ISRN GeometryVolume 2011 (2011), Article ID 530254, 7 pagesdoi:10.5402/2011/530254

Research Article

On the Fixed Points of Correspondences in Modular Spaces

Fatemeh Lael and Kourosh Nourouzi

Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran

Received 20 March 2011; Accepted 26 April 2011

Academic Editors: F. Balibrea and P. N. S. Wong

Copyright © 2011 Fatemeh Lael and Kourosh Nourouzi. 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.

We give two fixed point results for contractive and nonexpansive correspondences defined on modular spaces.

1. Introduction and Preliminaries

A modular on a real linear space is a real functional on which satisfies the conditions:(1) if and only if ,(2),(3), for all and , .

Then is called a modular space. Given a modular , a corresponding vector space is given as which is called modular linear space.

It is easy to see that for every modular if and , then , for all .

The theory of modular spaces was initiated by Nakano [1] in connection with the theory of ordered spaces. Musielak and Orlicz [2] redefined and generalized the notion of a modular space in order to obtain a generalization of the classical function spaces . Even if a metric is not defined, many results in metric fixed point theory can be reformulated in modular spaces, we refer, for instance, to [3–5].

In this work, we give some results on the existence of fixed points for contractive and nonexpansive correspondences on modular spaces.

We first recall some basic concepts of modular spaces. We refer to [6, 7] for more details on modular spaces.

Definition 1.1. Let be a subset of a modular space . (1)A sequence in is said to be convergent to a point and denoted by , if for every , there is a positive integer such that , for all . (2)The closure of is denoted by and defined as the set of all such that there is a sequence of which is convergent to . We say that is closed if . (3)A sequence in is said to be Cauchy, if for every , there is a positive integer such that , for all . (4) is said to be complete if each Cauchy sequence in is convergent to a point of . (5) is said to be compact if every sequence in has a convergent subsequence in .(6) is called sequentially bounded, if for each and each real sequence converging to zero we have , as .By a correspondence from a set to a set we mean a relation that assigns to each in a nonempty subset of . For any subset of and correspondence , an element is said to be a fixed point if . Also, .

Definition 1.2. Let be a subset of and let . We say that is a -contraction if for each and there is which satisfies the condition:

Definition 1.3 (see [8]). For a modular space , the function which is called growth function is defined on as follows:

2. Main Results

In the sequel, it is assumed that is a closed subset of complete modular space and is a correspondence with compact values.

Lemma 2.1. Let be a modular space satisfying . Then every convergent sequence in is a Cauchy sequence in .

Proof. Let be a convergent sequence in . Then for , there exists such that for all . Thus for every , . Hence is a Cauchy sequence.

Definition 2.2. Let be a subset of a modular space and let be given. A set is called an -net for if for every point there is a point of such that . The set is said to be totally bounded if for every there is a finite -net for .

By a relatively sequentially compact set in a modular space we mean that its closure is sequentially compact.

The following lemma is a counterpart of totally boundedness in metric spaces and has a same argument which is omitted here.

Lemma 2.3. Let A be a subset of a modular space satisfying . Then, (1)if A is totally bounded and X is complete, each sequence in A has a convergent subsequence in ;(2)if A is relatively sequentially compact, A is totally bounded.

Let be a modular space and let be a nonempty subset of . The diameter of is defined by and the set is bounded if and only if .

Definition 2.4. Let be a modular space and let be a bounded subset of . Then the Kuratowski constant of the set is defined as the greatest upper bound for that can be covered with a finite number of sets of diameter less than .

Theorem 2.5. Every -contraction with has a fixed point.

Proof. Choose and . Since is -contraction, there exists for which By induction, there exists a sequence in such that Put and choose . We claim that To see this, let and . By Definition 2.4, there exists a finite number of sets with diameter less than which covers . Let . By part (2) of Lemma 2.3, the totally boundedness of implies that for there exists a finite -net for . Let and . We will show that where Let , for some . Since , then for some . If , there exists such that . Also there exists such that Therefore, It implies that . Also for each , we have therefore, , that is, . Hence . Since for each , , we get . Consequently, for every It implies that and hence is totally bounded. By Lemma 2.3, has a convergent subsequence . Let . We have also therefore, . On the other hand, for each , there exists an such that therefore, . Since is a compact set, has a convergent subsequence. Let . Given , there exists such that for , It shows that which completes the proof.

Definition 2.6. A sequence in is said to be approximate fixed point sequence of , if for every there exists such that , as .We recall that a subset of a vector space is called star shaped, if there exists (the center of ) such that , for every , and .

Lemma 2.7. Suppose that satisfies for every , is star shaped with the center , and is nonexpansive, that is, for each and there exists such that Then, (1)for every , where , there exist such that ;(2)if is sequentially bounded, then has an approximate fixed point sequence.

Proof. (1) Let , with . We define by . If and , there exists such that . This implies that and Since , Theorem 2.5 implies that has a fixed point.(2) Let , . By part (1), for each , there exist and such that . Since is sequentially bounded, and , as . Therefore, as , that is, has an approximate fixed point sequence.

Theorem 2.8. Let be a complete modular linear space, and let be a compact and star shaped subset of . If satisfies  for every , and is nonexpansive, then has a fixed point.

Proof. First, we show that is sequentially bounded. To see this, if 's are real numbers converging to zero and , then every subsequence of has a convergent subsequence to zero. Choose subsequence of . Since is relatively compact, there exist and a subsequence of such that as . Taking so large that , we obtain Therefore, . It implies that . Otherwise there exists and a subsequence of such that , for all . This contradicts the fact that has a convergent subsequence to zero.By part (2) of Lemma 2.7, has an approximate fixed point sequence, that is, there exist and in such that and . The sequences and have convergent subsequences and , say , and . Since so . The nonexpansivity of implies that for each there exists such that Since is compact, consider a convergent subsequence of , . Again, As , we get .

References

H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950.

J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49–65, 1959.

A. Ait Taleb and E. Hanebaly, “A fixed point theorem and its application to integral equations in modular function spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 2, pp. 419–426, 2000.

T. Dominguez-Benavides, M. A. Khamsi, and S. Samadi, “Uniformly Lipschitzian mappings in modular function spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 46, no. 2, pp. 267–278, 2001.

M. A. Khamsi, W. M. Kozłowski, and S. Reich, “Fixed point theory in modular function spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 14, no. 11, pp. 935–953, 1990.

W. M. Kozlowski, Modular Function Spaces, vol. 122 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1988.

J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.

T. Dominguez-Benavides, M. A. Khamsi, and S. Samadi, “Asymptotically regular mappings in modular function spaces,” Scientiae Mathematicae Japonicae, vol. 53, no. 2, pp. 295–304, 2001.