Table of Contents
ISRN Geometry
Volume 2011, Article ID 530254, 7 pages
http://dx.doi.org/10.5402/2011/530254
Research Article

On the Fixed Points of Correspondences in Modular Spaces

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 𝑋 is a real functional 𝜌 on 𝑋 which satisfies the conditions:(1)𝜌(𝑥)=0 if and only if 𝑥=0,(2)𝜌(𝑥)=𝜌(𝑥),(3)𝜌(𝛼𝑥+𝛽𝑦)𝜌(𝑥)+𝜌(𝑦), for all 𝑥,𝑦𝑋 and 𝛼,𝛽0, 𝛼+𝛽=1.

Then (𝑋,𝜌) is called a modular space. Given a modular 𝜌, a corresponding vector space 𝑋𝜌 is given as 𝑋𝜌=𝑥𝑋lim𝜆0𝜌(𝜆𝑥)=0,(1.1) which is called modular linear space.

It is easy to see that for every modular 𝜌 if 𝛼,𝛽+ and 𝛼𝛽, then 𝜌(𝛼𝑥)𝜌(𝛽𝑥), for all 𝑥𝑋.

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 𝐿𝑝. 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 [35].

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 𝐵 be a subset of a modular space (𝑋,𝜌). (1)A sequence {𝑥𝑛} in 𝑋 is said to be convergent to a point 𝑥𝑋 and denoted by 𝑥𝑛𝑥, if for every 𝜖>0, there is a positive integer 𝑁 such that 𝜌(𝑥𝑛𝑥)<𝜖, for all 𝑛>𝑁. (2)The closure of 𝐵 is denoted by 𝐵 and defined as the set of all 𝑥𝑋 such that there is a sequence {𝑥𝑛} of 𝐵 which is convergent to 𝑥. We say that 𝐵 is closed if 𝐵=𝐵. (3)A sequence {𝑥𝑛} in 𝑋 is said to be Cauchy, if for every 𝜖>0, there is a positive integer 𝑁 such that 𝜌(𝑥𝑛𝑥𝑚)<𝜖, for all 𝑚,𝑛>𝑁. (4)𝐵 is said to be complete if each Cauchy sequence in 𝐵 is convergent to a point of 𝐵. (5)𝐵 is said to be compact if every sequence in 𝐵 has a convergent subsequence in 𝐵.(6)𝐵 is called sequentially bounded, if for each {𝑥𝑛}𝐵 and each real sequence {𝜖𝑛} converging to zero we have 𝜖𝑛𝑥𝑛0, as 𝑛.
By a correspondence 𝑓 from a set 𝑋 to a set 𝑌 we mean a relation that assigns to each 𝑥 in 𝑋 a nonempty subset 𝑓(𝑥) of 𝑌. For any subset 𝐶 of 𝑋 and correspondence 𝑓𝐶𝑋, an element 𝑥𝐶 is said to be a fixed point if 𝑥𝑓(𝑥). Also, 𝑓(𝐶)=𝑐𝐶𝑓(𝑐).

Definition 1.2. Let 𝐶 be a subset of 𝑋 and let 𝑘[0,1). We say that 𝑓𝐶𝐶 is a 𝑘-contraction if for each 𝑥,𝑦𝐶 and 𝑝𝑓(𝑥) there is 𝑞𝑓(𝑦) which satisfies the condition: 𝜌(𝑝𝑞)𝑘𝜌(𝑥𝑦).(1.2)

Definition 1.3 (see [8]). For a modular space (𝑋,𝜌), the function 𝑤𝜌 which is called growth function is defined on [0,) as follows: 𝑤𝜌(𝑡)=inf{𝑤𝜌(𝑡𝑥)𝑤𝜌(𝑥)𝑥𝑋,0<𝜌(𝑥)}.(1.3)

2. Main Results

In the sequel, it is assumed that 𝐵 is a closed subset of complete modular space (𝑋,𝜌) and 𝑓𝐵𝐵 is a correspondence with compact values.

Lemma 2.1. Let (𝑋,𝜌) be a modular space satisfying 𝑤𝜌(2)<. Then every convergent sequence in (𝑋,𝜌) is a Cauchy sequence in (𝑋,𝜌).

Proof. Let {𝑥𝑛} be a convergent sequence in (𝑋,𝜌). Then for 𝜖>0, there exists 𝑘 such that 𝜌𝑥𝑛<𝜖𝑥2𝑤𝜌,(2)(2.1) for all 𝑛>𝑘. Thus 𝜌𝑥𝑛+𝑟𝑥𝑛𝑤𝜌𝑥(2)𝜌𝑛+𝑟𝑥+𝑤𝜌𝑥(2)𝜌𝑛<𝜖𝑥2+𝜖2<𝜖,(2.2) for every 𝑛>𝑘, 𝑟. Hence {𝑥𝑛} is a Cauchy sequence.

Definition 2.2. Let 𝐶 be a subset of a modular space 𝑋 and let 𝜖>0 be given. A set 𝑀𝜖𝑋 is called an 𝜖-net for 𝐶 if for every point 𝑧𝐶 there is a point 𝑥 of 𝑀𝜖 such that 𝜌(𝑧𝑥)<𝜖. The set 𝐶 is said to be totally bounded if for every 𝜖 there is a finite 𝜖-net for 𝐶.

By a relatively sequentially compact set in a modular space (𝑋,𝜌) 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 𝑋 satisfying 𝑤𝜌(2)<. Then, (1)if A is totally bounded and X is complete, each sequence in A has a convergent subsequence in 𝑋;(2)if A is relatively sequentially compact, A is totally bounded.

Let (𝑋,𝜌) be a modular space and let 𝐴 be a nonempty subset of 𝑋. The diameter of 𝐴 is defined by 𝐷(𝐴)=sup𝑎,𝑏𝐴𝜌(𝑎𝑏),(2.3) and the set 𝐴 is bounded if and only if 𝐷(𝐴)<.

Definition 2.4. Let (𝑋,𝜌) be a modular space and let 𝐴 be a bounded subset of 𝑋. Then the Kuratowski constant 𝛼(𝐴) of the set 𝐴 is defined as the greatest upper bound for 𝜖>0 that 𝐴 can be covered with a finite number of sets of diameter less than 𝜖.

Theorem 2.5. Every 𝑘-contraction 𝑓𝐵𝐵 with 𝑤𝜌(2)2𝑘<1/2 has a fixed point.

Proof. Choose 𝑝0𝐵 and 𝑝1𝑓(𝑝0). Since 𝑓 is 𝑘-contraction, there exists 𝑝2𝑓(𝑝1) for which 𝜌𝑝1𝑝2𝑝𝑘𝜌0𝑝1.(2.4) By induction, there exists a sequence {𝑝𝑛} in 𝐵 such that 𝑝𝑛+1𝑝𝑓𝑛,𝜌𝑝𝑛+1𝑝𝑛𝑝𝑘𝜌𝑛𝑝𝑛1.(2.5) Put 𝑃={𝑝𝑛𝑛} and choose 𝑙(2𝑘𝑤𝜌(2)2,1). We claim that 𝛼(𝑓(𝑃))𝑙𝛼(𝑃).(2.6) To see this, let 𝑟>𝑟>0 and 𝑙𝛼(𝑃)<𝑟. By Definition 2.4, there exists a finite number of sets 𝑁𝑖(𝑖{1,2,,𝑛}) with diameter less than 𝑟/𝑙 which covers {𝑝𝑛}. Let 𝑥𝑖𝑁𝑖. By part (2) of Lemma 2.3, the totally boundedness of 𝑓(𝑥𝑖) implies that for 𝑟𝜖=2𝑤𝜌(2)212𝑘𝑤𝜌(2)2𝑙,(2.7) there exists a finite 𝜖-net 𝑀𝜖,𝑖𝐵 for 𝑓(𝑥𝑖). Let 𝑀𝜖,𝑖={𝑧𝑖1,𝑧𝑖2,,𝑧𝑖𝑛(𝑖)} and 𝑍=𝑛𝑖=1𝑀𝜖,𝑖. We will show that 𝑓(𝑃)𝑧𝑍𝐵(𝑧),𝐷(𝐵(𝑧))<𝑟,(2.8) where 𝑟𝐵(𝑧)=𝑥𝐵𝜌(𝑥𝑧)<2𝑤𝜌.(2)(2.9) Let 𝑦𝑓(𝑝𝑛), for some 𝑛. Since 𝑃𝑛𝑖=1𝑁𝑖, then 𝑝𝑛𝑁𝑖 for some 𝑖. If 𝑝𝑛,𝑥i𝑁𝑖, 𝑦𝑓(𝑝𝑛) there exists 𝑥𝑓(𝑥𝑖) such that 𝜌(𝑦𝑥)𝑘𝜌(𝑥𝑖𝑝𝑛). Also there exists 𝑧𝑖𝑗(𝑗{1,2,,𝑛(𝑖)}) such that 𝜌𝑥𝑧𝑖𝑗<𝑟2𝑤𝜌(2)212𝑘𝑤𝜌(2)2𝑙.(2.10) Therefore, 𝜌𝑦𝑧𝑖𝑗𝑤𝜌(2)𝜌(𝑦𝑥)+𝑤𝜌(2)𝜌𝑥𝑧𝑖𝑗𝑤𝜌𝑝(2)𝑘𝜌𝑛𝑥𝑖+𝑤𝜌(2)𝜌𝑥𝑧𝑖𝑗<𝑤𝜌𝑟(2)𝑘𝑙+𝑟𝑤𝜌(2)2𝑤𝜌(2)212𝑘𝑤𝜌(2)2𝑙<𝑟2𝑤𝜌(.2)(2.11) It implies that 𝑓(𝑃)𝑧𝑃𝐵(𝑧). Also for each 𝑧,𝑧𝐵(𝑧), we have 𝜌𝑧𝑧𝑤𝜌𝑧(2)𝜌𝑧+𝑤𝜌𝑧(2)𝜌𝑧𝑤𝜌𝑟(2)2𝑤𝜌(2)+𝑤𝜌𝑟(2)2𝑤𝜌(2)<𝑟,(2.12) therefore, 𝐷(𝐵(𝑧))<𝑟, that is, 𝛼(𝑓(𝑃))<𝑟. Hence 𝛼(𝑓(𝑃))𝑙𝛼(𝑃). Since for each 𝑛, 𝑝𝑛+1𝑓(𝑝𝑛), we get 𝛼(𝑃)𝛼(𝑓(𝑃)). Consequently, for every 𝑛𝛼(𝑃)𝛼(𝑓(𝑃))𝑙𝛼(𝑃)𝑙𝛼(𝑓(𝑃))𝑙𝑛𝛼(𝑃).(2.13) It implies that 𝛼(𝑃)=0 and hence 𝑃 is totally bounded. By Lemma 2.3, {𝑝𝑛} has a convergent subsequence {𝑝𝑛𝑙}. Let lim𝑙𝑝𝑛𝑙=𝑝. We have 𝜌𝑝𝑛𝑙+1𝑝𝑛𝑙𝑘𝑛𝑙𝜌𝑝1𝑝0,(2.14) also 𝜌𝑝𝑛𝑙+1𝑝𝑤𝜌𝑝(2)𝜌𝑛𝑙+1𝑝𝑛𝑙+𝑤𝜌𝑝(2)𝜌𝑛𝑙𝑝,(2.15) therefore, lim𝑙𝜌(𝑝𝑛𝑙+1𝑝)=0. On the other hand, for each 𝑙, there exists an 𝑥𝑛𝑙𝑓(𝑝) such that 𝜌𝑝𝑛𝑙+1𝑥𝑛𝑙𝑝𝑘𝜌𝑛𝑙𝑝,(2.16) therefore, lim𝑙𝜌(𝑝𝑛𝑙+1𝑥𝑛𝑙)=0. Since 𝑓(𝑝) is a compact set, {𝑥𝑛𝑙} has a convergent subsequence. Let lim𝑖𝑥𝑛𝑙𝑖=𝑥. Given 𝜖>0, there exists 𝑖0 such that for 𝑖>𝑖0, 𝜌𝑥𝑝3𝜌𝑥𝑥𝑛𝑙𝑖𝑥+𝜌𝑛𝑙𝑖𝑝𝑛𝑙𝑖+1𝑝+𝜌𝑛𝑙𝑖+1𝑝<𝜖.(2.17) It shows that 𝑥=𝑝 which completes the proof.

Definition 2.6. A sequence {𝑦𝑛} in 𝐶 is said to be approximate fixed point sequence of 𝑓𝐶𝐶, if for every 𝑛 there exists 𝑝𝑛𝑓(𝑦𝑛) such that 𝜌(𝑦𝑛𝑝𝑛)0, as 𝑛.
We recall that a subset 𝐶 of a vector space is called star shaped, if there exists 𝑧𝐶 (the center of 𝐶) such that 𝜆𝑥+(1𝜆)𝑧𝐶, for every 𝑥𝐶, and 𝜆[0,1].

Lemma 2.7. Suppose that 𝜌 satisfies 𝑤𝜌(𝛽)𝑤2𝜌(2)<1/2 for every 𝛽(0,1), 𝐵 is star shaped with the center 𝑧, and 𝑓𝐵𝐵 is nonexpansive, that is, for each 𝑥,𝑦𝐵 and 𝑝𝑓(𝑥) there exists 𝑞𝑓(𝑦) such that 𝜌(𝑝𝑞)𝜌(𝑥𝑦).(2.18) Then, (1)for every 𝛼,𝛽+, where 𝛼+𝛽=1, there exist 𝑝0𝐵 such that 𝑝0𝛼𝑧+𝛽𝑓(𝑝0);(2)if 𝑓(𝐵) is sequentially bounded, then 𝑓 has an approximate fixed point sequence.

Proof. (1) Let 𝛼,𝛽+, with 𝛼+𝛽=1. We define 𝑔𝐵𝐵 by 𝑔(𝑥)=𝛼𝑧+𝛽𝑓(𝑥). If 𝑥,𝑦𝐵 and 𝑝𝑓(𝑥), there exists 𝑞𝑓(𝑦) such that 𝜌(𝑝𝑞)𝜌(𝑥𝑦). This implies that 𝛼𝑧+𝛽𝑞𝑔(𝑦) and 𝜌(𝛼𝑧+𝛽𝑝𝛼𝑧𝛽𝑞)=𝜌(𝛽(𝑝𝑞))𝑤𝜌(𝛽)𝜌(𝑥𝑦).(2.19) Since 𝑤𝜌(𝛽)𝑤2𝜌(2)<1/2, Theorem 2.5 implies that 𝑔 has a fixed point.
(2) Let {𝑘𝑛𝑛}(0,1), 𝑘𝑛1. By part (1), for each 𝑛, there exist 𝑦𝑛𝐵 and 𝑝𝑛𝑓(𝑦𝑛) such that 𝑦𝑛=(1𝑘𝑛)𝑧+𝑘𝑛𝑝𝑛. Since 𝑓(𝐵) is sequentially bounded, 𝜌(2(1𝑘𝑛)𝑧)0 and 𝜌(2(1𝑘𝑛)𝑝𝑛)0, as 𝑛. Therefore, 𝜌𝑦𝑛𝑝𝑛2𝜌1𝑘𝑛𝑝𝑛2+𝜌1𝑘𝑛𝑧0,(2.20) as 𝑛, that is, 𝑓 has an approximate fixed point sequence.

Theorem 2.8. Let 𝑋𝜌 be a complete modular linear space, and let 𝐵 be a compact and star shaped subset of 𝑋𝜌. If 𝜌 satisfies 𝑤𝜌(𝛽)𝑤2𝜌(2)<1/2  for every 𝛽(0,1), and f𝐵𝐵 is nonexpansive, then 𝑓 has a fixed point.

Proof. First, we show that 𝐵 is sequentially bounded. To see this, if 𝜆𝑘's are real numbers converging to zero and {𝑝𝑘}𝐵, then every subsequence of {𝜆𝑘𝑝𝑘} has a convergent subsequence to zero. Choose subsequence {𝜆𝑘𝑛𝑝𝑘𝑛} of {𝜆𝑘𝑝𝑘}. Since 𝐵 is relatively compact, there exist 𝑝𝑋𝜌 and a subsequence {𝑘𝑛𝑖} of {𝑘𝑛} such that 𝜌(𝑝𝑘𝑛𝑖𝑝)0 as 𝑖. Taking 𝑖 so large that 2𝜆𝑘𝑛𝑖<1, we obtain 𝜌𝜆𝑛𝑘𝑖𝑝𝑛𝑘𝑖𝜌2𝜆𝑛𝑘𝑖𝑝𝑛𝑘𝑖𝑝+𝜌2𝜆𝑛𝑘𝑖𝑝𝑝𝜌𝑛𝑘𝑖𝑝+𝜌2𝜆𝑛𝑘𝑖𝑝.(2.21) Therefore, lim𝑖𝜆𝑛𝑘𝑖𝑝𝑛𝑘𝑖=0. It implies that lim𝑘𝜆𝑘𝑝𝑘=0. Otherwise there exists 𝜖>0 and a subsequence {𝜆𝑘𝑗𝑝𝑘𝑗} of {𝜆𝑘𝑝𝑘} such that 𝜌(𝜆𝑘𝑗𝑝𝑘𝑗)>𝜖, for all 𝑗. This contradicts the fact that {𝜆𝑘𝑗𝑝𝑘𝑗} has a convergent subsequence to zero.
By part (2) of Lemma 2.7, 𝑓 has an approximate fixed point sequence, that is, there exist {𝑦𝑛} and {𝑝𝑛} in 𝐵 such that 𝑝𝑛𝑓(𝑦𝑛) and 𝜌(𝑝𝑛𝑦𝑛)0. The sequences {𝑝𝑛} and {𝑦𝑛} have convergent subsequences {𝑝𝑛𝑘} and {𝑦𝑛𝑘}, say lim𝑘𝑝𝑛𝑘=𝑝, and lim𝑘𝑦𝑛𝑘=𝑦. Since 𝜌𝑝𝑦3𝜌𝑝𝑝𝑛𝑘𝑝+𝜌𝑛𝑘𝑦𝑛𝑘𝑦+𝜌𝑛𝑘,𝑦(2.22) so 𝑦=𝑝. The nonexpansivity of 𝑓 implies that for each 𝑘 there exists 𝑧𝑛𝑘𝑓(𝑦) such that 𝜌𝑧𝑛𝑘𝑝𝑛𝑘𝑦𝜌𝑛𝑘.𝑦(2.23) Since 𝑓(𝑦) is compact, consider a convergent subsequence of {𝑧𝑛𝑘}, 𝑧𝑛𝑘𝑖𝑧. Again, 𝜌𝑦𝑧3𝜌𝑦𝑝𝑛𝑘𝑖𝑝+𝜌𝑛𝑘𝑖𝑧𝑛𝑘𝑖𝑧+𝜌𝑛𝑘𝑖𝑧𝜌𝑦𝑝𝑛𝑘𝑖𝑦+𝜌𝑛𝑘𝑖𝑧𝑦+𝜌𝑛𝑘𝑖.𝑧(2.24) As 𝑖, we get 𝑝𝑓(𝑝).

References

  1. H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950.
  2. J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49–65, 1959. View at Google Scholar · View at Zentralblatt MATH
  3. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. W. M. Kozlowski, Modular Function Spaces, vol. 122 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1988.
  7. J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.
  8. 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. View at Google Scholar · View at Zentralblatt MATH