific Bul-letin Series A Applied Mathematics and Physics vol 76pp 135ndash146 2014

[7] B A Bin Dehaish and M A Khamsi ldquoBrowder and Gohdefixed point theorem for monotone nonexpansive mappingsrdquoFixed Point 1eory and Applications vol 2016 no 1 2016

[8] A A N Abdou and M A Khamsi ldquoFixed point theorems inmodular vector spacesrdquo1e Journal of Nonlinear Sciences andApplications vol 10 no 8 pp 4046ndash4057 2017

[9] P Kumam ldquoFixed point theorems for nonexpansive map-pings in modular spacesrdquo Archivum Mathematicum vol 40pp 345ndash353 2004

[10] K Chaira and S Lazaiz ldquoBest proximity pair and fixed pointresults for noncyclic mappings in modular spacesrdquo ArabJournal of Mathematical Sciences vol 24 no 2 p 147 2018

[11] B A B Dehaish and M A Khamsi ldquoOn monotone mappingsin modular function spacesrdquo Journal of Nonlinear Sciencesand Applications vol 9 no 8 pp 5219ndash5228 2016

[12] F E Browder ldquoNonexpansive nonlinear operators in aBanach spacerdquo Proceedings of the National Academy of Sci-ences vol 54 no 4 pp 1041ndash1044 1965

8 International Journal of Mathematics and Mathematical Sciences