Open Access

# Fixed Points of Multivalued Maps in Modular Function Spaces

Fixed Point Theory and Applications20092009:786357

DOI: 10.1155/2009/786357

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.

## 1. Introduction and Preliminaries

The well-known Banach fixed point theorem on complete metric spaces (specifically, each contraction self-map of a complete metric space has a unique fixed point) has been extended and generalized in different directions. For example, see Edelstein [1, 2], Kasahara [3], Rhoades [4], Siddiq and Ansari [5], and others. One of its generalizations is for nonexpansive single-valued maps on certain subsets of a Banach space. Indeed, these fixed points are not necessarily unique. See, for example, Browder [68] and Kirk [9]. Fixed point theorems for contractive and nonexpansive multivalued maps have also been established by several authors. Let denote the Hausdorff metric on the space of all bounded nonempty subsets of a metric space . A multivalued map (where denotes the collection of all nonempty subsets of ) with bounded subsets as values is called contractive [10] if
(1.1)

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 .

The definition of is then extended to by
(1.2)

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.

The modular defines a corresponding modular space, that is, the vector space given by
(1.3)
When is convex, the formula
(1.4)
defines a norm in the modular space which is frequently called the Luxemburg norm. We can also consider the space
(1.5)

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 .

(7)A subset of is called -bounded if
(1.6)

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.

Let be as aforementioned. We define a growth function by
(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]).

Let be a convex function modular satisfying the -type condition. Then
(1.8)

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

Lemma 1.10 (see [16]).

Let be a function modular satisfying the -condition and let be a sequence in such that , and there exists such that . Then,
(1.9)
Moreover, one has
(1.10)

## 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 say that a multivalued map is -contractive-type if there exists such that for any and for any , there exists such that
(2.1)
and -nonexpansive-type if for any and for any , there exists such that
(2.2)

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.

Let . Without loss of generality, assume that is not a fixed point of . Then there exists such that . Hence . Since is -contractive-type, then there exists such that
(2.3)
By induction, one can easily construct a sequence such that and
(2.4)
for any . In particular we have
(2.5)
Without loss of generality, we may assume , otherwise is a fixed point of . Hence
(2.6)
Using Lemma 1.9, we get
(2.7)
Using the properties of , we get
(2.8)
So
(2.9)
which implies
(2.10)
Since and , then . This forces to be -Cauchy. Hence the sequence -converges to some . Since satisfies the -condition, then -converges to . Since is -closed, then . Let us prove that is indeed a fixed point of . Since is a -contractive-type mapping, then for any , there exists such that
(2.11)

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.

Let be a -closed convex subset of the modular function space . Let be -nonexpansive-type map. Then there exists an approximate fixed points sequence in , that is, for any there exists such that
(2.12)
In particular one has , where
(2.13)

Proof.

Let and let be a fixed point in . For each , define a map
(2.14)
Note that is nonempty and -closed subset of because is -closed and is convex. Since is a -nonexpansive-type map, for each and for any , there exists such that
(2.15)
Since is convex we get
(2.16)
which implies
(2.17)
In other words, the map is a -contractive-type. Theorem 2.1 implies the existence of a fixed point of , thus there exists such that
(2.18)
In particular, we have
(2.19)
where is the -diameter of . Note that since is -bounded, then . If we choose , for and write and , we get
(2.20)

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.

Proposition 2.3 ensures the existence of a sequence in and a sequence such that and . Without loss of generality we may assume that -a.e. converges to and -a.e. converges to . Lemma 1.10 implies
(2.21)
Hence . Since is a -nonexpansive-type map, then there exists a sequence such that
(2.22)
for all . Since is -compact, we may assume that is -convergent to some . Lemma 1.10 implies
(2.23)
Since satisfies the -condition, then
(2.24)
(see, [20]). Since , we get
(2.25)
which implies
(2.26)

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 ;

(2)for any , is lower semicontinuous; that is, if -converges to , then
(3.1)

(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.

Let be a -modular and associated to , that is, for any and any , there exists such that . Fix and . By induction one can construct a sequence such that and
(3.2)
for every . In particular we have , for every . Using the properties of , we get
(3.3)
for any . Lemma 3.1 implies that the sequence is -Cauchy. Hence -converges to some . Using the lower semicontinuity of , we get
(3.4)
for any . Since and is weakly -contractive-type map, there exists such that
(3.5)
for any . Lemma 3.1 implies that - converges to as well. Since is -closed, then , that is, is a fixed point of . Let us complete the proof by showing that . Since , there exists such that . By induction we can construct a sequence in such that and , for any . So we have , for any . Lemma 3.1 implies that is -Cauchy. Hence - converges to some . Using the lower semicontinuity of we get
(3.6)
Hence . Then for any , we have
(3.7)

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 [2123] 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

(1)
Department of Mathematics, King Abdulaziz University

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