On the Fixed Points of Correspondences in Modular Spaces

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 𝑋_𝜌 is given as 𝑋_𝜌=𝑥∈𝑋∶lim_𝜆→0𝜌(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 𝐿^𝑝. 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 {𝑥_𝑛} in 𝑋 is said to be convergent to a point 𝑥∈𝑋 and denoted by 𝑥_𝑛→𝑥, if for every 𝜖>0, there is a positive integer 𝑁 such that 𝜌(𝑥_𝑛−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 {𝑥_𝑛} of 𝐵 which is convergent to 𝑥. We say that 𝐵 is closed if 𝐵=𝐵. (3)A sequence {𝑥_𝑛} in 𝑋 is said to be Cauchy, if for every 𝜖>0, there is a positive integer 𝑁 such that 𝜌(𝑥_𝑛−𝑥_𝑚)<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 {𝑥_𝑛}⊂𝐵 and each real sequence {𝜖_𝑛} converging to zero we have 𝜖_𝑛𝑥_𝑛→0, 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 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 𝑞∈𝑓(𝑦) which satisfies the condition: 𝜌(𝑝−𝑞)≤𝑘𝜌(𝑥−𝑦).(1.2)

Definition 1.3 (see [8]). For a modular space (𝑋,𝜌), the function 𝑤_𝜌 which is called growth function is defined on [0,∞) as follows: 𝑤_𝜌(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 𝑤_𝜌(2)<∞. Then every convergent sequence in (𝑋,𝜌) is a Cauchy sequence in (𝑋,𝜌).

Proof. Let {𝑥_𝑛} be a convergent sequence in (𝑋,𝜌). Then for 𝜖>0, there exists 𝑘∈ℕ such that 𝜌𝑥_𝑛−𝑥<𝜖2𝑤_𝜌(2),(2.1) for all 𝑛>𝑘. Thus 𝜌𝑥_𝑛+𝑟−𝑥_𝑛≤<𝑤𝜖_𝜌(2)𝜌𝑥_𝑛+𝑟−𝑥+𝑤_𝜌(2)𝜌𝑥_𝑛−𝑥<𝜖2,+2𝜖(2.2) for every 𝑛>𝑘, 𝑟∈ℕ. Hence {𝑥_𝑛} 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 𝑀_𝜖⊆𝑋 is called an 𝜖-net for 𝐶 if for every point 𝑧∈𝐶 there is a point 𝑥 of 𝑀_𝜖 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 𝑤_𝜌(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)=sup_{𝑎,𝑏∈𝐴}𝜌(𝑎−𝑏),(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 𝑤_𝜌(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≤𝑘𝜌𝑝₀−𝑝₁.(2.4) By induction, there exists a sequence {𝑝_𝑛} in 𝐵 such that 𝜌𝑝_𝑛+1𝑝_𝑛+1∈𝑓𝑝_𝑛,−𝑝_𝑛≤𝑘𝜌𝑝_𝑛−𝑝_𝑛−1.(2.5) Put 𝑃={𝑝_𝑛∶𝑛∈ℕ} and choose 𝑙∈(2𝑘𝑤_𝜌(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 𝑁_𝑖(𝑖∈{1,2,…,𝑛}) with diameter less than 𝑟/𝑙 which covers {𝑝_𝑛}. Let 𝑥_𝑖∈𝑁_𝑖. By part (2) of Lemma 2.3, the totally boundedness of 𝑓(𝑥_𝑖) implies that for 𝜖=𝑟2𝑤_𝜌(2)²1−2𝑘𝑤^𝜌(2)²𝑙,(2.7) there exists a finite 𝜖-net 𝑀_𝜖,𝑖⊆𝐵 for 𝑓(𝑥_𝑖). Let 𝑀_𝜖,𝑖={𝑧^𝑖1,𝑧^𝑖2,…,𝑧^{𝑖𝑛(𝑖)}} and 𝑍=⋃^𝑛𝑖=1𝑀_𝜖,𝑖. We will show that 𝑓(𝑃)⊆_𝑧∈𝑍𝐵(𝑧),𝐷(𝐵(𝑧))<𝑟,(2.8) where 𝐵(𝑧)=𝑥∈𝐵∶𝜌(𝑥−𝑧)<𝑟2𝑤_𝜌(2).(2.9) Let 𝑦∈𝑓(𝑝_𝑛), for some 𝑛∈ℕ. Since 𝑃⊆⋃^𝑛𝑖=1𝑁_𝑖, then 𝑝_𝑛∈𝑁_𝑖 for some 𝑖. If 𝑝_𝑛,𝑥_i∈𝑁_𝑖, 𝑦∈𝑓(𝑝_𝑛) there exists 𝑥∈𝑓(𝑥_𝑖) such that 𝜌(𝑦−𝑥)≤𝑘𝜌(𝑥_𝑖−𝑝_𝑛). Also there exists 𝑧^𝑖𝑗(𝑗∈{1,2,…,𝑛(𝑖)}) such that 𝜌𝑥−𝑧^𝑖𝑗<𝑟2𝑤_𝜌(2)²1−2𝑘𝑤^𝜌(2)²𝑙.(2.10) Therefore, 𝜌𝑦−𝑧^𝑖𝑗≤≤𝑤𝑤^𝜌_𝜌(2)𝜌(𝑦−𝑥)+𝑤_𝜌(2)𝜌𝑥−𝑧^𝑖𝑗<𝑤_𝜌(2)𝑘𝜌𝑝_𝑛−𝑥_𝑖+𝑤_𝜌(2)𝜌𝑥−𝑧^𝑖𝑗(2)𝑘𝑟𝑙+𝑟𝑤^𝜌(2)2𝑤_𝜌(2)²1−2𝑘𝑤^𝜌(2)²<𝑟𝑙2𝑤_𝜌(2).(2.11) It implies that 𝑓(𝑃)⊆⋃_𝑧∈𝑃𝐵(𝑧). Also for each 𝑧′,𝑧^∈𝐵(𝑧), we have 𝜌𝑧^−𝑧^≤≤𝑤𝑤_𝜌𝜌(2)𝜌𝑧^−𝑧+𝑤_𝜌(2)𝜌𝑧^−𝑧(2)𝑟2𝑤_𝜌(2)+𝑤^𝜌(2)𝑟<𝑟,2𝑤_𝜌(2)(2.12) therefore, 𝐷(𝐵(𝑧))<𝑟′, that is, 𝛼(𝑓(𝑃))<𝑟′. Hence 𝛼(𝑓(𝑃))≤𝑙𝛼(𝑃). Since for each 𝑛, 𝑝_𝑛+1∈𝑓(𝑝_𝑛), we get 𝛼(𝑃)≤𝛼(𝑓(𝑃)). Consequently, for every 𝑛∈ℕ𝛼(𝑃)≤𝛼(𝑓(𝑃))≤𝑙𝛼(𝑃)≤𝑙𝛼(𝑓(𝑃))≤⋯≤𝑙^𝑛𝛼(𝑃).(2.13) It implies that 𝛼(𝑃)=0 and hence 𝑃 is totally bounded. By Lemma 2.3, {𝑝_𝑛} has a convergent subsequence {𝑝_𝑛𝑙}. Let lim_𝑙→∞𝑝_𝑛𝑙=𝑝. We have 𝜌𝑝_𝑛𝑙+1−𝑝_𝑛𝑙≤𝑘^𝑛𝑙𝜌𝑝₁−𝑝₀,(2.14) also 𝜌𝑝_𝑛𝑙+1−𝑝≤𝑤_𝜌(2)𝜌𝑝_𝑛𝑙+1−𝑝_𝑛𝑙+𝑤^𝜌(2)𝜌𝑝_𝑛𝑙−𝑝,(2.15) therefore, lim_𝑙→∞𝜌(𝑝_𝑛𝑙+1−𝑝)=0. On the other hand, for each 𝑙, there exists an 𝑥_𝑛𝑙∈𝑓(𝑝) such that 𝜌𝑝_𝑛𝑙+1−𝑥_𝑛𝑙≤𝑘𝜌𝑝^𝑛_𝑙−𝑝,(2.16) therefore, lim_𝑙→∞𝜌(𝑝_𝑛𝑙+1−𝑥_𝑛𝑙)=0. Since 𝑓(𝑝) is a compact set, {𝑥_𝑛𝑙} has a convergent subsequence. Let lim_𝑖→∞𝑥_𝑛𝑙𝑖=𝑥. Given 𝜖>0, there exists 𝑖₀∈ℕ such that for 𝑖>𝑖₀, 𝜌𝑥−𝑝3≤𝜌𝑥−𝑥^𝑛𝑙𝑖+𝜌𝑥^𝑛_𝑙𝑖<𝜖.−𝑝_{𝑛𝑙𝑖+1}+𝜌𝑝^𝑛_𝑙𝑖⁺¹−𝑝(2.17) It shows that 𝑥=𝑝 which completes the proof.

Definition 2.6. A sequence {𝑦_𝑛} in 𝐶 is said to be approximate fixed point sequence of 𝑓∶𝐶↠𝐶, if for every 𝑛∈ℕ there exists 𝑝_𝑛∈𝑓(𝑦_𝑛) such that 𝜌(𝑦_𝑛−𝑝_𝑛)→0, as 𝑛→∞.
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 𝑤_𝜌(𝛽)𝑤^2𝜌(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 𝑞∈𝑓(𝑦) such that 𝜌(𝑝−𝑞)≤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 𝑞∈𝑓(𝑦) such that 𝜌(𝑝−𝑞)≤𝜌(𝑥−𝑦). This implies that 𝛼𝑧+𝛽𝑞∈𝑔(𝑦) and 𝜌(𝛼𝑧+𝛽𝑝−𝛼𝑧−𝛽𝑞)=𝜌(𝛽(𝑝−𝑞))≤𝑤_𝜌(𝛽)𝜌(𝑥−𝑦).(2.19) Since 𝑤_𝜌(𝛽)𝑤^2𝜌(2)<1/2, Theorem 2.5 implies that 𝑔 has a fixed point.
(2) Let {𝑘_𝑛∶𝑛∈ℕ}⊂(0,1), 𝑘_𝑛→1. By part (1), for each 𝑛∈ℕ, there exist 𝑦_𝑛∈𝐵 and 𝑝_𝑛∈𝑓(𝑦_𝑛) such that 𝑦_𝑛=(1−𝑘_𝑛)𝑧+𝑘_𝑛𝑝_𝑛. Since 𝑓(𝐵) is sequentially bounded, 𝜌(2(1−𝑘_𝑛)𝑧)→0 and 𝜌(2(1−𝑘_𝑛)𝑝_𝑛)→0, as 𝑛→∞. Therefore, 𝜌𝑦_𝑛−𝑝_𝑛≤𝜌21−𝑘_𝑛𝑝_𝑛+𝜌21−𝑘_𝑛𝑧⟶0,(2.20) as 𝑛→∞, that is, 𝑓 has an approximate fixed point sequence.

Theorem 2.8. Let 𝑋_𝜌 be a complete modular linear space, and let 𝐵 be a compact and star shaped subset of 𝑋_𝜌. If 𝜌 satisfies 𝑤_𝜌(𝛽)𝑤^2𝜌(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 𝜆_𝑘's are real numbers converging to zero and {𝑝_𝑘}⊆𝐵, then every subsequence of {𝜆_𝑘𝑝_𝑘} has a convergent subsequence to zero. Choose subsequence {𝜆_𝑘𝑛𝑝_𝑘𝑛} of {𝜆_𝑘𝑝_𝑘}. Since 𝐵 is relatively compact, there exist 𝑝∈𝑋_𝜌 and a subsequence {𝑘_𝑛𝑖} of {𝑘_𝑛} such that 𝜌(𝑝_𝑘𝑛𝑖−𝑝)→0 as 𝑖→∞. Taking 𝑖 so large that 2𝜆_𝑘𝑛𝑖<1, we obtain 𝜌𝜆_𝑛𝑘𝑖𝑝_𝑛𝑘𝑖≤≤𝜌𝜌2𝑝𝜆^𝑛_{𝑘𝑖𝑛𝑘𝑖}𝑝^𝑛_𝑘𝑖−𝑝+𝜌2𝜆_𝑛𝑘𝑖𝑝−𝑝+𝜌2𝜆_𝑛𝑘𝑖𝑝.(2.21) Therefore, lim_𝑖→∞𝜆_𝑛𝑘𝑖𝑝_𝑛𝑘𝑖=0. It implies that lim_𝑘→∞𝜆_𝑘𝑝_𝑘=0. Otherwise there exists 𝜖>0 and a subsequence {𝜆_𝑘𝑗𝑝_𝑘𝑗} of {𝜆_𝑘𝑝_𝑘} such that 𝜌(𝜆_𝑘𝑗𝑝_𝑘𝑗)>𝜖, for all 𝑗. This contradicts the fact that {𝜆_𝑘𝑗𝑝_𝑘𝑗} has a convergent subsequence to zero.
By part (2) of Lemma 2.7, 𝑓 has an approximate fixed point sequence, that is, there exist {𝑦_𝑛} and {𝑝_𝑛} in 𝐵 such that 𝑝_𝑛∈𝑓(𝑦_𝑛) and 𝜌(𝑝_𝑛−𝑦_𝑛)→0. The sequences {𝑝_𝑛} and {𝑦_𝑛} have convergent subsequences {𝑝_𝑛𝑘} and {𝑦_𝑛𝑘}, say lim_𝑘→∞𝑝_𝑛𝑘=𝑝, and lim_𝑘→∞𝑦_𝑛𝑘=𝑦. Since 𝜌𝑝−𝑦3≤𝜌𝑝−𝑝^𝑛𝑘+𝜌𝑝_𝑛𝑘−𝑦_𝑛𝑘+𝜌𝑦_𝑛𝑘−𝑦,(2.22) so 𝑦=𝑝. The nonexpansivity of 𝑓 implies that for each 𝑘 there exists 𝑧_𝑛𝑘∈𝑓(𝑦) such that 𝜌𝑧_𝑛𝑘−𝑝_𝑛𝑘≤𝜌𝑦_𝑛𝑘−𝑦.(2.23) Since 𝑓(𝑦) is compact, consider a convergent subsequence of {𝑧_𝑛𝑘}, 𝑧_𝑛𝑘𝑖→𝑧. Again, 𝜌𝑦−𝑧3≤𝜌𝑦−𝑝^𝑛𝑘𝑖+𝜌𝑝^𝑛_𝑘𝑖≤𝜌𝑦−𝑝_𝑛𝑘𝑖−𝑧_𝑛𝑘𝑖+𝜌𝑧^𝑛_𝑘𝑖−𝑧+𝜌𝑦^𝑛_𝑘𝑖−𝑦+𝜌𝑧_𝑛𝑘𝑖−𝑧.(2.24) As 𝑖→∞, we get 𝑝∈𝑓(𝑝).