# Fixed Points of Multivalued Maps in Modular Function Spaces

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

**2009**:786357

**DOI: **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.

## 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

() 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:

(ii) for every scalar with and , and

In addition, if the following property is satisfied,

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.

(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:

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

(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.

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:

(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.

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

## References

- Edelstein M:
**An extension of Banach's contraction principle.***Proceedings of the American Mathematical Society*1961,**12**(1):7–10.MathSciNetMATHGoogle Scholar - Edelstein M:
**On fixed and periodic points under contractive mappings.***Journal of the London Mathematical Society*1962,**37:**74–79. 10.1112/jlms/s1-37.1.74MathSciNetView ArticleMATHGoogle Scholar - Kasahara S:
**On some generalizations of the Banach contraction theorem.***Publications of the Research Institute for Mathematical Sciences*1976,**12**(2):427–437. 10.2977/prims/1195190723MathSciNetView ArticleMATHGoogle Scholar - Rhoades BE:
**A comparison of various definitions of contractive mappings.***Transactions of the American Mathematical Society*1977,**226:**257–290.MathSciNetView ArticleMATHGoogle Scholar - Siddiqi AH, Ansari QH:
**An iterative method for generalized variational inequalities.***Mathematica Japonica*1989,**34**(3):475–481.MathSciNetMATHGoogle Scholar - Browder FE:
**On a theorem of Beurling and Livingston.***Canadian Journal of Mathematics*1965,**17:**367–372. 10.4153/CJM-1965-037-2MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Fixed-point theorems for noncompact mappings in Hilbert space.***Proceedings of the National Academy of Sciences of the United States of America*1965,**53**(6):1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Nonexpansive nonlinear operators in a Banach space.***Proceedings of the National Academy of Sciences of the United States of America*1965,**54**(4):1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleMATHGoogle Scholar - Kirk WA:
**A fixed point theorem for mappings which do not increase distances.***The American Mathematical Monthly*1965,**72**(9):1004–1006. 10.2307/2313345MathSciNetView ArticleMATHGoogle Scholar - Nadler SB Jr.:
**Multi-valued contraction mappings.***Pacific Journal of Mathematics*1969,**30**(2):475–488.MathSciNetView ArticleMATHGoogle Scholar - Markin JT:
**A fixed point theorem for set valued mappings.***Bulletin of the American Mathematical Society*1968,**74:**639–640. 10.1090/S0002-9904-1968-11971-8MathSciNetView ArticleMATHGoogle Scholar - Lami Dozo E:
**Multivalued nonexpansive mappings and Opial's condition.***Proceedings of the American Mathematical Society*1973,**38**(2):286–292.MathSciNetView ArticleMATHGoogle Scholar - Husain T, Tarafdar E:
**Fixed point theorems for multivalued mappings of nonexpansive type.***Yokohama Mathematical Journal*1980,**28**(1–2):1–6.MathSciNetMATHGoogle Scholar - Husain T, Latif A:
**Fixed points of multivalued nonexpansive maps.***International Journal of Mathematics and Mathematical Sciences*1991,**14**(3):421–430. 10.1155/S0161171291000558MathSciNetView ArticleMATHGoogle Scholar - Khamsi MA, Kozlowski WM, Reich S:
**Fixed point theory in modular function spaces.***Nonlinear Analysis: Theory, Methods & Applications*1990,**14**(11):935–953. 10.1016/0362-546X(90)90111-SMathSciNetView ArticleMATHGoogle Scholar - Khamsi MA:
**Fixed point theory in modular function spaces.**In*Recent Advances on Metric Fixed Point Theory (Seville, 1995), Ciencias*.*Volume 48*. Universidad de Sevilla, Seville, Spain; 1996:31–57.Google Scholar - Dhompongsa S, Domínguez Benavides T, Kaewcharoen A, Panyanak B:
**Fixed point theorems for multivalued mappings in modular function spaces.***Scientiae Mathematicae Japonicae*2006,**63**(2):161–169.MathSciNetMATHGoogle Scholar - Kozlowski WM:
*Modular Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 122*. Marcel Dekker, New York, NY, USA; 1988:x+252.Google Scholar - Dominguez Benavides T, Khamsi MA, Samadi S:
**Uniformly Lipschitzian mappings in modular function spaces.***Nonlinear Analysis: Theory, Methods & Applications*2001,**46**(2):267–278. 10.1016/S0362-546X(00)00117-6MathSciNetView ArticleMATHGoogle Scholar - Dominguez-Benavides T, Khamsi MA, Samadi S:
**Asymptotically regular mappings in modular function spaces.***Scientiae Mathematicae Japonicae*2001,**53**(2):295–304.MathSciNetMATHGoogle Scholar - Kada O, Suzuki T, Takahashi W:
**Nonconvex minimization theorems and fixed point theorems in complete metric spaces.***Mathematica Japonica*1996,**44**(2):381–391.MathSciNetMATHGoogle Scholar - Suzuki T, Takahashi W:
**Fixed point theorems and characterizations of metric completeness.***Topological Methods in Nonlinear Analysis*1996,**8**(2):371–382.MathSciNetMATHGoogle Scholar - Suzuki T:
**Generalized distance and existence theorems in complete metric spaces.***Journal of Mathematical Analysis and Applications*2001,**253**(2):440–458. 10.1006/jmaa.2000.7151MathSciNetView ArticleMATHGoogle Scholar

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