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

Authors: Aunyarat Bunyawat; Suthep Suantai
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce an iterative method for finding a common fixed
point of a countable family of multivalued quasi-nonexpansive mapping{Ti} 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 Ti. Some strong convergence theorems of the proposed method
are also obtained for the following cases: all Ti are continuous and one of Ti is
hemicompact, and the domain K is compact.


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## Body

## 1. Introduction

LetX 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 that(1.1)d(x,k)=d(x,K),
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.LetX 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
(1.2)H(A,B)=max⁡{sup⁡x∈Ad(x,B),sup⁡y∈Bd(y,A)},
for all A,B∈CB(X).An elementp∈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 mappingT:K→K is said to be quasi-nonexpansive if ∥Tx-p∥≤∥x-p∥ for all x∈K and p∈F(T).A multivalued mappingT: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 mappingT 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 mapT:K→CB(K) is called hemicompact if, for any sequence {xn} in K such that d(xn,Txn)→0 as n→∞, there exists a subsequence {xnk} of {xn} such that xnk→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 (ɛ,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 {Ti:i∈ℕ} of multivalued quasi-nonexpansive mapping in a uniformly convex Banach space.Let{Ti} 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=1∞F(Ti)≠∅ and let p∈F. For x1∈K, we define(1.3)xn+1=αn,0xn+∑i=1∞αn,ixn,i,
where the sequences {αn,i}⊂[0,1) satisfying ∑i=0∞αn,i=1 and xn,i∈Tixn such that d(p,xn,i)=d(p,Tixn) 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 Ti.


## 2. Preliminaries

Before to say the main theorem, we need the following lemmas.Lemma 2.1 (see [24]).
Suppose thatX is a uniformly convex Banach space and 0<p≤tn≤q<1 for all positive integers n. Also suppose that {xn} and {yn} are two sequences of X such that limsupn→∞∥xn∥≤r, limsupn→∞∥yn∥≤r, and limn→∞∥tnxn+(1-tn)yn∥=r hold for some r≥0. Then limsupn→∞∥xn-yn∥=0.Lemma 2.2 (see [25]).
LetE be a uniformly convex Banach space. For arbitrary r>0, let Br(0)∶={x∈E:∥x∥≤r}. Then, for any given sequence {xn}n=1∞⊂Br(0) and for any given sequence {λn}n=1∞ of positive numbers such that ∑i=1∞λi=1, there exists a continuous strictly increasing convex function
(2.1)g:[0,2r]⟶R,g(0)=0,
such that for any positive integers i, j with i<j, the following inequality holds:
(2.2)‖∑n=1∞λnxn‖2≤∑n=1∞λn‖xn‖2-λiλjg(‖xi-xj‖).


## 3. Main Results

We first prove that the sequence{xn} generated by (1.3) is an approximating fixed point sequence of each Ti(i∈ℕ).Lemma 3.1.
LetK be a nonempty bounded and closed convex subset of a uniformly convex Banach space X. For i∈ℕ, let {Ti} be a sequence of multivalued quasi-nonexpansive mappings from K into P(K) with F∶=⋂i=1∞F(Ti)≠∅and p∈F. Let {xn} be a sequence defined by (1.3). Then(i)
∥xn+1-p∥≤∥xn-p∥,(ii)
limn→∞∥xn-p∥ exists.Proof.
By (1.3), we have
(3.1)‖xn+1-p‖≤αn,0‖xn-p‖+∑i=1∞αn,i‖xn,i-p‖=αn,0‖xn-p‖+∑i=1∞αn,id(Tixn,p)≤αn,0‖xn-p‖+∑i=1∞αn,iH(Tixn,Tip)≤αn,0‖xn-p‖+∑i=1∞αn,i‖xn-p‖=‖xn-p‖.
So (i) is obtained. (ii) follows from (i).Theorem 3.2.
LetK be a nonempty bounded and closed convex subset of a uniformly convex Banach space X. For i∈ℕ, let {Ti} be a sequence of multivalued quasi-nonexpansive mappings from K into P(K) with F∶=⋂i=1∞F(Ti)≠∅ and p∈F. Let {xn} be a sequence defined by (1.3) with liminfn→∞αn,0αn,j>0 for all j∈ℕ. Then limn→∞d(xn,Tixn)=0 for all i∈ℕ.Proof.
Forj∈ℕ, by Lemma 2.2, we get
(3.2)‖xn+1-p‖2=‖αn,0(xn-p)+∑i=1∞αn,i(xn,i-p)‖2≤αn,0‖xn-p‖2+∑i=1∞αn,i‖xn,i-p‖2-αn,0αn,jg(‖xn-xn,j‖)=αn,0‖xn-p‖2+∑i=1∞αn,i(d(Tixn,p))2-αn,0αn,jg(‖xn-xn,j‖)≤αn,0‖xn-p‖2+∑i=1∞αn,i(H(Tixn,Tip))2-αn,0αn,jg(‖xn-xn,j‖)≤αn,0‖xn-p‖2+∑i=1∞αn,i‖xn-p‖2-αn,0αn,jg(‖xn-xn,j‖)=‖xn-p‖2-αn,0αn,jg(‖xn-xn,j‖).
Thus, 0<αn,0αn,jg(∥xn-xn,j∥)≤∥xn-p∥2-∥xn+1-p∥2. It follows that limn→∞g(∥xn-xn,j∥)=0. By property of g, we have limn→∞∥xn-xn,j∥=0. Thus limn→∞d(xn,Tjxn)=0 for i∈ℕ.Theorem 3.3.
LetX be a uniformly convex real Banach space and K be a bounded and closed convex subset of X. For i∈ℕ, let {Ti} be a sequence of multivalued quasi-nonexpansive and continuous mappings from K into P(K) with F∶=⋂i=1∞F(Ti)≠∅ and p∈F. Let {xn} be a sequence defined by (1.3) with liminfn→∞αn,0αn,j>0 for all j∈ℕ. Assume that one of Ti is hemicompact. Then {xn} converges strongly to a common fixed point of {Ti}.Proof.
Suppose thatTi is hemicompact for some i0∈ℕ. By Theorem 3.2, we have limn→∞d(xn,Tixn)=0 for all i∈ℕ. Then there exists a subsequence {xnk} of {xn} such that limk→∞xnk=q∈K. From continuity of Ti, we get d(xnk,Tixnk)→d(q,Tiq). This implies that d(q,Tiq)=0 and q∈F. Since limn→∞∥xn-q∥ exists, it follows that {xn} converges strongly to q.Theorem 3.4.
LetX be a uniformly convex real Banach space and K be a compact convex subset of X. For i∈ℕ, let {Ti} be a sequence of multivalued quasi-nonexpansive mappings from K into P(K) with F∶=⋂i=1∞F(Ti)≠∅ and p∈F. Let {xn} be a sequence defined by (1.3) with liminfn→∞αn,0αn,j>0 for all j∈ℕ. Then {xn} converges strongly to a common fixed point of {Ti}.Proof.
From the compactness ofK, there exists a subsequence {xnk}n=k∞ of {xn}n=1∞ such that limk→∞∥xnk-q∥=0 for some q∈K. Thus, it follows by Theorem 3.2 that,
(3.3)d(q,Tiq)≤d(q,xnk)+d(xnk,Tixnk)+H(Tixnk,Tiq)≤2‖xnk-q‖+d(xnk,Tixnk)⟶0ask⟶∞.
Hence q∈F. By Lemma 3.1(ii), limn→∞∥xn-q∥ exists. Hence limn→∞xn=q. The proof is complete.



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*Source: 435790-2012-03-15.xml*