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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 672069, 8 pages
http://dx.doi.org/10.1155/2013/672069
Research Article

One-Local Retract and Common Fixed Point in Modular Metric Spaces

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 8 July 2013; Accepted 19 August 2013

Academic Editor: Mohamed A. Khamsi

Copyright © 2013 Afrah A. N. Abdou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we introduced and study the concept of one-local retract in modular metric space. In particular, we investigate the existence of common fixed points of modular nonexpansive mappings defined on nonempty -closed -bounded subset of modular metric space.

1. Introduction

The purpose of this paper is to give an outline of a common fixed-point theory for nonexpansive mappings (i.e., mappings with the modular Lipschitz constant 1) on some subsets of modular metric spaces which are natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii, and many other spaces. Modular metric spaces were introduced in [1, 2]. The main idea behind this new concept is the physical interpretation of the modular. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) “field of (generalized) velocities” to each “time” (the absolute value of) an average velocity is associated in such a way that in order to cover the “distance” between points it takes time to move from to with velocity . But the way we approached the concept of modular metric spaces is different. Indeed we look at these spaces as the nonlinear version of the classical modular spaces introduced by Nakano [3] on vector spaces and Musielak-Orlicz spaces introduced by Musielak [4] and Orlicz [5].

In recent years, there was an uptake interest in the study of electrorheological fluids, sometimes referred to as “smart fluids” (for instance, lithium polymethacrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces, and , where is a fixed constant is not adequate, but rather the exponent should be able to vary [6, 7]. One of the most interesting problems in this setting is the famous Dirichlet energy problem [8, 9]. The classical technique used so far in studying this problem is to convert the energy function, naturally defined by a modular, to a convoluted and complicated problem which involves a norm (the Luxemburg norm). The modular metric approach is more natural and has not been used extensively.

In many cases, particularly in applications to integral operators, approximation, and fixed point results, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts. In recent years, there was a strong interest to study the fixed point property in modular function spaces after the first paper [10] was published in 1990. More recently, the authors presented a fixed point result for pointwise nonexpansive and asymptotic pointwise nonexpansive acting in modular functions spaces [11]. The theory of nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s (see, e.g., Belluce and Kirk [12], Browder [13], Bruck [14], and Lim [15]), and generalized to other metric spaces (see e.g., [1618]), and modular function spaces (see e.g., [10]). The corresponding fixed-point results were then extended to larger classes of mappings like pointwise contractions, asymptotic pointwise contractions [1822], and asymptotic pointwise nonexpansive mappings [11]. In [23], Penot presented an abstract version of Kirk’s fixed point theorem [24] for nonexpansive mappings. Many results of fixed point in metric spaces were developed after Penot’s formulation. Using Penot’s work, the author in [25] proved some results in metric spaces with uniform normal structure similar to the ones known in Banach spaces. In [26], Khamsi introduced the concept of one-local retract in metric spaces and proved that any commutative family of nonexpansive mappings defined on a metric space with a compact and normal convexity structure has a common fixed point. Recently in [27], the authors introduced the concept of one-local retract in modular function spaces and proved the existence of common fixed points for commutative mappings.

In this paper, we study the concept of one-local retract in more general setting in modular metric space; therefore, we prove the existence of common fixed points for a family of modular nonexpansive mappings defined on nonempty -closed -bounded subsets in modular metric space.

For more on metric fixed point theory, the reader may consult the book [28] and for modular function spaces the book [29].

2. Basic Definitions and Properties

Let be a nonempty set. Throughout this paper for a function , we will write for all and .

Definition 1 (see [1, 2]). A function is said to be modular metric on if it satisfies the following axioms: (i) if and only if , for all ; (ii), for all , and ; (iii), for all and .If, instead of (i), we have only the condition (i′) then is said to be a pseudomodular (metric) on . A modular metric on is said to be regular if the following weaker version of (i) is satisfied: Finally, is said to be convex if, for and , it satisfies the inequality

Note that, for a metric pseudomodular on a set , and any , the function is nonincreasing on . Indeed, if , then

Definition 2 (see [1, 2]). Let be a pseudomodular on . Fix . The two sets: are said to be modular spaces (around ).

It is clear that but this inclusion may be proper in general. It follows from [1, 2] that if is a modular on , then the modular space can be equipped with a (nontrivial) metric, generated by and given by for any . If is a convex modular on , according to [1, 2] the two modular spaces coincide, that is , and this common set can be endowed with the metric given by for any . These distances will be called Luxemburg distances (see example below for the justification).

Definition 3. Let be a modular metric space. (1)The sequence in is said to be -convergent to if and only if , as . will be called the -limit of . (2)The sequence in is said to be -Cauchy if , as . (3)A subset of is said to be -closed if the -limit of a -convergent sequence of always belongs to .(4)A subset of is said to be -complete if any -Cauchy sequence in is a -convergent sequence and its -limit is in . (5)Let and . The -distance between and is defined as (6)A subset of is said to be -bounded if we have

In general if , for some , then we may not have , for all . Therefore, as it is done in modular function spaces, we will say that satisfies condition if this is the case; that is , for some implies , for all . In [1, 2], one will find a discussion about the connection between -convergence and metric convergence with respect to the Luxemburg distances. In particular, we have for any and . And in particular we have that -convergence and -convergence are equivalent if and only if the modular satisfies the -condition. Moreover, if the modular is convex, then we know that and are equivalent which implies that for any and [1, 2]. Another question that arises in this setting is the uniqueness of the -limit. Assume is regular, and let be a sequence such that   -converges to and . Then we have for any . Our assumptions will imply . Since is regular, we get ; that is, the -limit of a sequence is unique.

Let be a modular metric space. Throughout the rest of this work, we will assume that satisfies the Fatou property; that is, if   -converges to and   -converges to , then we must have For any and , we define the modular ball Note that if satisfies the Fatou property, then modular balls (-balls) are -closed. An admissible subset of is defined as an intersection of modular balls. We say is an admissible subset of if where , , and is an arbitrary index set. Denote by the family of admissible subsets of . Note that is stable by intersection. At this point we will need to define the concept of Chebyshev center and radius in modular metric spaces. Let be a nonempty -bounded subset. For any , define The Chebyshev radius of is defined by Obviously we have , for any . The Chebyshev center of is defined as

Definition 4. Let be a modular metric space. Let be a nonempty subset of . (i)We will say that is compact if any family of elements of has a nonempty intersection provided , for any finite subset . (ii)We will say that is normal if  for any , not reduced to one point, -bounded, we have .

Remark 5. Note that if is compact, then is -complete.

Definition 6. Let be a modular metric space. Let be a nonempty subset of . A mapping is said to be -nonexpansive if For such mapping we will denote by the set of its fixed points; that is, .

In [1, 2] the author defined Lipschitzian mappings in modular metric spaces and proved some fixed point theorems. Our definition is more general. Indeed, in the case of modular function spaces, it is proved in [10] that if and only if , for any . Next we give an example, which first appeared in [10], of a mapping which is -nonexpansive in our sense but fails to be nonexpansive with respect to .

Example 7. Let . Define the Musielak-Orlicz function modular on the space of all Lebesgue measurable functions by Let be the set of all measurable functions such that . Consider the map Clearly, . In [10], it was proved that, for every and for all , we have This inequality clearly implies that is -nonexpansive. On the other hand, if we take , then which clearly implies that is not -nonexpansive.

Next we present the analog of Kirk’s fixed point theorem [24] in modular metric spaces.

Theorem 8 (see [30]). Let be a modular metric space and be a nonempty -closed -bounded subset of . Assume that the family is normal and compact. Let be -nonexpansive. Then has a fixed point.

3. One-Local Retract Subsets in Modular Metric Spaces

Let be a nonempty subset of . A nonempty subset of is said to be a one-local retract of if, for every family of -balls centered in such that , it is the case that . It is immediate that each -nonexpansive retract of is a one-local retract (but not conversely). Recall that is a -nonexpansive retract of if there exists a -nonexpansive map such that , for every .

The result in [26] may be stated in modular metric spaces as follows.

Theorem 9. Let be a modular metric space and be a nonempty -closed -bounded subset of . Assume that is normal and compact. Then, for any -nonexpansive mapping , the fixed point set is nonempty one-local retract of .

Proof. Theorem 8 shows that is nonempty. Let us complete the proof by showing that it is a one-local retract of . Let be any family of -closed balls such that , for  any , and Let us prove that . Since and is -nonexpansive, then . Clearly, and is nonempty. Then we have . Therefore, is compact and normal. Theorem 8 will imply that has a fixed point in which will imply

Now, we discuss some properties of one-local retract subsets.

Theorem 10. Let be a modular metric space. Let be a nonempty -closed -bounded subset of . Let be a nonempty subset of . The following are equivalent. (i) is a one-local retract of . (ii) is a -nonexpansive retract of , for every .

Proof. Let us prove . Let . We may assume that does not belong to . In order to construct a -nonexpansive retract , we only need to find such that Since and , then Since is one-local retract of , we get Any point in will work as .

Next, we prove that . In order to prove that is a one-local retract of , let be any family of -closed balls such that , for any , and Let us prove that . Let . If , we have nothing to prove. Assume otherwise that does not belong to . Property implies the existence of a -nonexpansive retract . It is easy to check that , which completes the proof of our theorem.

For the rest of this work, we will need the following technical result.

Lemma 11. Let be a modular metric space and be a nonempty -closed -bounded subset of . Let be a nonempty one-local retract of . Set . Then (i), for any ; (ii); (iii).

Proof. Let us first prove . Fix . Since , we get . On the other hand we have . The definition of implies . Hence , which implies Next, we prove (ii). Let . We have . Using (i), we get Hence, . Next, let . We have . Hence, . Hence Since is a one-local retract of , we get Let . Then it is easy to see that . Hence . Since was arbitrary taken in , we get which implies Finally, let us prove (iii). Since , we get Now, for any , we have Hence This implies Since was taken arbitrary in , we get The definition of implies So for any , we have Hence which implies

As an application of this lemma we have the following result.

Theorem 12. Let be a modular metric space and be a nonempty -closed -bounded subset of . Assume that is normal and compact. If is a nonempty one-local retract of , then is compact and normal.

Proof. Using the definition of one-local retract, it is easy to see that is compact. Let us show that is normal. Let be nonempty and reduced to one point. Set Then from Lemma 11, we get Since , then we must have because is normal. Therefore, we have which completes the proof of our claim.

The following result has found many application in metric spaces. Most of the ideas in its proof go back to Baillon’s work [31].

Theorem 13. Let be a modular metric space and be a nonempty -closed -bounded subset of . Assume that is normal and compact. Let be a decreasing family of one-local retracts of , where is totally ordered. Then is not empty and is one-local retract of .

Proof. Consider the family is not empty since . will be ordered by inclusion; that is, if and only if for any . From Theorem 12, we know that is compact, for every . Therefore, satisfies the hypothesis of Zorn’s Lemma. Hence for every , there exists a minimal element such that . We claim that if is minimal, then there exists such that , for every . Assume not, that is, , for every . Fix . For every , set Consider, where The family is decreasing since . Let . Then , since and . Hence the family is decreasing. On the other hand if , then since . Hence . Therefore, we have . Since is minimal, then . Hence Let and . Since , then Because , then we have which implies Since , then Therefore, we have Using the definition of Chebyshev radius , we get Let and set . Then since . Hence, Since is one-local retract of , then Since , then we have Let , then . Hence, , which implies Hence, . Therefore, we have Since , for every , set to the Chebyshev center of , that is, , for every . Since , for every , then the family is decreasing. Indeed, let and . Then we have . Since we proved that then which implies that . Therefore, we have . Since and is minimal, we get . Therefore, we have for every . This contradicts the fact that is normal for every . Hence there exists such that The proof of our claim is therefore complete. Then we have , for every . This clearly implies that . In order to complete the proof, we need to show that is one-local retract of . Let be a family of -balls centered in such that . Set Since is a one-local retract of and the family is centered in , then is not empty and . Therefore, Let be a minimal element of . The above proof shows that The proof of our theorem is complete.

The next theorem will be useful to prove the main result of the next section.

Theorem 14. Let be a modular metric space and be a nonempty -closed -bounded subset of . Assume that is normal and compact. Let be a family of one-local retracts of such that for any finite subset of . Then is not empty and is one-local retract of .

Proof. Consider the family of subsets such that, for any finite subset (empty or not), we have that is nonempty one-local retract of . Note that is not empty since any finite subset of is in . Using Theorem 13, we can show that satisfies the hypothesis of Zorn’s lemma. Hence has a maximal element . Assume . Let . Obviously we have . This is a clear contradiction with the maximality of . Therefore we have ; that is, is not empty and is a one-local retract of .

4. Common Fixed Point Result

In this section we discuss the existence of a common fixed point of a family of commutative -nonexpansive mappings in modular metric space which either generalize or improve the corresponding recent common fixed point results of [26, 27].

First, we will need to discuss the case of finite families.

Theorem 15. Let be a modular metric space and be a nonempty -closed -bounded subset of . Assume that is normal and compact. Let be a family of commutative -nonexpansive mappings defined on . Then the family has a common fixed point. Moreover, the common fixed point set is a one-local retract of .

Proof. First, let us prove that is not empty. Using Theorem 9, is nonempty one-local retract of , and then Theorem 12 implies that is compact and normal. On the other hand since and are commutative, we have Hence has a fixed point in . If we restrict ourselves to , the common fixed point set of and , then one can prove in an identical argument that has a fixed point in . Step by step, we can prove that the common fixed point set of is not empty. The same argument used to prove that the fixed point set of -nonexpansive map is a one-local retract can be reduced here to prove that is one-local retract.

The following result extends [26, Theorem 8] to the setting of modular metric space.

Theorem 16. Let be a modular metric space and let be a nonempty -closed -bounded subset of . Assume that is normal and compact. Let be a family of commutative -nonexpansive mappings defined on . Then the family has a common fixed point. Moreover, the common fixed point set is a one-local retract of .

Proof. Let . Theorem 15 implies that, for every , the set of common fixed point set of the mappings , , is nonempty one-local retract of . Clearly the family is decreasing and satisfies the assumptions of Theorem 14. Therefore, we deduced that is nonempty and is a one-local retract of .

Acknowledgment

The author would like to thank Professor Mohamed A. Khamsi with whom the author had many fruitful discussions regarding this work.

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