International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 530254 | https://doi.org/10.5402/2011/530254

Fatemeh Lael, Kourosh Nourouzi, "On the Fixed Points of Correspondences in Modular Spaces", International Scholarly Research Notices, vol. 2011, Article ID 530254, 7 pages, 2011. https://doi.org/10.5402/2011/530254

On the Fixed Points of Correspondences in Modular Spaces

Academic Editor: F. Balibrea
Received20 Mar 2011
Accepted26 Apr 2011
Published23 Jun 2011

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,𝑦∈𝑋 and 𝛼,𝛽≥0, 𝛼+𝛽=1.

Then (𝑋,𝜌) is called a modular space. Given a modular 𝜌, a corresponding vector space 𝑋𝜌 is given as 𝑋𝜌=𝑥∈𝑋∶lim𝜆→0𝜌(𝜆𝑥)=0,(1.1) which is called modular linear space.

It is easy to see that for every modular 𝜌 if 𝛼,𝛽∈ℝ+ and 𝛼≤𝛽, then 𝜌(𝛼𝑥)≤𝜌(𝛽𝑥), for all 𝑥∈𝑋.

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 [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 𝐵 be a subset of a modular space (𝑋,𝜌). (1)A sequence {𝑥𝑛} in 𝑋 is said to be convergent to a point 𝑥∈𝑋 and denoted by 𝑥𝑛→𝑥, if for every 𝜖>0, there is a positive integer 𝑁 such that 𝜌(𝑥𝑛−𝑥)<𝜖, for all 𝑛>𝑁. (2)The closure of 𝐵 is denoted by 𝐵 and defined as the set of all 𝑥∈𝑋 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(𝑥𝑛−𝑥𝑚)<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 𝜖𝑛𝑥𝑛→0, as ğ‘›â†’âˆž.
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 𝑓∶𝐶↠𝑋, an element 𝑥∈𝐶 is said to be a fixed point if 𝑥∈𝑓(𝑥). Also, ⋃𝑓(𝐶)=𝑐∈𝐶𝑓(𝑐).

Definition 1.2. Let 𝐶 be a subset of 𝑋 and let 𝑘∈[0,1). We say that 𝑓∶𝐶↠𝐶 is a 𝑘-contraction if for each 𝑥,𝑦∈𝐶 and 𝑝∈𝑓(𝑥) there is ğ‘žâˆˆğ‘“(𝑦) which satisfies the condition: 𝜌(ğ‘âˆ’ğ‘ž)≤𝑘𝜌(𝑥−𝑦).(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{𝑤∶𝜌(𝑡𝑥)≤𝑤𝜌(𝑥)∶𝑥∈𝑋,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 𝑓∶𝐵↠𝐵 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∈ℕ such that 𝜌𝑥𝑛<𝜖−𝑥2𝑤𝜌,(2)(2.1) for all 𝑛>𝑘. Thus 𝜌𝑥𝑛+𝑟−𝑥𝑛≤𝑤𝜌𝑥(2)𝜌𝑛+𝑟−𝑥+𝑤𝜌𝑥(2)𝜌𝑛<𝜖−𝑥2+𝜖2<𝜖,(2.2) for every 𝑛>𝑘, 𝑟∈ℕ. 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 𝑀𝜖⊆𝑋 is called an 𝜖-net for 𝐶 if for every point 𝑧∈𝐶 there is a point 𝑥 of 𝑀𝜖 such that 𝜌(𝑧−𝑥)<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)=supğ‘Ž,𝑏∈𝐴𝜌(ğ‘Žâˆ’ğ‘),(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 𝑓∶𝐵↠𝐵 with 𝑤𝜌(2)2𝑘<1/2 has a fixed point.

Proof. Choose 𝑝0∈𝐵 and 𝑝1∈𝑓(𝑝0). Since 𝑓 is 𝑘-contraction, there exists 𝑝2∈𝑓(𝑝1) for which 𝜌𝑝1−𝑝2𝑝≤𝑘𝜌0−𝑝1.(2.4) By induction, there exists a sequence {𝑝𝑛} in 𝐵 such that 𝑝𝑛+1𝑝∈𝑓𝑛,𝜌𝑝𝑛+1−𝑝𝑛𝑝≤𝑘𝜌𝑛−𝑝𝑛−1.(2.5) Put 𝑃={𝑝𝑛∶𝑛∈ℕ} 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 𝑥𝑖∈𝑁𝑖. By part (2) of Lemma 2.3, the totally boundedness of 𝑓(𝑥𝑖) implies that for 𝑟𝜖=2𝑤𝜌(2)21−2𝑘𝑤𝜌(2)2𝑙,(2.7) there exists a finite 𝜖-net 𝑀𝜖,𝑖⊆𝐵 for 𝑓(𝑥𝑖). Let 𝑀𝜖,𝑖={𝑧𝑖1,𝑧𝑖2,…,𝑧𝑖𝑛(𝑖)} and ⋃𝑍=𝑛𝑖=1𝑀𝜖,𝑖. We will show that 𝑓(𝑃)⊆𝑧∈𝑍𝐵(𝑧),𝐷(𝐵(𝑧))<𝑟,(2.8) where 𝑟𝐵(𝑧)=𝑥∈𝐵∶𝜌(𝑥−𝑧)<2𝑤𝜌.(2)(2.9) Let 𝑦∈𝑓(𝑝𝑛), for some 𝑛∈ℕ. Since ⋃𝑃⊆𝑛𝑖=1𝑁𝑖, then 𝑝𝑛∈𝑁𝑖 for some 𝑖. If 𝑝𝑛,𝑥i∈𝑁𝑖, 𝑦∈𝑓(𝑝𝑛) there exists 𝑥∈𝑓(𝑥𝑖) such that 𝜌(𝑦−𝑥)≤𝑘𝜌(𝑥𝑖−𝑝𝑛). Also there exists 𝑧𝑖𝑗(𝑗∈{1,2,…,𝑛(𝑖)}) such that 𝜌𝑥−𝑧𝑖𝑗<𝑟2𝑤𝜌(2)21−2𝑘𝑤𝜌(2)2𝑙.(2.10) Therefore, 𝜌𝑦−𝑧𝑖𝑗≤𝑤𝜌(2)𝜌(𝑦−𝑥)+𝑤𝜌(2)𝜌𝑥−𝑧𝑖𝑗≤𝑤𝜌𝑝(2)𝑘𝜌𝑛−𝑥𝑖+𝑤𝜌(2)𝜌𝑥−𝑧𝑖𝑗<𝑤𝜌𝑟(2)𝑘𝑙+𝑟𝑤𝜌(2)2𝑤𝜌(2)21−2𝑘𝑤𝜌(2)2𝑙<𝑟2𝑤𝜌(.2)(2.11) It implies that ⋃𝑓(𝑃)⊆𝑧∈𝑃𝐵(𝑧). Also for each 𝑧′,ğ‘§î…žî…žâˆˆğµ(𝑧), we have ğœŒî€·ğ‘§î…žâˆ’ğ‘§î…žî…žî€¸â‰¤ğ‘¤ğœŒî€·ğ‘§(2)ğœŒî…žî€¸âˆ’ğ‘§+𝑤𝜌𝑧(2)ğœŒî…žî…žî€¸âˆ’ğ‘§â‰¤ğ‘¤ğœŒğ‘Ÿ(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+1∈𝑓(𝑝𝑛), we get 𝛼(𝑃)≤𝛼(𝑓(𝑃)). Consequently, for every 𝑛∈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 limğ‘™â†’âˆžğ‘ğ‘›ğ‘™=𝑝. We have 𝜌𝑝𝑛𝑙+1−𝑝𝑛𝑙≤𝑘𝑛𝑙𝜌𝑝1−𝑝0,(2.14) also 𝜌𝑝𝑛𝑙+1−𝑝≤𝑤𝜌𝑝(2)𝜌𝑛𝑙+1−𝑝𝑛𝑙+𝑤𝜌𝑝(2)𝜌𝑛𝑙−𝑝,(2.15) therefore, limğ‘™â†’âˆžğœŒ(𝑝𝑛𝑙+1−𝑝)=0. On the other hand, for each 𝑙, there exists an 𝑥𝑛𝑙∈𝑓(𝑝) such that 𝜌𝑝𝑛𝑙+1−𝑥𝑛𝑙𝑝≤𝑘𝜌𝑛𝑙−𝑝,(2.16) therefore, limğ‘™â†’âˆžğœŒ(𝑝𝑛𝑙+1−𝑥𝑛𝑙)=0. Since 𝑓(𝑝) is a compact set, {𝑥𝑛𝑙} has a convergent subsequence. Let limğ‘–â†’âˆžğ‘¥ğ‘›ğ‘™ğ‘–=𝑥. Given 𝜖>0, there exists 𝑖0∈ℕ such that for 𝑖>𝑖0, 𝜌𝑥−𝑝3≤𝜌𝑥−𝑥𝑛𝑙𝑖𝑥+𝜌𝑛𝑙𝑖−𝑝𝑛𝑙𝑖+1𝑝+𝜌𝑛𝑙𝑖+1−𝑝<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 𝑓∶𝐶↠𝐶, if for every 𝑛∈ℕ there exists 𝑝𝑛∈𝑓(𝑦𝑛) such that 𝜌(𝑦𝑛−𝑝𝑛)→0, as ğ‘›â†’âˆž.
We recall that a subset 𝐶 of a vector space is called star shaped, if there exists 𝑧∈𝐶 (the center of 𝐶) such that 𝜆𝑥+(1−𝜆)𝑧∈𝐶, for every 𝑥∈𝐶, 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 𝑓∶𝐵↠𝐵 is nonexpansive, that is, for each 𝑥,𝑦∈𝐵 and 𝑝∈𝑓(𝑥) there exists ğ‘žâˆˆğ‘“(𝑦) such that 𝜌(ğ‘âˆ’ğ‘ž)≤𝜌(𝑥−𝑦).(2.18) Then, (1)for every 𝛼,𝛽∈ℝ+, where 𝛼+𝛽=1, there exist 𝑝0∈𝐵 such that 𝑝0∈𝛼𝑧+𝛽𝑓(𝑝0);(2)if 𝑓(𝐵) is sequentially bounded, then 𝑓 has an approximate fixed point sequence.

Proof. (1) Let 𝛼,𝛽∈ℝ+, with 𝛼+𝛽=1. We define 𝑔∶𝐵↠𝐵 by 𝑔(𝑥)=𝛼𝑧+𝛽𝑓(𝑥). If 𝑥,𝑦∈𝐵 and 𝑝∈𝑓(𝑥), there exists ğ‘žâˆˆğ‘“(𝑦) such that 𝜌(ğ‘âˆ’ğ‘ž)≤𝜌(𝑥−𝑦). This implies that 𝛼𝑧+ğ›½ğ‘žâˆˆğ‘”(𝑦) and 𝜌(𝛼𝑧+ğ›½ğ‘âˆ’ğ›¼ğ‘§âˆ’ğ›½ğ‘ž)=𝜌(𝛽(ğ‘âˆ’ğ‘ž))≤𝑤𝜌(𝛽)𝜌(𝑥−𝑦).(2.19) Since 𝑤𝜌(𝛽)𝑤2𝜌(2)<1/2, Theorem 2.5 implies that 𝑔 has a fixed point.
(2) Let {𝑘𝑛∶𝑛∈ℕ}⊂(0,1), 𝑘𝑛→1. By part (1), for each 𝑛∈ℕ, there exist 𝑦𝑛∈𝐵 and 𝑝𝑛∈𝑓(𝑦𝑛) such that 𝑦𝑛=(1−𝑘𝑛)𝑧+𝑘𝑛𝑝𝑛. Since 𝑓(𝐵) is sequentially bounded, 𝜌(2(1−𝑘𝑛)𝑧)→0 and 𝜌(2(1−𝑘𝑛)𝑝𝑛)→0, as ğ‘›â†’âˆž. Therefore, 𝜌𝑦𝑛−𝑝𝑛2≤𝜌1−𝑘𝑛𝑝𝑛2+𝜌1−𝑘𝑛𝑧⟶0,(2.20) as ğ‘›â†’âˆž, 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 f∶𝐵↠𝐵 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 𝑝∈𝑋𝜌 and a subsequence {𝑘𝑛𝑖} of {𝑘𝑛} such that 𝜌(𝑝𝑘𝑛𝑖−𝑝)→0 as ğ‘–â†’âˆž. Taking 𝑖 so large that 2𝜆𝑘𝑛𝑖<1, we obtain 𝜌𝜆𝑛𝑘𝑖𝑝𝑛𝑘𝑖≤𝜌2𝜆𝑛𝑘𝑖𝑝𝑛𝑘𝑖−𝑝+𝜌2𝜆𝑛𝑘𝑖𝑝𝑝≤𝜌𝑛𝑘𝑖−𝑝+𝜌2𝜆𝑛𝑘𝑖𝑝.(2.21) Therefore, limğ‘–â†’âˆžğœ†ğ‘›ğ‘˜ğ‘–ğ‘ğ‘›ğ‘˜ğ‘–=0. It implies that limğ‘˜â†’âˆžğœ†ğ‘˜ğ‘ğ‘˜=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 𝑝𝑛∈𝑓(𝑦𝑛) and 𝜌(𝑝𝑛−𝑦𝑛)→0. The sequences {𝑝𝑛} and {𝑦𝑛} have convergent subsequences {𝑝𝑛𝑘} and {𝑦𝑛𝑘}, say limğ‘˜â†’âˆžğ‘ğ‘›ğ‘˜=𝑝, and limğ‘˜â†’âˆžğ‘¦ğ‘›ğ‘˜=𝑦. Since 𝜌𝑝−𝑦3≤𝜌𝑝−𝑝𝑛𝑘𝑝+𝜌𝑛𝑘−𝑦𝑛𝑘𝑦+𝜌𝑛𝑘,−𝑦(2.22) so 𝑦=𝑝. The nonexpansivity of 𝑓 implies that for each 𝑘 there exists 𝑧𝑛𝑘∈𝑓(𝑦) such that 𝜌𝑧𝑛𝑘−𝑝𝑛𝑘𝑦≤𝜌𝑛𝑘.−𝑦(2.23) Since 𝑓(𝑦) is compact, consider a convergent subsequence of {𝑧𝑛𝑘}, 𝑧𝑛𝑘𝑖→𝑧. Again, 𝜌𝑦−𝑧3≤𝜌𝑦−𝑝𝑛𝑘𝑖𝑝+𝜌𝑛𝑘𝑖−𝑧𝑛𝑘𝑖𝑧+𝜌𝑛𝑘𝑖−𝑧≤𝜌𝑦−𝑝𝑛𝑘𝑖𝑦+𝜌𝑛𝑘𝑖𝑧−𝑦+𝜌𝑛𝑘𝑖.−𝑧(2.24) As ğ‘–â†’âˆž, we get 𝑝∈𝑓(𝑝).

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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.


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