Open Access

Fixed Points of Multivalued Maps in Modular Function Spaces

Fixed Point Theory and Applications20092009:786357

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

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.

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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Copyright

© M. A. Kutbi and A. Latif. 2009

This article is published under license to BioMed Central Ltd. 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.