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

Abstract

In this paper, we obtain some fixed point results for noncyclic monotone -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.

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 -condition have a fixed point.

In 2016, Bin Dehaish and Khamsi proved in [7] the following theorems.

Theorem 1. (see [7]). Let be a (UUC1) and be a nonempty convex -closed -bounded subset of not reduced to one point. Let be a monotone -nonexpansive mapping and -continuous. Assume there exists such that and are comparable. Then, has a fixed point.

Theorem 2. (see [7]). Assume that is UUCED and uniformly continuous. Assume that satisfies the property (R). Let be a nonempty convex -closed -bounded subset of not reduced to one point. Let be a monotone -nonexpansive mapping and -continuous. Assume there exists such that and are comparable. Then, 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 mapping is said to be noncyclic provided that and , where 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 such thatthat is, is a best proximity pair.
In this paper, we discuss some best proximity pair results for the class of monotone noncyclic relatively -nonexpansive mappings in the framework of modular spaces equipped with a partial order defined by a -closed convex cone , that is, for any and in the modular space one has if and only if in . 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 -nonexpansive mappings to the case of noncyclic monotone relatively -nonexpansive mappings in modular spaces.
To illustrate the effectiveness of our main results, we give an application to an integral equation which involves noncyclic monotone -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 modular . The aim of this work is to extend some fixed point theorems for nonexansive mappings to best proximity pair for monotone noncyclic relatively -nonexpansive mappings in modular spaces.

The paper is organized as follows. In Section 3, 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, stands for a linear vector space on the field . Let us start with some preliminaries and notations.

Definition 1. (see [8]). A function is called a modular if the following holds:(1)  if and only if (2) (3) , for any and for any and in If (3) is replaced by (3′) for any and and in , then is called a convex modular.
The modular space is defined as . Throughout this paper, we will assume that the modular is convex. The Luxemburg norm in is defined asAssociated to a modular, we introduce some basic notions needed throughout this work.

Definition 2. (see [8]). Let be a modular defined on a vector space :(1)We say that a sequence is -convergent to if and only if . Note that the limit is unique.(2)A sequence is called -Cauchy if as .(3)We say that is -complete if and only if any -Cauchy sequence is -convergent.(4)A subset of is said -closed if the -limit of a -convergent sequence of always belong to .(5)A subset of is said -bounded if we have (6)A subset of is said -sequentially compact if any sequence of has a subsequence -convergent to a point .(7)We say that satisfies the Fatou property if whenever -converges to for any , , and in .Let us note that -convergence does not imply -Cauchy condition. Also, does not imply in general , for every .

Definition 3. Let be a modular and be a nonempty subset of the modular space . A mapping is said to be(a)Monotone, if for any , such that .(b)Monotone -nonexpansive, if is monotone andwhenever , and .
Recall that is said to be -continuous if -converges to whenever -converges to . It is not true that a monotone -nonexpansive mapping is -continuous, since this result is not true in general when is a norm.
Let and be nonempty subsets of a modular space . We adopt the notation .
A pair is said to satisfy a property if both and satisfy that property. For instance, is -closed (resp. convex, -bounded) if and only if and are -closed (resp. convex, -bounded). A pair is not reduced to one point means that and are not singletons.
Recall the definition of the modular uniform convexity.

Definition 4. (see [8]). Let be a modular and and . Define, for ,If , letIf , we set .(i)We say that is uniformly convex (UCi) if for every and , and we have (ii)We say that is unique uniformly convex (UUCi) if for all and , and there exists such that (iii)We say that is strictly convex (SC), if for every , such that and , and we have The following proposition characterizes the relationship between the above notions

Proposition 1. (see [8]).(a) implies for (b) for and (c) implies implies (d) implies

Definition 5. (see [9]). Let be a modular. We say that the modular space satisfies the property (R) if and only if for every decreasing sequence of nonempty -closed convex and -bounded subsets of has a nonempty intersection.

Lemma 1. (see [8]). Let be a convex modular satisfying the Fatou property. Assume that is -complete and is (UUC₂). Then, satisfies the property (R).

Definition 6 (see [8]). Let be a sequence in and be a nonempty subset of . The function defined by is called a -type function.
Note that the -type function is convex since is convex. A sequence is called a minimizing sequence of if
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]). Let be a convex modular (UUC1) satisfying the Fatou property and a -compete modular space. Let be a nonempty -closed convex subset of . Consider the -type function generated by a sequence in . Assume that . Then, all the minimizing sequences of are -convergent to the same limit.

4. Main Results

A subset is a pointed -closed convex cone, if is a nonempty -closed subset of satisfying the following properties:(i)(ii) , for all (iii)

Using , we define an ordering on by .

Let be a convex modular and be a monotone mapping, where is a nonempty convex subset of the modular space . Let and . Consider Krasnoselskii-Ishikawa iteration sequence in defined by for all . Assume that (resp. ). By the definition of the partial ordering , we obtain (resp. ).

Since is monotone, one has (resp. ). By induction, we prove that

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

Definition 7. Let be a nonempty pair of a modular space . A mapping is said to be monotone noncyclic relatively -nonexpansive if it satisfies the following conditions:(i)For all , implies that (ii) and (iii)For all and such that , we have

Definition 8 (see [10]). Let be a pair of a modular space and be a noncyclic mapping. A pair is said to be a best proximity pair for the noncyclic mapping if

Definition 9. A space is said to satisfy the property , if and with for all ; then, .
We use to denote the ordered -proximal pair obtained from byNote that if is nonempty if and only if is nonempty. Indeed, if then there exists such that and . Thus, for this we take , we have and , that is, . As the same we prove the converse.

Proposition 2. Let be a convex modular satisfying the Fatou property and satisfies the property (P). Let be a nonempty convex pair of . Assume that is nonempty:(i)If is -sequentially compact, then is -sequentially compact(ii) is convex(iii)(iv)If is a noncyclic monotone relatively -nonexpansive mapping, then and , that is, and are -invariant

Proof. It is quite easy to verify (iii):(i)Let be a sequence in . Then, there exists a sequence in such that and for all . Since is -sequentially compact, there exists subsequences and of and , respectively, such that for all , and -converge to and , respectively. From property (P), one has . Since satisfies the Fatou property, then Thus, . Therefore, is -sequentially compact. Likewise, we prove that is also -sequentially compact.(ii)Let , , and set . Since , , then there exists , such that , and . Moreover, and . Hence, . Since is convex, then Then, . Hence, . Therefore, is convex. Using the same argument we prove that is convex.(iii)Let , then there exists such that and . Since is monotone noncyclic relatively -nonexpansive, then and . Thus, . Therefore, . As the same way, we obtain .

Theorem 3. Let be a convex modular satisfying the Fatou property and be a -complete modular space. Let be a nonempty convex and -bounded pair of such that the subset is nonempty -closed. Let be a monotone noncyclic relatively -nonexpansive mapping and -continuous on . Assume that there exists such that . Then, has a best proximity pair.

Proof. Let such that . Consider the sequence defined by for all and . Set for any . The sequence is a decreasing sequence of nonempty -closed convex -bounded subsets.
Since is -invariant, then . Using the convexity of , one has . Assume that , then . Again the convexity of , implies that . Therefore, for all . Hence, for all , there exists such that and . Thus, there exists such that . Therefore, is nonempty for all . Moreover, since then is a decreasing sequence.
Let be a sequence in -converges to . We have , then and , for all . Since is -closed, one has . Moreover, . Thus, since is -closed. Hence, . Therefore, is -closed for all .
As and are convex, it is easy to see that is convex for all . Furthermore, is -bounded for all .
By Lemma 1, satisfies the property (R). Then, is nonempty and -closed convex.
Moreover, . In fact, let ; then, for all . Since is monotone and , one has for all . Moreover, since is -invariant. Hence, .
Consider the type function generated by the sequence , that is, for all . Let be a minimizing sequence of . By Lemma 2, -converges to a point since is -closed. Otherwise,Hence, . Then, is also a minimizing sequence of . By Lemma 2, -converges to . As is -continuous on , then -converges to . Therefore, . Then, there exists such that and , for all . By the definition of , there exists such that and . In order to finish the proof, we show that is a fixed point of on . We haveThen, . Moreover,Thus, . Since is (UUC2), it is (SC). Hence, , which implies that . Therefore, is a best proximity pair of the mapping .

To illustrate Theorem 3, we consider the following example.

Example 1. Let , we define the modular by for all . The modular is convex satisfying the Fatou property and (UUC1), and is a -complete modular space. Consider the -closed convex cone . Let and . The pair is -closed convex and -bounded in with . Moreover, and are -closed. Furthermore, we have(i)For all , there exists such that (ii)For the other cases the elements of and are not comparableConsider the mapping defined byThe mapping is monotone noncyclic relatively -nonexpansive -continuous on , and for we have . Then, has a best proximity pair , where .
The following corollary is an immediate consequence of Theorem 3, and it suffices to take ; therefore, and . It will be considered as a generalization of Browder and Göhde fixed point theorem for monotone -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. Let be a convex modular satisfying the Fatou property and be a -complete modular space. Let be a nonempty -closed convex -bounded subset of and be a monotone -nonexpansive mapping and -continuous. If there exists such that , then has a fixed point.

Particular case of normed vector spaces:

Remark 1. (i)If is a normed vector space, then the norm is continuous and satisfies the Fatou property. Moreover, if is a uniformly convex Banach space, then is reflexive. Hence, it satisfies the property (R).(ii)If is reflexive and is a nonempty closed and bounded pair of , then is also closed convex pair of .First case: if is empty, then there is nothing to prove.
Second case: assume that is nonempty. Let be a sequence in such that -converges to . There exists a sequence in such that and , for all . Since is reflexive and is bounded, then there exists a subsequence in which converges weakly to . Since converges weakly to , thenThus, and . Hence, . Therefore, is closed. Using the same argument, we prove that is also closed.
In the context of uniformly convex normed vector spaces, Theorem 3 is given as follows.

Theorem 5. Let be a nonempty convex and bounded pair of a partially ordered uniformly convex Banach space such that nonempty and closed. Let be a noncyclic monotone relatively nonexpansive mapping. Assume that there exists such that . Then, has a best proximity pair.
In order to weaken the assumptions of Theorem 3, we introduce the notions of uniform continuity and uniform convexity in every direction.

Definition 10. A modular is said to be uniformly continuous, if for any and there exists such that whenever and .

Definition 11. Let be a modular. We say that is uniformly convex in every direction (UCED) if for any and a non null , we haveWe say that is unique uniform convexity in every direction (UUCED) if there exists , for and nonnull in , such thatThe following proposition characterizes relationships between uniform convexity, uniform convexity in every direction, and strict convexity of a modular.

Proposition 3. (a) (resp. ) implies (resp. ) for (b) implies (c) implies

Proof. It is quite easy to show (a) and (b). To prove (c), let , such that . If , there is nothing to prove. Otherwise, we assume that and we consider . Hence, . Since is then , which impliesThus,That is, .
As an example of a modular which is (UUCED) but it is not (UUC1), consider the function defined on by

Abdou and Khamsi [8] proved that this convex modular is (UUC2) which imply that 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 Lemma 3.5 in [7].

Lemma 3. Let be a convex modular uniformly continuous and (UUCED). Assume that satisfies the property (R). Let be a nonempty -closed convex and -bounded subset of and be a nonempty -closed convex subset of . Let be a sequence in and consider the -type function defined byThen, has a unique minimum point in .

Theorem 6. Let be a convex modular (UUCED) and is uniformly continuous. Assume that satisfies the property (R). Let be a nonempty convex and -bounded pair in such that is nonempty and -closed. Let be a monotone noncyclic relatively -nonexpansive mapping. Assume that there exists such that . Then, has a best proximity pair.

Proof. Let such that . Consider the sequence given by , where . For any , set . Similar to Theorem 3, one has is nonempty, -closed convex, and -invariant.
Consider the type function generated by the sequence , that is, for all . By Lemma 3, we know that has a unique minimum point . Since is monotone noncyclic relatively -nonexpansive, we havewhich implies that , since . Therefore, is also a minimum point of in . The uniqueness of the minimum point of implies that .

Since , then . Thus, there exists such that and . Using the same idea as in the proof of Theorem 3, we show that is a fixed point of on . Therefore, , , and , that is, is a best proximity pair.

In the case , therefore and , 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. Let be a convex modular (UUCED) and uniformly continuous. Assume that satisfies the property (R). Let be a nonempty -closed convex and -bounded subset of . Let be a monotone -nonexpansive mapping. If there exists such that , then 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. Let be the space of measurable and square-integrable functions on . Recall that is a -complete modular space, denoted , where is a modular. Next, we consider the following integral equation:where such that for all and is measurable in both variables and for every such that

Recall that if and only if , where . Consider the -closed convex subsets and of given by and where , , , and are in . Since, for all then is -bounded. Moreover, and are -closed.

Theorem 8. Let the mapping be defined byAssume that the following conditions are fulfilled: (c₁) , for all , and  (c₂) and  (c₃) fulfill the following monotonicity condition , for all such that  (c₄) , whenever , such that Then, there exists and such that , , , and .

Proof. is a (UUC1) convex modular, then is (UUCED). Moreover, satisfies the property since is -complete modular space and is (UUC2) (from (d) of Proposition 1) and satisfies the Fatou property.
is uniformly continuous. Indeed, let and , and we prove that there exists such thatwhere and . For all , we have . Then,If , thenSince , this is equivalent to . Therefore, . We take . Hence, is uniformly continuous.
The condition () implies that and . Indeed, let Then, . In addition,Then, . Hence, . Using the same arguments, we show that . From condition , one has . By condition () and since on , we conclude that is monotone on .
Consider . Condition () implies that . Hence, by Theorem 6, there exists and such that , , , and , that is, a best proximity pair.

Example 2. Take , for all and .
Consider the functions , , , , and defined on by , , , , and , for all . Denote the sets and . Let the mapping defined byWe will check that the mapping satisfies conditions . () For every the inequality holds, and . () It easy to see that then and  () We have for all and such that . () Let and such that , we haveLet , for any such that we have . Thus,for all . Since , thenwhere and such that , which implies that is a monotone relatively -nonexpansive mapping. Thus, , , and , that is, is a best proximity pair of .

6. Conclusions

Throughout the paper, we have discussed some best proximity pair results for the class of monotone noncyclic relatively -nonexpansive mappings in the framework of modular spaces equipped with a partial order defined by a -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 -nonexpansive mappinngs.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.