# Fixed Point Theorems for Noncyclic Monotone Relativelyρ-Nonexpansive Mappings in Modular Spaces

Authors: Nour-eddine El Harmouchi; Karim Chaira; El Miloudi Marhrani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

In this paper, we obtain some fixed point results for noncyclic monotoneρ-nonexpansive mappings in uniformly convex modular spaces and uniformly convex in every direction modular spaces. As an application, we prove the existence of the solution of an integral equation.


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## Body

## 1. Introduction

The notion of modular spaces, as a generalization of Banach spaces, was firstly introduced by Nakano [1] in connection with the theory of ordered spaces. These spaces were developed and generalized by Orlicz and Musielak [2].It is well known that fixed point theory is an active field of research, a powerful tool in solving integral and differential equations. Following the publications of Ran and Reuring [3] and Neito and Rrodriguez-Lopez [4], fixed point theory in partially ordered modular spaces has recently received a good attention from researchers.Alfuraidan et al. [5] gave a modular version of Ran and Reuring fixed point theorem and proved the existence of fixed point of monotone contraction mappings in modular function spaces. Thereafter, they gave an extension of their main results for pointwise monotone contractions. In [6], Gordji et al. have proved that any quasi-contraction mappings in partially ordered modular spaces without Δ2-condition have a fixed point.In 2016, Bin Dehaish and Khamsi proved in [7] the following theorems.Theorem 1.
(see [7]). Let ρ∈ℛ be a (UUC1) and C be a nonempty convex ρ-closed ρ-bounded subset of Lρ not reduced to one point. Let T:C⟶C be a monotone ρ-nonexpansive mapping and ρ-continuous. Assume there exists f0∈C such that f0 and Tf0 are comparable. Then, T has a fixed point.Theorem 2.
(see [7]). Assume that ρ∈ℛ is UUCED and uniformly continuous. Assume that Lρ satisfies the property (R). Let C be a nonempty convex ρ-closed ρ-bounded subset of Lρ not reduced to one point. Let T:C⟶C be a monotone ρ-nonexpansive mapping and ρ-continuous. Assume there exists f0∈C such that f0 and Tf0 are comparable. Then, T has a fixed point.
These theorems are a generalization to modular function spaces of Browder and Göhde fixed point theorem for monotone nonexpansive mappings in Banach spaces.
A mappingT:A∪B⟶A∪B is said to be noncyclic provided that TA⊆A and TB⊆B, where A,B is a nonempty pair in a modular space. When we consider this type of mappings, it is interesting to ask if it is possible to find a pair x∗,y∗∈A×B such that(1)Tx∗=x∗,Ty∗=y∗,ρx∗−y∗=dρA,B,that is, x∗,y∗ is a best proximity pair.
In this paper, we discuss some best proximity pair results for the class of monotone noncyclic relativelyρ-nonexpansive mappings in the framework of modular spaces equipped with a partial order defined by a ρ-closed convex cone P, that is, for any x and y in the modular space Xρ one has x⪯y if and only if y−x in P. We took this ordering because in the case of modular spaces the order intervals are not convex closed as in modular function spaces (see Theorem 2.4 in [7]). Our results generalize Theorems 1 and 2 and others for monotone ρ-nonexpansive mappings to the case of noncyclic monotone relatively ρ-nonexpansive mappings in modular spaces.
To illustrate the effectiveness of our main results, we give an application to an integral equation which involves noncyclic monotoneρ-nonexpansive mappings.


## 2. Materials and Methods

The materials used in this study are obtained from journal articles and books existing on the Internet. The main tools used in this manuscript are the uniform convexity and the uniform convexity in every direction of a modularρ. The aim of this work is to extend some fixed point theorems for nonexansive mappings to best proximity pair for monotone noncyclic relatively ρ-nonexpansive mappings in modular spaces.The paper is organized as follows. In Section3, we begin with recollection of some basic definitions and lemmas with corresponding references that will be used in the sequel. Thereafter, we give the main results of the paper with some examples. To validate the utility of our results, we give in Section 4 an application to an integral equation.


## 3. Preliminaries

Throughout this work,X stands for a linear vector space on the field ℝ. Let us start with some preliminaries and notations.Definition 1.
(see [8]). A function ρ:X⟶0,+∞ is called a modular if the following holds:(1)
ρx=0 if and only if x=0(2)
ρ−x=ρx(3)
ραx+1−αy≤ρx+ρy, for any α∈0,1 and for any x and y in X
If (3) is replaced by (3′)ραx+1−αy≤αρx+1−αρy for any α∈0,1 and x and y in X, then ρ is called a convex modular.
The modular space is defined asXρ=x∈X:limλ⟶0ρλx=0. Throughout this paper, we will assume that the modular ρ is convex. The Luxemburg norm in Xρ is defined as(2)xρ=infλ>0:ρxλ≤1.
Associated to a modular, we introduce some basic notions needed throughout this work.Definition 2.
(see [8]). Let ρ be a modular defined on a vector space X:(1)
We say that a sequencexnn∈ℕ⊂Xρ is ρ-convergent to x∈Xρ if and only if limn⟶∞ρxn−x=0. Note that the limit is unique.(2)
A sequencexnn⊂Xρ is called ρ-Cauchy if ρxn−xm⟶0 as n,m⟶+∞.(3)
We say thatXρ is ρ-complete if and only if any ρ-Cauchy sequence is ρ-convergent.(4)
A subsetC of Xρ is said ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belong to C.(5)
A subsetC of Xρ is said ρ-bounded if we have δρC=supρx−y:x,y∈C<∞.(6)
A subsetK of Xρ is said ρ-sequentially compact if any sequence xnn of C has a subsequence ρ-convergent to a point x∈C.(7)
We say thatρ satisfies the Fatou property if ρx−y≤limn⟶+∞ρx−yn whenever ynnρ-converges to y for any x, y, and yn in Xρ.
Let us note thatρ-convergence does not imply ρ-Cauchy condition. Also, xn⟶ρx does not imply in general λxn⟶ρλx, for every λ>1.Definition 3.
Letρ be a modular and C be a nonempty subset of the modular space Xρ. A mapping T:C⪯C is said to be(a)
Monotone, ifTx⪯Ty for any x, y∈C such that x⪯y.(b)
Monotoneρ-nonexpansive, if T is monotone and(3)ρTx−Ty⪯ρx−y,whenever x, y∈Xρ and x⪯y.
Recall thatT:C⟶C is said to be ρ-continuous if Txnnρ-converges to Tx whenever xnnρ-converges to x. It is not true that a monotone ρ-nonexpansive mapping is ρ-continuous, since this result is not true in general when ρ is a norm.
LetA and B be nonempty subsets of a modular space Xρ. We adopt the notation dρA,B=infρx−y:x∈A,y∈B.
A pairA,B is said to satisfy a property if both A and B satisfy that property. For instance, A,B is ρ-closed (resp. convex, ρ-bounded) if and only if A and B are ρ-closed (resp. convex, ρ-bounded). A pair A,B is not reduced to one point 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 r>0 and ε>0. Define, for i∈1,2,(4)Dir,ε=x,y∈Xρ×Xρ:ρx≤r,ρy≤r,ρx−yi≥rε.
IfDir,ε≠∅, let(5)δir,ε=inf1−1rρx+y2:x,y∈Di.
IfDir,ε=∅, we set δir,ε=1.(i)
We say thatρ is uniformly convex (UCi) if for every r>0 and ε>0, and we have δir,ε>0(ii)
We say thatρ is unique uniformly convex (UUCi) if for all s≥0 and ε>0, and there exists ηs,ε>0 such that δir,ε>ηs,ε,forr>s(iii)
We say thatρ is strictly convex (SC), if for every x, y∈Xρ such that ρx=ρy and ρx+y/2=ρx+ρy/2, and we have x=y
The following proposition characterizes the relationship between the above notionsProposition 1.
(see [8]).(a)
UUCi implies UCi for i=1,2(b)
δ1r,ε≤δ2r,ε for r>0 and ε>0(c)
UC1 implies UC2 implies SC(d)
UUC1 implies UUC2Definition 5.
(see [9]). Let ρ be a modular. We say that the modular space Xρ satisfies the property (R) if and only if for every decreasing sequence Cnn∈ℕ of nonempty ρ-closed convex and ρ-bounded subsets of Xρ has a nonempty intersection.Lemma 1.
(see [8]). Let ρ be a convex modular satisfying the Fatou property. Assume that Xρ is ρ-complete and ρ is (UUC2). Then, Xρ satisfies the property (R).Definition 6 (see [8]).
Letxnn be a sequence in Xρ and K be a nonempty subset of Xρ. The function τ:K⟶0,∞ defined by τx=limsupn⟶∞ρxn−x is called a ρ-type function.
Note that theρ-type function τ is convex since ρ is convex. A sequence cnn⊂K is called a minimizing sequence of τ if limn⟶+∞τcn=infx∈Kτx.
The following result, found in [8], plays a crucial role in the proof of many fixed point results in modular spaces.Lemma 2 (see [8]).
Letρ be a convex modular (UUC1) satisfying the Fatou property and Xρ a ρ-compete modular space. Let C be a nonempty ρ-closed convex subset of Xρ. Consider the ρ-type function τ:C⟶0,+∞ generated by a sequence xnn in Xρ. Assume that τ0=infτx:x∈C<∞. Then, all the minimizing sequences of τ are ρ-convergent to the same limit.


## 4. Main Results

A subsetP⊂Xρ is a pointed ρ-closed convex cone, if P is a nonempty ρ-closed subset of Xρ satisfying the following properties:(i)
P+P⊂P(ii)
λP⊂P , for all λ∈ℝ+(iii)
P∩−P=0UsingP, we define an ordering on Xρ by x⪯yif and only ify−x∈P.Letρ be a convex modular and T:C⟶C be a monotone mapping, where C is a nonempty convex subset of the modular space Xρ. Let x0∈C and λ∈0,1. Consider Krasnoselskii-Ishikawa iteration sequence xnn∈ℕ in C defined by xn+1=1−λxn+λTxn for all n∈ℕ. Assume that x0⪯Tx0 (resp. Tx0⪯x0). By the definition of the partial ordering ⪯, we obtain x0⪯x1⪯Tx0 (resp. Tx0⪯x1⪯x0).SinceT is monotone, one has Tx0⪯Tx1 (resp. Tx1⪯Tx0). By induction, we prove that(6)xn⪯xn+1⪯Txn⪯Txn+1resp.Txn+1⪯Txn⪯xn+1⪯xn,for alln∈ℕ.Let us introduce the class of mappings for which the problem of fixed point will be considered.Definition 7.
LetA,B be a nonempty pair of a modular space Xρ. A mapping T:A∪B⟶A∪B is said to be monotone noncyclic relatively ρ-nonexpansive if it satisfies the following conditions:(i)
For allx,y∈A∪B2, x⪯y implies that Tx⪯Ty(ii)
TA⊆A and TB⊆B(iii)
For allx∈A and y∈B such that x⪯y, we have ρTx−Ty≤ρx−yDefinition 8 (see [10]).
LetA,B be a pair of a modular space Xρ and T:A∪B⟶A∪B be a noncyclic mapping. A pair x,y∈A×B is said to be a best proximity pair for the noncyclic mapping T if(7)Tx=x,Ty=y,ρx−y=dρA,B.Definition 9.
A spaceXρ is said to satisfy the property p, if xn⟶ρx and yn⟶ρy with xn⪯yn for all n∈ℕ; then, x⪯y.
We useA0⪯,B0⪯ to denote the ordered ρ-proximal pair obtained from A,B by(8)A0⪯=x∈A:ρx−y=dρA,Bandx⪯yfor somey∈B,B0⪯=y∈B:ρx−y=dρA,Bandx⪯yfor somex∈A.
Note that ifA0⪯ is nonempty if and only if B0⪯ is nonempty. Indeed, if x0∈A0⪯ then there exists y∈B such that x0⪯y and ρx0−y=dρA,B. Thus, for this y we take x=x0∈A, we have x⪯y and ρx−y=dρA,B, that is, y∈B0⪯. As the same we prove the converse.Proposition 2.
Letρ be a convex modular satisfying the Fatou property and Xρ satisfies the property (P). Let A,B be a nonempty convex pair of Xρ. Assume that A0⪯ is nonempty:(i)
IfA,B is ρ-sequentially compact, then A0⪯,B0⪯ is ρ-sequentially compact(ii)
A0⪯,B0⪯ is convex(iii)
dρA0⪯,B0⪯=dρA,B(iv)
IfT:C⟶C is a noncyclic monotone relatively ρ-nonexpansive mapping, then TA0⪯⊆A0⪯ and TB0⪯⊆B0⪯, that is, A0⪯ and B0⪯ are T-invariantProof.
It is quite easy to verify (iii):(i)
Letxnn be a sequence in A0⪯. Then, there exists a sequence ynn in B such that xn⪯yn and ρxn−yn=dρA,B for all n≥0. Since A,B is ρ-sequentially compact, there exists subsequences xnkk and ynkk of xnn and ynn, respectively, such that xnk⪯¯ynk for all k≥0, and ρ-converge to x∈A and y∈B, respectively.From property (P), one has x⪯¯y. Since ρ satisfies the Fatou property, then(9)dρA,B≤ρx−y≤liminfk⟶∞ρxnk−ynk=dρA,B.Thus,ρx−y=dρA,B. Therefore, A0⪯ is ρ-sequentially compact. Likewise, we prove that B0⪯ is also ρ-sequentially compact.(ii)
Letx1, x2∈A0⪯, δ∈0,1 and set z=δx1+1−δx2. Since x1, x2∈A0⪯, then there exists y1, y2∈B such that x1⪯y1, x2⪯y2 and ρx1−y1=ρx2−y2=dρA,B. Moreover, δx1⪯δy1 and 1−δx2⪯1−δy2. Hence, z=δx1+1−δx2⪯z′=δy1+1−δy2. Since ρ is convex, then(10)dρA,B≤ρz−z′=ρδx1+1−δx2−δx1+1−δx2≤δρx1−y1+1−δρx2−y2≤δdρA,B+1−δdρA,B=dρA,B.Then,ρz−z′=dρA,B. Hence, z∈A0⪯. Therefore, A0⪯ is convex. Using the same argument we prove that B0⪯ is convex.(iii)
Letx∈A0⪯, then there exists y∈B such that x⪯y and ρx−y=dρA,B. Since T is monotone noncyclic relatively ρ-nonexpansive, then Tx⪯Ty and dρA,B≤ρTx−Ty≤ρx−y=dρA,B. Thus, ρTx−Ty=dρA,B. Therefore, Tx∈A0⪯. As the same way, we obtain TB0⪯⊆B0⪯.Theorem 3.
Letρ be a convex modular UUC1 satisfying the Fatou property and Xρ be a ρ-complete modular space. Let A,B be a nonempty convex and ρ-bounded pair of Xρ such that the subset B0⪯ is nonempty ρ-closed. Let T:A∪B⟶A∪B be a monotone noncyclic relatively ρ-nonexpansive mapping and ρ-continuous on B. Assume that there exists x0∈A0⪯ such that x0⪯Tx0. Then, T has a best proximity pair.Proof.
Letx0∈A0⪯ such that x0⪯Tx0. Consider the sequence xnn defined by xn+1=λTxn+1−λxn for all n≥0 and λ∈0,1. Set Cn=y∈B0⪯:xn⪯y for any n≥0. The sequence Cnn∈ℕ is a decreasing sequence of nonempty ρ-closed convex ρ-bounded subsets.
SinceA0⪯ is T-invariant, then Tx0∈A0⪯. Using the convexity of A0⪯, one has x1=λTx0+1−λx0∈A0⪯. Assume that xn∈A0⪯, then Txn∈A0⪯. Again the convexity of A0⪯, implies that xn+1=λTxn+1−λxn∈A0⪯. Therefore, xn∈A0⪯ for all n≥0. Hence, for all n≥0, there exists y∈B such that xn⪯y and ρxn−y=dρA,B. Thus, there exists y∈B0⪯ such that xn⪯y. Therefore, Cn is nonempty for all n≥0. Moreover, since xn⪯xn+1 then Cnn∈ℕ is a decreasing sequence.
Letcpp≥0 be a sequence in Cnρ-converges to b∈B0⪯. We have cp∈Cn, then cp−xn∈P and cp∈B0⪯, for all p≥0. Since B0⪯ is ρ-closed, one has b∈B0⪯. Moreover, cp−xn⟶ρb−xn. Thus, b−xn∈P since P is ρ-closed. Hence, b∈Cn. Therefore, Cn is ρ-closed for all n≥0.
AsP and B0⪯ are convex, it is easy to see that Cn is convex for all n≥0. Furthermore, Cn is ρ-bounded for all n≥0.
By Lemma1, Xρ satisfies the property (R). Then, C∞=∩n≥0Cn is nonempty and ρ-closed convex.
Moreover,TC∞⊆C∞. In fact, let z∈C∞; then, xn⪯z for all n≥0. Since T is monotone and xn⪯Txn, one has xn⪯Tz for all n≥0. Moreover, Tz∈B0⪯ since B0⪯ is T-invariant. Hence, TC∞⊆C∞.
Consider the type functionτ:C∞⟶0,+∞ generated by the sequence xnn∈ℕ, that is, τz=limsupn⟶+∞ρxn−z for all z∈C∞. Let zpp∈ℕ be a minimizing sequence of τ. By Lemma 2, zppρ-converges to a point y∈C∞ since C∞ is ρ-closed. Otherwise,(11)τTzp=limsupn⟶+∞ρxn+1−Tzp≤1−λlimsupn⟶+∞ρxn−Tzp+λlimsupn⟶+∞ρxn−zp≤1−λτTzp+λτzp.
Hence,τTzp≤τzp. Then, Tzpp≥0 is also a minimizing sequence of τ. By Lemma 2, Tzppρ-converges to y. As T is ρ-continuous on B, then Tzppρ-converges to Ty. Therefore, Ty=y. Then, there exists y∈B0⪯ such that Ty=y and xn⪯y, for all n∈ℕ. By the definition of B0⪯, there exists x∈A such that x⪯y and ρx−y=dρA,B. In order to finish the proof, we show that x is a fixed point of T on A. We have(12)dρA,B≤ρTx−y=ρTx−Ty≤ρx−y=dρA,B.
Then,ρTx−y=dρA,B. Moreover,(13)dρA,B≤ρTx−y2+x−y2≤ρTx−y2+ρx−y2=dρA,B.
Thus,ρTx−y+x−y/2=dρA,B. Since ρ is (UUC2), it is (SC). Hence, Tx−y=x−y, which implies that Tx=x. Therefore, x,y is a best proximity pair of the mapping T.To illustrate Theorem3, we consider the following example.Example 1.
LetX=ℝ2, we define the modular ρ:X⟶0,+∞ by ρx=x12+x22 for all x=x1,x2∈ℝ2. The modular ρ is convex satisfying the Fatou property and (UUC1), and Xρ is a ρ-complete modular space. Consider the ρ-closed convex cone P=x1,x2∈Xρ:x1,x2≥0,0. Let A=−1,−2;−1,2 and B=1,−4;1,0. The pair A,B is ρ-closed convex and ρ-bounded in Xρ with dρA,B=4. Moreover, A0⪯=−1,−2;−1,0 and B0⪯=1,−2;1,0 are ρ-closed. Furthermore, we have(i)
For allx∈−1,−2;−1,0, there exists y∈1,−2;1,0 such that x⪯¯y(ii)
For the other cases the elements ofA and B are not comparable
Consider the mappingT:A∪B⟶A∪B defined by(14)Tx1,x2=−1,0,ifx1,x2∈−1,−2;−1,0,x1,x2,ifx1,x2∈−1,0;−1,2,Ty1,y2=1,0,ify1,y2∈B.
The mappingT is monotone noncyclic relatively ρ-nonexpansive ρ-continuous on A∪B, and for x0=−1,−2∈A0≤ we have x0⪯¯Tx0. Then, T has a best proximity pair 1,0,−1,0, where dρA,B=ρ1,0−−1,0=4.
The following corollary is an immediate consequence of Theorem3, and it suffices to take A=B; therefore, A=A0⪯ and B=B0⪯. It will be considered as a generalization of Browder and Göhde fixed point theorem for monotone ρ-nonexpansive mappings in modular spaces (see [7, 11, 12]). This corollary result has already been mentioned by Bin Dehaish and Khamsi (see Theorem 1) [7] in the framework of modular function spaces.Corollary 4.
Letρ be a convex modular UUC1 satisfying the Fatou property and Xρ be a ρ-complete modular space. Let C be a nonempty ρ-closed convex ρ-bounded subset of Xρ and T:C⟶C be a monotone ρ-nonexpansive mapping and ρ-continuous. If there exists x0∈C such that x0⪯Tx0, then T has a fixed point.Particular case of normed vector spaces:Remark 1.
(i)
IfX,⋅ is a normed vector space, then the norm ⋅ is continuous and satisfies the Fatou property. Moreover, if X,⋅ is a uniformly convex Banach space, then X,⋅ is reflexive. Hence, it satisfies the property (R).(ii)
IfX,⋅ is reflexive and A,B is a nonempty closed and bounded pair of X, then A0⪯,B0⪯ is also closed convex pair of X.
First case: ifA0⪯,B0⪯ is empty, then there is nothing to prove.
Second case: assume thatA0⪯,B0⪯ is nonempty. Let xnn be a sequence in A0⪯ such that xnρ-converges to x∈A. There exists a sequence ynn in B such that xn⪯yn and xn−yn=dA,B, for all n≥0. Since X is reflexive and ynn is bounded, then there exists a subsequence yφnn in B which converges weakly to y∈B. Since xφnn converges weakly to x, then(15)x−y≤liminfn⟶∞xφn−yφn.
Thus,x−y=dA,B and x⪯y. Hence, x∈A0⪯. Therefore, A0⪯ is closed. Using the same argument, we prove that B0⪯ is also closed.
In the context of uniformly convex normed vector spaces, Theorem3 is given as follows.Theorem 5.
LetA,B be a nonempty convex and bounded pair of a partially ordered uniformly convex Banach space X such that B0⪯ nonempty and closed. Let T:A∪B⟶A∪B be a noncyclic monotone relatively nonexpansive mapping. Assume that there exists x0∈A0⪯ such that x0⪯Tx0. Then, T has a best proximity pair.
In order to weaken the assumptions of Theorem3, we introduce the notions of uniform continuity and uniform convexity in every direction.Definition 10.
A modularρ is said to be uniformly continuous, if for any ε>0 and R>0 there exists η>0 such that ρy−ρx+y≤ε, whenever ρx≤η and ρy≤R.Definition 11.
Letρ be a modular. We say that ρ is uniformly convex in every direction (UCED) if for any r>0 and a non null z∈Xρ, we have(16)δr,z=inf1−1rρx+z2:ρx≤r,ρx+z≤r>0.
We say thatρ is unique uniform convexity in every direction (UUCED) if there exists ηs,z>0, for s≥0 and z nonnull in Xρ, such that(17)δr,z>ηs,z,  forr>s.
The following proposition characterizes relationships between uniform convexity, uniform convexity in every direction, 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 SCProof.
It is quite easy to show (a) and (b). To prove (c), letx, y∈Xρ such that x≠y. If ρx≠ρy, there is nothing to prove. Otherwise, we assume that ρx=ρy=r>0 and we consider z=y−x. Hence, ρx+z=ρy=r. Since ρ is UCED then δr,z>0, which implies(18)1−1rρx+z2≥δr,z>0.
Thus,(19)ρx+y2=ρx+y−x2≤1−δr,zr<r.
That is,ρx+y/2<r=ρx+ρy/2.
As an example of a modular which is (UUCED) but it is not (UUC1), consider the function ρ defined on X=ℝℕ by(20)ρx=ρxn=∑n=1∞xnn+1.Abdou and Khamsi [8] proved that this convex modular is (UUC2) which imply that ρ is (UUCED), but it is not (UUC1).The following lemma plays an important role in the proof of the next fixed point theorem. To prove it, we use the same ideas as those used to prove Lemma3.5 in [7].Lemma 3.
Letρ be a convex modular uniformly continuous and (UUCED). Assume that Xρ satisfies the property (R). Let C be a nonempty ρ-closed convex and ρ-bounded subset of Xρ and K be a nonempty ρ-closed convex subset of C. Let xkk∈ℕ be a sequence in C and consider the ρ-type function τ:K⟶0,+∞ defined by(21)τy=limsupk⟶+∞ρy−xk.
Then,τ has a unique minimum point in K.Theorem 6.
Letρ be a convex modular (UUCED) and is uniformly continuous. Assume that Xρ satisfies the property (R). Let A,B be a nonempty convex and ρ-bounded pair in Xρ such that B0⪯ is nonempty and ρ-closed. Let T:A∪B⟶A∪B be a monotone noncyclic relatively ρ-nonexpansive mapping. Assume that there exists x0∈A0⪯ such that x0⪯Tx0. Then, T has a best proximity pair.Proof.
Letx0∈A0⪯ such that x0⪯Tx0. Consider the sequence xnn≥0 given by xn+1=1−λxn+λTxn, where α∈0,1. For any n≥0, set Bn=y∈B0⪯:xn⪯y. Similar to Theorem 3, one has B∞=∩n≥0Bn is nonempty, ρ-closed convex, and T-invariant.
Consider the type functionτ:B∞⟶0,+∞ generated by the sequence xnn∈ℕ, that is, τz=limsupn⟶+∞ρxn−z for all z∈B∞. By Lemma 3, we know that τ has a unique minimum point y∈B∞. Since T is monotone noncyclic relatively ρ-nonexpansive, we have(22)τTy=limsupnρxn+1−Ty≤limsupn1−λρxn−Ty+λρTxn−Ty≤1−λlimsupnρxn−Ty+λlimsupnρTxn−Ty≤1−λτTy+λτy,which implies that τTy≤τy, since λ∈0,1. Therefore, Ty is also a minimum point of τ in B∞. The uniqueness of the minimum point of τ implies that Ty=y.Sincey∈B∞, then y∈B0⪯. Thus, there exists x∈A such that x⪯y and ρx−y=dρA,B. Using the same idea as in the proof of Theorem 3, we show that x is a fixed point of T on A. Therefore, Tx=x, Ty=y, and ρx−y=dρA,B, that is, x,y is a best proximity pair.In the caseA=B, therefore A=A0⪯ and B=B0⪯, we obtain the following corollary. The result of this corollary has already been mentioned by Khamsi and Bin Dehaish (see Theorem 2) [7] in modular function spaces.Corollary 7.
Letρ be a convex modular (UUCED) and uniformly continuous. Assume that Xρ satisfies the property (R). Let C be a nonempty ρ-closed convex and ρ-bounded subset of Xρ. Let T:C⟶C be a monotone ρ-nonexpansive mapping. If there exists x0∈C such that x0⪯Tx0, then T has a fixed point.


## 5. Application

In this section, we present an application of our main results to prove the existence of the solution for nonlinear integral equations. LetX=L20,1,ℝ be the space of measurable and square-integrable functions on 0,1. Recall that X is a ρ-complete modular space, denoted Xρ, where ρf=∫0,1ft2dt is a modular. Next, we consider the following integral equation:(23)xt=signxtgt+∫01Ft,s,xsds,t∈0,1,where(24)signxt=1,ifxt≥0,−1,ifxt<0,g∈Xρ such that gt≥0 for all t∈0,1 and F:0,1×0,1×ℝ⟶ℝ is measurable in both variables t and s for every x such that(25)∫01Ft,s,xsds<∞,∀t∈0,1.Recall thatx⪯y if and only if y−x∈P, where P=x∈Xρ:xt≥0a.e.. Consider the ρ-closed convex subsets A and B of Xρ given by A=x∈Xρ:γ⪯x⪯η and B=y∈Xρ:ϕ⪯y⪯ψ where γ, η, ϕ, and ψ are in Xρ. Since, ρx−y≤maxρη−γ+ρψ−ϕ,ρϕ−η+ρψ−γ<∞ for all x,y∈A∪B then A,B is ρ-bounded. Moreover, A0⪯=η and B0⪯=ϕ are ρ-closed.Theorem 8.
Let the mappingJ be defined by(26)Jxt=signxtgt+∫01Ft,s,xsds,t∈0,1.
Assume that the following conditions are fulfilled:(c1) ηt<0<ϕt, for all t∈0,1, and ρη−ϕ>0(c2) γt≤−gt+∫01Ft,s,γsdsηt≥−gt+∫01Ft,s,ηsds and ϕt≤gt+∫01Ft,s,ϕsdsψt≥gt+∫01Ft,s,ψsds(c3) F fulfill the following monotonicity condition 0≤Ft,s,xs−Ft,s,ys, for all x,y∈Xρ such that y⪯x(c4) ρJx−Jy≤ρx−y, whenever x, y∈A∪B such that x⪯y
Then, there existsx∗∈A and y∗∈B such that x∗⪯y∗, Jx∗=x∗, Jy∗=y∗, and ρx∗−y∗=dρA,B.Proof.
ρ is a (UUC1) convex modular, then ρ is (UUCED). Moreover, Xρ satisfies the property ℛ since Xρ is ρ-complete modular space and ρ is (UUC2) (from (d) of Proposition 1) and satisfies the Fatou property.
ρ is uniformly continuous. Indeed, let ε>0 and r>0, and we prove that there exists α>0 such that(27)ρy−ρx+y≤ε,where ρx≤α and ρy≤r. For all t∈0,1, we have yt2−xt+yt2≤xt2+2ytxt. Then,(28)ρy−ρx+y≤∫0,1xt2dt+2∫0,1ytxtdt≤ρx+2ρxρy≤ρxρx+2ρy.
Ifρx≤1, then(29)ρxρx+2ρy≤ρx1+2r.
Sinceρx1+2r≤ε, this is equivalent to ρx≤ε/2r+1. Therefore, ρx≤ε/2r+12. We take α=min1,ε/2r+12. Hence, ρ is uniformly continuous.
The condition (c2) implies that JA⊆A and JB⊆B. Indeed, let x∈A(30)Jxt−γt=−gt+∫01Ft,s,xsds−γt≥∫01Ft,s,xs−Ft,s,γsds≥0.
Then,γ⪯Jx. In addition,(31)ηt−Jxt=ηt−−gt+∫SFt,s,xsds≥∫SFt,s,ηs−Ft,s,xsds≥0.
Then,Jx⪯η. Hence, JA⊆A. Using the same arguments, we show that JB⊆B. From condition c1, one has dρA,B>0. By condition (c3) and since 0⪯g on A∪B, we conclude that J is monotone on A∪B.
Considerx0=γ. Condition (c2) implies that x0⪯Jx0. Hence, by Theorem 6, there exists x∗∈A and y∗∈B such that x∗⪯y∗, Jx∗=x∗, Jy∗=y∗, and ρx∗−y∗=dρA,B, that is, a best proximity pair.Example 2.
TakeFt,s,xs=ts/3xs, for all t,s∈0,12 and x∈Xρ.
Consider the functionsγ, η, ϕ, ψ, and g defined on Xρ by γt=−2t, ηt=−t, ϕt=t, ψt=2t, and gt=8/9t, for all t∈0,1. Denote the sets A=x∈Xρ:γ⪯x⪯η and B=y∈Xρ:ϕ⪯y⪯ψ. Let the mapping J:A∪B⟶A∪B defined by(32)Jxt=signxtgt+∫01Ft,s,xsds,t∈0,1.
We will check that the mappingJ satisfies conditions c1−c4.(c1) For every t∈0,1 the inequality ηt<0<ϕt holds, and dρA,B=ρϕ−η=∫012t2dt=4/3.(c2) It easy to see that(33)γt≤−gt+∫01Ft,s,γsds,ηt≥−gt+∫01Ft,s,ηsds,ϕt≤gt+∫01Ft,s,ϕsds,ψt≥gt+∫01Ft,s,ψsds,thenJA⊂A and JB⊂B(c3) We have 0≤Ft,s,ys−Ft,s,xs=ts/3ys−xs, for all t,s∈0,12 and x,y∈Xρ×Xρ such that x≤y.(c4) Let x∈A and y∈B such that y⪯x, we have(34)ρJy−Jx=∫01∫01Ft,s,ys−Ft,s,ysds+2gt2dt=∫012gt21+∫01ts/3ys−xsds2gt2dt.
Letu0∈0,1, for any u∈ℝ such that u≥uo we have uo1+u≤u1+uo. Thus,(35)∫uozuo1+udu≤∫uozu1+uoduuo1+z2−1+uo2≤1+uoz2−uo21+z2≤1+uo1+uo+z2−uo2uo1+z2≤1+uo1+1uoz2,for all z≥u0. Since ∫01ts/3ys−xsds/2gt≥1/8for allt∈0,1, then(36)ρJy−Jx≤∫012gt21+181+11/8∫01ts/3ys−xsds2gt2dt≤98∫012gt21+932∫01sys−xsds2dt≤32271+932∫01s2ds∫01ys−xs2ds≤32271+332ρys−xs≤89dρA,B+19ρys−xs≤ρys−xs,where x∈A and y∈B such that y⪯x, which implies that J is a monotone relatively ρ-nonexpansive mapping. Thus, Jη=η, Jϕ=ϕ, and ρϕ−β=dρA,B, that is, η,ϕ is a best proximity pair of J.


## 6. Conclusions

Throughout the paper, we have discussed some best proximity pair results for the class of monotone noncyclic relativelyρ-nonexpansive mappings in the framework of modular spaces equipped with a partial order defined by a ρ-closed convex cone. Our results are extensions of several results as in relevant items from the reference section of this paper, as well as in the literature in general. In order to show the utility of our main results, we prove the existence of the solution of an integral equation which involves monotone noncyclic relatively ρ-nonexpansive mappinngs.



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