- Research Article
- Open Access

# Fixed Points of Multivalued Maps in Modular Function Spaces

- Marwan A. Kutbi
^{1}and - Abdul Latif
^{1}Email author

**2009**:786357

https://doi.org/10.1155/2009/786357

© M. A. Kutbi and A. Latif. 2009

**Received:**7 February 2009**Accepted:**14 April 2009**Published:**12 May 2009

## Abstract

The purpose of this paper is to study the existence of fixed points for contractive-type and nonexpansive-type multivalued maps in the setting of modular function spaces. We also discuss the concept of
-modular function and prove fixed point results for *weakly*-modular contractive maps in modular function spaces. These results extend several similar results proved in metric and Banach spaces settings.

## Keywords

- Modular Space
- Fixed Point Theorem
- Lower Semicontinuity
- Growth Function
- Unique Fixed Point

## 1. Introduction and Preliminaries

for all and for a fixed number . If the Lipschitz constant , then is called a multivalued nonexpansive mapping [11]. Nadler [10], Markin [11], Lami-Dozo [12], and others proved fixed point theorems for these maps under certain conditions in the setting of metric and Banach spaces. Note that an element is called a fixed point of a multivalued map if . Among others, without using the concept of the Hausdorff metric, Husain and Tarafdar [13] introduced the notion of a nonexpansive-type multivalued map and proved a fixed point theorem on compact intervals of the real line. Using such type of notions Husain and Latif [14] extended their result to general Banach space setting.

The fixed point results in modular function spaces were given by Khamsi et al. [15]. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces. For instance, fixed point theorems are proved in [15, 16] for nonexpansive maps.

In this paper, we define nonexpansive-type and contractive-type multivalued maps in modular function spaces, investigate the existence of fixed points of such mappings, and prove similar results found in [17].

Now, we recall some basic notions and facts about modular spaces as formulated by Kozlowski [18]. For more details the reader may consult [15, 16].

Let be a nonempty set and let be a nontrivial -algebra of subsets of . Let be a -ring of subsets of , such that for any and

Let us assume that there exists an increasing sequence of sets such that . By we denote the linear space of all simple functions with supports from . By we will denote the space of all measurable functions, that is, all functions such that there exists a sequence , and for all . By we denote the characteristic function of the set .

Definition 1.1.

A functional is called a function modular if

() for any ,

() whenever for any , and ,

() is a -subadditive measure for every ,

() as decreases to for every , where ,

()if there exists such that , then for every , and

()for any is order continuous on , that is, if and decreases to .

For the sake of simplicity we write instead of .

Definition 1.2.

A set is said to be -null if for every A property is said to hold -almost everywhere ( -a.e.) if the set is -null.

Definition 1.3.

A modular function is called -finite if there exists an increasing sequence of sets such that and It is easy to see that the functional is a modular and satisfies the following properties:

(i) if and only if -a.e.,

(ii) for every scalar with and , and

(iii) if , and .

In addition, if the following property is satisfied,

(iii)' if , and, ,

we say that is a convex modular.

Definition 1.4.

A function modular is said to satisfy the -condition if as whenever decreases to and as

We know from [18] that when satisfies the -condition.

Definition 1.5.

A function modular is said to satisfy the -type condition if there exists such that for any we have

In general, -type condition and -condition are not equivalent, even though it is obvious that -type condition implies -condition on the modular space

Definition 1.6.

Let be a modular space.

(1)The sequence is said to be -convergent to if as .

(2)The sequence is said to be -a.e. convergent to if the set is -null.

(3)The sequence is said to be -Cauchy if as and go to .

(4)A subset of is called -closed if the -limit of a -convergent sequence of always belongs to .

(5)A subset of is called -a.e. closed if the -a.e. limit of a -a.e. convergent sequence of always belongs to .

(6)A subset of is called -a.e. compact if every sequence in has a -a.e. convergent subsequence in .

We recall two basic results (see [15]) in the theory of modular spaces.

(i)If there exists a number such that then there exists a subsequence of such that -a.e.

(ii)(Lebesgue's Theorem) If , -a.e. and there exists a function such that -a.e. for all then

We know, by [15, 16] that under -condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the -type condition. In the sequel we will assume that the modular function is convex and satisfies the -type condition.

Definition 1.7.

We have the following:

Lemma 1.8 (see [19]).

Let be as aforementioned. Then the growth function has the following properties:

(1) , ,

(2) is a convex, strictly increasing function. So, it is continuous,

(3) ,

(4) ; where is the function inverse of .

The following lemma shows that the growth function can be used to give an upper bound for the norm of a function.

Lemma 1.9 (see [19]).

The next lemma will be of major interest throughout this work.

Lemma 1.10 (see [16]).

## 2. Fixed Points of Contractive-Type and Nonexpansive-Type Maps

In the sequel we assume that is a convex, -finite modular function satisfying the -type condition, and is a nonempty -bounded subset of the modular function space . We denote that is a collection of all nonempty -closed subsets of , and is a collection of all nonempty -compact subsets of .

We have the following fixed point theorem (for which a similar result may be found in [17]).

Theorem 2.1.

Let be a nonempty -closed subset of the modular function space . Then any -contractive-type map has a fixed point, that is, there exists such that .

Proof.

Hence converges to 0. Since satisfies the -condition, we have converges to 0. Since -converges to , then -converges to . Hence -converges to . Since is -closed and , we get .

Remark 2.2.

Consider the multivalued map , where is a nonempty -closed subset of . Then it is easy to show that is a -contractive-type map. The set of all fixed point of is exactly the set . In particular, -contractive-type maps may not have a unique fixed point.

As an application of the above theorem, we have the following result.

Proposition 2.3.

Proof.

for any , which implies .

Using the above result, we are now ready to prove the main fixed point result for -nonexpansive-type multivalued maps.

Theorem 2.4.

Let be a nonempty -closed convex subset of the modular function space . Assume that is -a.e. compact. Then each -nonexpansive-type map has a fixed point.

Proof.

Hence or . Hence ; that is, is a fixed point of .

Proposition 2.3 and Theorem 2.4 are also hold if we assume that is starshaped instead of Convex. (A set is called starshaped if there exists such that provided and )

## 3. Fixed Points of -Contractive-Type Maps

In [21] the authors introduced the concept of -distance in metric spaces which they connected to the existence of fixed point of single and multivalued maps (see also [22]). Similarly we extend their definition and results to modular spaces. Indeed let be a convex, -finite modular function. A function is called -modular on the modular function space if the following are satisfied:

(1) for any ;

(3)for any , there exists such that and imply .

As it was done in [21], we need the following technical lemma.

Lemma 3.1.

Let be -modular on the modular function space . Let and be sequences in , and let and be sequences in converging to 0, and . Then the following hold:

(1)if and , for all , then ; in particular if and , then ;

(2)if and , for any , then -converges to ;

(3)if for any with , then is a -Cauchy sequence;

(4)if for any , then is a -Cauchy sequence.

The proof is easy and similar to the one given in [21]. Now we are ready to give the first fixed point result in this setting. Let be a nonempty -closed subset of the modular function space . We say that a multivalued map is weakly -contractive-type map if there exists -modular on and such that for any and any , there exists such that .

Theorem 3.2.

Let be a nonempty -closed subset of the modular function space . Then each weakly -contractive-type map has a fixed point , and .

Proof.

Lemma 3.1 implies , or .

Note that in the proof above we did not use the -condition. The reason behind is that satisfies the triangle inequality. If is single valued, then we have little more information about the fixed point. Indeed, let be a nonempty -closed subset of the modular function space . The map is called a weakly -contractive type map if there exists -modular on and such that for any .

Theorem 3.3.

Let be a nonempty -closed subset of the modular function space . Then each weakly -contractive type map has a unique fixed point , and .

Proof.

Theorem 3.2 ensures the existence of a fixed point , that is, and . Let us show that is the only fixed point of . Assume that is another fixed point of . Then we must have . Combining this with , Lemma 3.1 implies .

Similar extensions of the results as found in [21–23] may be proved in our setting.

## Declarations

### Acknowledgments

The authors thank the referees for their valuable comments and suggestions. The authors would also like to thank Professor M.A. Khamsi for productive discussion and cooperation regarding this work.

## Authors’ Affiliations

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