Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces

Abstract

We introduce an iterative method for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mapping {𝑇_𝑖} in a uniformly convex Banach space. We prove that under certain control conditions, the iterative sequence generated by our method is an approximating fixed point sequence of each 𝑇_𝑖. Some strong convergence theorems of the proposed method are also obtained for the following cases: all 𝑇_𝑖 are continuous and one of 𝑇_𝑖 is hemicompact, and the domain 𝐾 is compact.

1. Introduction

Let 𝑋 be a real Banach space. A subset 𝐾 of 𝑋 is called proximinal if for each 𝑥∈𝑋, there exists an element 𝑘∈𝐾 such that𝑑(𝑥,𝑘)=𝑑(𝑥,𝐾),(1.1) where 𝑑(𝑥,𝐾)=inf{‖𝑥−𝑦‖∶𝑦∈𝐾} is the distance from the point 𝑥 to the set 𝐾. It is clear that every closed convex subset of a uniformly convex Banach space is proximinal.

Let 𝑋 be a uniformly convex real Banach space, 𝐾 be a nonempty closed convex subset of 𝑋, 𝐶𝐵(𝐾) be a family of nonempty closed bounded subsets of 𝐾, and 𝑃(𝐾) be a nonempty proximinal bounded subsets of 𝐾. The Hausdorff metric on 𝐶𝐵(𝑋) is defined by 𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑(𝑦,𝐴),(1.2) for all 𝐴,𝐵∈𝐶𝐵(𝑋).

An element 𝑝∈𝐾 is called a fixed point of a single valued mapping 𝑇 if 𝑝=𝑇𝑝 and of a multivalued mapping 𝑇 if 𝑝∈𝑇𝑝. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇).

A single valued mapping 𝑇∶𝐾→𝐾 is said to be quasi-nonexpansive if ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖ for all 𝑥∈𝐾 and 𝑝∈𝐹(𝑇).

A multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is said to be:(i)quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝐻(𝑇𝑥,𝑇𝑝)≤‖𝑥−𝑝‖ for all 𝑥∈𝐾 and 𝑝∈𝐹(𝑇),(ii)nonexpansive if 𝐻(𝑇𝑥,𝑇𝑦)≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾.

It is well known that every nonexpansive multivalued mapping 𝑇 with 𝐹(𝑇)≠∅ is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive. It is clear that if 𝑇 is a quasi-nonexpansive multivalued mapping, then 𝐹(𝑇) is closed.

A map 𝑇∶𝐾→𝐶𝐵(𝐾) is called hemicompact if, for any sequence {𝑥_𝑛} in 𝐾 such that 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)→0 as 𝑛→∞, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑝∈𝐾. We note that if 𝐾 is compact, then every multivalued mapping 𝑇 is hemicompact.

In 1969, Nadler [1] proved a fixed point theorem for multivalued contraction mappings and convergence of a sequence. They extended theorems on the stability of fixed points of single-valued mappings and also given a counterexample to a theorem about (𝜀,𝜆)-uniformly locally expansive (single-valued) mappings. Later in 1997, Hu et al. [2] obtained common fixed point of two nonexpansive multivalued mappings satisfying certain contractive condition.

In 2005, Sastry and Babu [3] proved the convergence theorem of multivalued maps by defining Ishikawa and Mann iterates and gave example which shows that the limit of the sequence of Ishikawa iterates depends on the choice of the fixed point 𝑝 and the initial choice of 𝑥₀. In 2007, there is paper which generalized results of Sastry and Babu [3] to uniformly convex Banach spaces by Panyanak [4] and proved a convergence theorem of Mann iterates for a mapping defined on a noncompact domain.

Later in 2008, Song and Wang [5] shown that strong convergence for Mann and Ishikawa iterates of multivalued nonexpansive mapping 𝑇 under some appropriate conditions. In 2009, Shahzad and Zegeye [6] proved strong convergence theorems of quasi-nonexpansive multivalued mapping for the Ishikawa iteration. They also constructed an iteration scheme which removes the restriction of 𝑇 with 𝑇𝑝={𝑝} for any 𝑝∈𝐹(𝑇) which relaxed compactness of the domain of 𝑇.

Recently, Abbas et al. [7] established weak- and strong-convergence theorems of two multivalued nonexpansive mappings in a real uniformly convex Banach space by one-step iterative process to approximate common fixed points under some basic boundary conditions.

A fixed points and common fixed points theorem of multivalued maps in uniformly convex Banach space or in complete metric spaces or in convex metric spaces have been intensively studied by many authors; for instance, see [8–23].

In this paper, we generalize and modify the iteration of Abbas et al. [7] from two mapping to the infinite family mappings {𝑇_𝑖∶𝑖∈ℕ} of multivalued quasi-nonexpansive mapping in a uniformly convex Banach space.

Let {𝑇_𝑖} be a countable family of multivalued quasi-nonexpansive mapping from a bounded and closed convex subset 𝐾 of a Banach space into 𝑃(𝐾) with 𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)≠∅ and let 𝑝∈𝐹. For 𝑥₁∈𝐾, we define𝑥_𝑛+1=𝛼_𝑛,0𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝑥_𝑛,𝑖,(1.3) where the sequences {𝛼_𝑛,𝑖}⊂[0,1) satisfying ∑^∞𝑖=0𝛼_𝑛,𝑖=1 and 𝑥_𝑛,𝑖∈𝑇_𝑖𝑥_𝑛 such that 𝑑(𝑝,𝑥_𝑛,𝑖)=𝑑(𝑝,𝑇_𝑖𝑥_𝑛) for 𝑖∈ℕ. The main purpose of this paper is to prove strong convergence of the iterative scheme (1.3) to a common fixed point of 𝑇_𝑖.

2. Preliminaries

Before to say the main theorem, we need the following lemmas.

Lemma 2.1 (see [24]). Suppose that 𝑋 is a uniformly convex Banach space and 0<𝑝≤𝑡_𝑛≤𝑞<1 for all positive integers 𝑛. Also suppose that {𝑥_𝑛} and {𝑦_𝑛} are two sequences of 𝑋 such that limsup_𝑛→∞‖𝑥_𝑛‖≤𝑟, limsup_𝑛→∞‖𝑦_𝑛‖≤𝑟, and lim_𝑛→∞‖𝑡_𝑛𝑥_𝑛+(1−𝑡_𝑛)𝑦_𝑛‖=𝑟 hold for some 𝑟≥0. Then limsup_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.2 (see [25]). Let 𝐸 be a uniformly convex Banach space. For arbitrary 𝑟>0, let 𝐵_𝑟(0)∶={𝑥∈𝐸∶‖𝑥‖≤𝑟}. Then, for any given sequence {𝑥_𝑛}^∞𝑛=1⊂𝐵_𝑟(0) and for any given sequence {𝜆_𝑛}^∞𝑛=1 of positive numbers such that ∑^∞𝑖=1𝜆_𝑖=1, there exists a continuous strictly increasing convex function 𝑔∶[0,2𝑟]⟶ℝ,𝑔(0)=0,(2.1) such that for any positive integers 𝑖, 𝑗 with 𝑖<𝑗, the following inequality holds: ‖‖‖‖^∞_𝑛=1𝜆_𝑛𝑥_𝑛‖‖‖‖²≤^∞_𝑛=1𝜆_𝑛‖‖𝑥_𝑛‖‖²−𝜆_𝑖𝜆_𝑗𝑔‖‖𝑥_𝑖−𝑥_𝑗‖‖.(2.2)

3. Main Results

We first prove that the sequence {𝑥_𝑛} generated by (1.3) is an approximating fixed point sequence of each 𝑇_𝑖(𝑖∈ℕ).

Lemma 3.1. Let 𝐾 be a nonempty bounded and closed convex subset of a uniformly convex Banach space 𝑋. For 𝑖∈ℕ, let {𝑇_𝑖} be a sequence of multivalued quasi-nonexpansive mappings from 𝐾 into 𝑃(𝐾) with 𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)≠∅and 𝑝∈𝐹. Let {𝑥_𝑛} be a sequence defined by (1.3). Then(i)‖𝑥_𝑛+1−𝑝‖≤‖𝑥_𝑛−𝑝‖,(ii)lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists.

Proof. By (1.3), we have ‖‖𝑥_𝑛+1−𝑝‖‖≤𝛼_𝑛,0‖‖𝑥_𝑛−𝑝‖‖+^∞_𝑖=1=𝛼_𝑛,0𝛼_𝑛,𝑖‖‖𝑥_𝑛,𝑖−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖+^∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝑑𝑇_𝑖𝑥_𝑛,𝑝‖‖𝑥_𝑛−𝑝‖‖+^∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝐻𝑇_𝑖𝑥_𝑛,𝑇_𝑖𝑝‖‖𝑥_𝑛−𝑝‖‖+^∞_𝑖=1=‖‖𝑥𝛼_𝑛,𝑖‖‖𝑥_𝑛−𝑝_𝑛‖‖−𝑝‖‖.(3.1) So (i) is obtained. (ii) follows from (i).

Theorem 3.2. Let 𝐾 be a nonempty bounded and closed convex subset of a uniformly convex Banach space 𝑋. For 𝑖∈ℕ, let {𝑇_𝑖} be a sequence of multivalued quasi-nonexpansive mappings from 𝐾 into 𝑃(K) with 𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)≠∅ and 𝑝∈𝐹. Let {𝑥_𝑛} be a sequence defined by (1.3) with liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑗>0 for all 𝑗∈ℕ. Then lim_𝑛→∞𝑑(𝑥_𝑛,𝑇_𝑖𝑥_𝑛)=0 for all 𝑖∈ℕ.

Proof. For 𝑗∈ℕ, by Lemma 2.2, we get ‖‖𝑥_𝑛+1−𝑝‖‖²=‖‖‖‖𝛼_𝑛,0𝑥_𝑛−𝑝+^∞_𝑖=1𝛼_𝑛,𝑖𝑥_𝑛,𝑖−𝑝‖‖‖‖²≤𝛼_𝑛,0‖‖𝑥_𝑛−𝑝‖‖²+^∞_𝑖=1=𝛼_𝑛,0𝛼_𝑛,𝑖‖‖𝑥_𝑛,𝑖−𝑝‖‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝑥_𝑛−𝑥_𝑛,𝑗‖‖‖‖𝑥_𝑛−𝑝‖‖²+^∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝑑𝑇_𝑖𝑥_𝑛,𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝑥_𝑛−𝑥_𝑛,𝑗‖‖‖‖𝑥_𝑛−𝑝‖‖²+^∞_𝑖=1≤𝛼_𝑛,0𝛼_𝑛,𝑖𝐻𝑇_𝑖𝑥_𝑛,𝑇_𝑖𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝑥_𝑛−𝑥_𝑛,𝑗‖‖‖‖𝑥_𝑛−𝑝‖‖²+^∞_𝑖=1=‖‖𝑥_𝑛𝛼_𝑛,𝑖‖‖𝑥_𝑛−𝑝‖‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝑥_𝑛−𝑥_𝑛,𝑗‖‖−𝑝‖‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝑥_𝑛−𝑥_𝑛,𝑗‖‖.(3.2) Thus, 0<𝛼_𝑛,0𝛼_𝑛,𝑗𝑔(‖𝑥_𝑛−𝑥_𝑛,𝑗‖)≤‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛+1−𝑝‖². It follows that lim_𝑛→∞𝑔(‖𝑥_𝑛−𝑥_𝑛,𝑗‖)=0. By property of 𝑔, we have lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛,𝑗‖=0. Thus lim_𝑛→∞𝑑(𝑥_𝑛,𝑇_𝑗𝑥_𝑛)=0 for 𝑖∈ℕ.

Theorem 3.3. Let 𝑋 be a uniformly convex real Banach space and 𝐾 be a bounded and closed convex subset of 𝑋. For 𝑖∈ℕ, let {𝑇_𝑖} be a sequence of multivalued quasi-nonexpansive and continuous mappings from 𝐾 into 𝑃(𝐾) with 𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)≠∅ and 𝑝∈𝐹. Let {𝑥_𝑛} be a sequence defined by (1.3) with liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑗>0 for all 𝑗∈ℕ. Assume that one of 𝑇_𝑖 is hemicompact. Then {𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑖}.

Proof. Suppose that 𝑇_𝑖 is hemicompact for some 𝑖₀∈ℕ. By Theorem 3.2, we have lim_𝑛→∞𝑑(𝑥_𝑛,𝑇_𝑖𝑥_𝑛)=0 for all 𝑖∈ℕ. Then there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that lim_𝑘→∞𝑥_𝑛𝑘=𝑞∈𝐾. From continuity of 𝑇_𝑖, we get 𝑑(𝑥_𝑛𝑘,𝑇_𝑖𝑥_𝑛𝑘)→𝑑(𝑞,𝑇_𝑖𝑞). This implies that 𝑑(𝑞,𝑇_𝑖𝑞)=0 and 𝑞∈𝐹. Since lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists, it follows that {𝑥_𝑛} converges strongly to 𝑞.

Theorem 3.4. Let 𝑋 be a uniformly convex real Banach space and 𝐾 be a compact convex subset of 𝑋. For 𝑖∈ℕ, let {𝑇_𝑖} be a sequence of multivalued quasi-nonexpansive mappings from 𝐾 into 𝑃(𝐾) with 𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)≠∅ and 𝑝∈𝐹. Let {𝑥_𝑛} be a sequence defined by (1.3) with liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑗>0 for all 𝑗∈ℕ. Then {𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑖}.

Proof. From the compactness of 𝐾, there exists a subsequence {𝑥_𝑛𝑘}^∞𝑛=𝑘 of {𝑥_𝑛}^∞𝑛=1 such that lim_𝑘→∞‖𝑥_𝑛𝑘−𝑞‖=0 for some 𝑞∈𝐾. Thus, it follows by Theorem 3.2 that, 𝑑𝑞,𝑇_𝑖𝑞≤≤𝑑2‖‖𝑥𝑞,_𝑛𝑘𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘,𝑇_𝑖𝑥_𝑛𝑘+H𝑇_𝑖𝑥_𝑛𝑘,𝑇_𝑖𝑞−𝑞‖‖+𝑑𝑥_𝑛𝑘,𝑇_𝑖𝑥_𝑛𝑘⟶0as𝑘⟶∞.(3.3) Hence 𝑞∈𝐹. By Lemma 3.1(ii), lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists. Hence lim_𝑛→∞𝑥_𝑛=𝑞. The proof is complete.

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The first author is supported by the Graduate School, Chiang Mai University, Thailand.