# Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms

Authors: Seyit Temir; Adem Kiliçman
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

LetX be a real Banach space and K a nonempty closed convex subset of X. Let Ti:K→K(i=1,2,…,m) be m asymptotically nonexpansive mappings with sequence {kn}⊂[1,∞), ∑n=1∞(kn-1)<∞, and ℱ=⋂i=1m‍F(Ti)≠∅, where F is the set of fixed points of Ti. Suppose that {ain}n=1∞,  {bin}n=1∞,  i=1,2,…,m are appropriate sequences in [0,1] and {uin}n=1∞,  i=1,2,…,m are bounded sequences in K such that ∑n=1∞bin<∞ for i=1,2,…,m. We give {xn} defined by x1∈K,xn+1=(1-a1n-b1n)yn+m-2+a1nT1nyn+m-2+b1nu1n,yn+m-2=(1-a2n-b2n)yn+m-3+a2nT2nyn+m-3+b2nu2n,…,yn+2=(1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nTm-2nyn+1+b(m-2)nu(m-2)n,yn+1=(1-a(m-1)n-b(m-1)n)yn+a(m-1)nTm-1nyn+b(m-1)nu(m-1)n,yn=(1-amn-bmn)xn+amnTmnxn+bmnumn,m≥2,n≥1. The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite family of asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature.


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## Body

## 1. Introduction

LetK be a nonempty subset of a real Banach space X and let T:K→K be a mapping. Let F(T)={x∈K:Tx=x} be the set of fixed points of T.A mappingT:K→K is called nonexpansiveif
(1)∥Tx-Ty∥≤∥x-y∥
for all x,y∈K. Similarly, T is called asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with limn→∞kn=1 such that
(2)∥Tnx-Tny∥≤kn∥x-y∥
for all x,y∈K and n≥1. The mapping T is called uniformly L-Lipschitzianif there exists a positive constant L such that
(3)∥Tnx-Tny∥≤L∥x-y∥
for all x,y∈K and n≥1.It is easy to see that ifT is asymptotically nonexpansive, then it is uniformly L-Lipschitzian with the uniform Lipschitz constant L=sup{kn:n≥1}.The class of asymptotically nonexpansive mappings which is an important generalization of the class nonexpansive maps was introduced by Goebel and Kirk [1]. They proved that every asymptotically nonexpansive self-mapping of a nonempty closed convex bounded subset of a uniformly convex Banach space has a fixed point.The main tool for approximation of fixed points of generalizations of nonexpansive mappings remains iterative technique. Iterative techniques for nonexpansive self-mappings in Banach spaces including Mann type (one-step), Ishikawa type (two-step), and three-step iteration processes have been studied extensively by various authors; see, for example, ([2–8]).Recently, Chidume and Ali [9] defined (4) and constructed the sequence for the approximation of common fixed points of finite families of asymptotically nonexpansive mappings. Yıldırım and Özdemir [10] introduced an iteration scheme for approximating common fixed points of a finite family of asymptotically quasi-nonexpansive self-mappings and proved some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces. Quan et al. [11] studied sufficient and necessary conditions for finite step iterative schemes with mean errors for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces to converge to a common fixed point. Peng [12] proved the convergence of finite step iterative schemes with mean errors for asymptotically nonexpansive mappings in Banach spaces. More recently Kızıltunç and Temir [13] introduced and studied a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces.In [9], the authors introduced an iterative process for a finite family of asymptotically nonexpansive mappings as follows:
(4)x1∈K,xn+1=(1-a1n)xn+a1nT1nyn+m-2,yn+m-2=(1-a2n)xn+a2nT2nyn+m-3,⋮yn+1=(1-a(m-1)n)xn+a(m-1)nTm-1nyn,yn=(1-amn)xn+amnTmnxn,ifm≥2,n≥1,
where T1,T2,…,Tm:K→K are m asymptotically nonexpansive mappings and {ain}⊂[0,1] for i=1,…,m.Inspired and motivated by these facts, it is our purpose in this paper to construct an iteration scheme for approximating common fixed points of finite family of asymptotically nonexpansive mappings and study weak and some strong convergence theorems for such mappings in real Banach spaces.LetX be a real Banach space and K a nonempty closed convex subset of X. Let Ti:K→K(i=1,2,…,m) be m asymptotically nonexpansive mappings with sequence {kn}⊂[1,∞), ∑n=1∞(kn-1)<∞, and ℱ=⋂i=1mF(Ti)≠∅. Suppose that {ain}n=1∞,  {bin}n=1∞, i=1,2,…,m are appropriate sequences in [0,1] and {uin}n=1∞, i=1,2,…,m are bounded sequences in K such that ∑n=1∞bin<∞ for i=1,2,…,m. Let {xn} be defined by
(5)x1∈K,xn+1=(1-a1n-b1n)yn+m-2+a1nT1nyn+m-2+b1nu1n,yn+m-2=(1-a2n-b2n)yn+m-3+a2nT2nyn+m-3+b2nu2n,⋮yn+2=(1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nTm-2nyn+1+b(m-2)nu(m-2)n,yn+1=(1-a(m-1)n-b(m-1)n)yn+a(m-1)nTm-1nyn+b(m-1)nu(m-1)n,yn=(1-amn-bmn)xn+amnTmnxn+bmnumn,m≥2,n≥1.


## 2. Preliminaries

LetX be a real Banach space, K a nonempty closed convex subset of X, and F(T) the set of fixed points of T. A Banach space X is said to be uniformly convex if the modulus of convexity of X(6)δ(ε)=inf{1-∥x+y∥2:∥x∥=∥y∥=1,∥x-y∥=ε}>0
for all 0<ε≤2 (i.e., δ:(0,2]→[0,1]). Recall that a Banach space X is said to satisfy Opial’s condition if, for each sequence {xn} in X, the condition xn⇀x implies that
(7)liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥
for all y∈X with y≠x. It is well known that all lr spaces for 1<r<∞ have this property. However, the Lr spaces do not have unless r=2.A mappingT:K→K is said to be semicompact if, for any bounded sequence {xn} in K such that ∥xn-Txn∥→0 as n→∞, there exists a subsequence say {xnj} of {xn} such that {xnj} converges strongly to some x* in K. T is said to be completely continuous if for every bounded sequence {xn} in K, there exists a subsequence say {xnj} of {xn} such that the sequence {Txnj} converges strongly to some element of the range of T.The following lemmas were given in [14, 15], respectively, and we need them to prove our main results.Lemma 1.
Let{sn}, {tn}, and {σn} be sequences of nonnegative real numbers satisfying the following conditions: for all n≥1, sn+1≤(1+σn)sn+tn, where ∑n=1∞σn<∞ and ∑n=1∞tn<∞. Then (i)
limn→∞sn exists;(ii)
in particular, if{sn} has a subsequence {snj} converging to 0, then limn→∞sn=0.Lemma 2.
Letp>1 and C>0 be two fixed numbers. Then a Banach space X is uniformly convex if and only if there exists a continuous, strictly increasing, convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(8)∥λx+(1-λ)y∥p≤λ∥x∥p+(1-λ)∥y∥pbcbcbcbcbcbcbcbcb-wp(λ)g(∥x-y∥)
for all x,y∈BC∶={x∈X:∥x∥≤C}, and λ∈[0,1], where wp(λ)=λ(1-λ)p+λp(1-λ).The following lemmas were proved in [3].Lemma 3.
LetX be a uniformly convex Banach space and BC∶={x∈X:∥x∥≤C},C>0. Then there exists a continuous, strictly increasing, convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(9)∥λx+μy+νz∥2≤λ∥x∥2+μ∥y∥2∥λx+μy+νz∥2≤+ν∥z∥2-(λμ)g(∥x-y∥)
for all x,y,z∈BC and λ,μ,ν∈[0,1] with λ+μ+ν=1.Lemma 4.
LetX be a uniformly convex Banach space, K a nonempty closed convex subset of X, and T:K→K an asymptotically nonexpansive mapping. Then I-T (I is identity mapping) is demiclosed at zero; that is, if xn→x* weakly and xn-Txn→0 strongly, then x*∈F(T), where F(T) is the set of fixed points of T.Definition 5.
A family{Ti:i∈{1,…,m}} of asymptotically nonexpansive mappings on K with ℱ=⋂i=1mF(Ti)≠∅ is said to satisfy condition (A) on K if there exists a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0,f(r)>0, for all r∈(0,∞) such that max1≤i≤m∥x-Tix∥≥f(d(x,F)) for all x∈K.


## 3. Main Results

In this section, we prove weak and strong convergence of the iterative sequence generated by iterative scheme (5) to a common element of the sets of fixed points of a finite family of asymptotically nonexpansive mappings in a real Banach space.Lemma 6.
LetX be a real Banach space and K a nonempty closed convex subset of X. Let Ti:K→K(i=1,2,…,m) be m asymptotically nonexpansive mappings with sequence {kn}⊂[1,∞), ∑n=1∞(kn-1)<∞, and ℱ=⋂i=1mF(Ti)≠∅. Suppose that {ain}n=1∞, {bin}n=1∞, i=1,2,…,m are appropriate sequences in [0,1] and {uin}n=1∞, i=1,2,…,m are bounded sequences in K such that ∑n=1∞bin<∞ for i=1,2,…,m. Let {xn} be given by (5). Then {xn} is bounded and limn→∞∥xn-p∥ exists for p∈ℱ.Proof.
For any givenp∈ℱ, since {uin}n=1∞, i=1,2,…,m are bounded sequences in K, let
(10)M=supn≥1,i=1,2,…,m∥uin-p∥.
For each n≥1, using (5), we have
(11)∥yn-p∥=∥((1-amn-bmn)xn+amnTmnxn+bmnumn)-p∥≤(1-amn-bmn)∥xn-p∥+amn∥Tmnxn-p∥+bmn∥umn-p∥≤(1-amn-bmn)∥xn-p∥+amnkn∥xn-p∥+bmn∥umn-p∥≤kn∥xn-p∥+bmnM,∥yn+1-p∥=∥((1-a(m-1)n-b(m-1)n)yn+a(m-1)nT(m-1)nyn+b(m-1)nu(m-1)n((1-a(m-1)n-b(m-1)n)yn+a(m-1)nT(m-1)nyn)-p∥≤(1-a(m-1)n-b(m-1)n)∥yn-p∥+a(m-1)n×∥T(m-1)nyn-p∥+b(m-1)n∥u(m-1)n-p∥≤(1-a(m-1)n-b(m-1)n)∥yn-p∥+a(m-1)nkn×∥yn-p∥+b(m-1)n∥u(m-1)n-p∥≤kn∥yn-p∥+b(m-1)nM≤kn2∥xn-p∥+bmnMkn+b(m-1)nM,∥yn+2-p∥=∥((1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nT(m-2)nyn+1+b(m-2)nu(m-2)n((1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nT(m-2)nyn+1)-p((1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nT(m-2)nyn+1∥≤(1-a(m-2)n-b(m-2)n)∥yn+1-p∥+a(m-2)n∥T(m-2)nyn+1-p∥+b(m-2)n∥u(m-2)n-p∥≤(1-a(m-2)n-b(m-2)n)∥yn+1-p∥+a(m-2)nkn∥yn+1-p∥+b(m-2)n∥u(m-2)n-p∥≤kn∥yn+1-p∥+b(m-2)nM≤kn3∥xn-p∥+bmnMkn2+knb(m-1)nM+b(m-2)nM,⋮∥yn+m-2-p∥=∥((1-a2n-b2n)yn+m-3+a2nT2nyn+m-3+b2nu2n)-p∥≤(1-a2n-b2n)∥yn+m-3-p∥+a2n×∥T2nyn+m-3-p∥+b2n∥u2n-p∥≤(1-a2n-b2n)∥yn+m-3-p∥+a2nkn×∥yn+m-3-p∥+b2n∥u2n-p∥≤kn∥yn+m-3-p∥+b2nM≤kn(m-1)∥xn-p∥+kn(m-2)bmnM+kn(m-3)b(m-1)nM+⋯+knb3nM+b2nM,∥xn+1-p∥=∥((1-a1n-b1n)yn+m-2+a1nT1nyn+m-2+b1nu1n)-p∥≤(1-a1n-b1n)∥yn+m-2-p∥+a1n×∥T1nyn+m-2-p∥+b1n∥u1n-p∥≤(1-a1n-b1n)∥yn+m-2-p∥+a1nkn×∥yn+m-2-p∥+b1n∥u1n-p∥≤kn∥yn+m-2-p∥+b1nM≤knm∥xn-p∥+kn(m-1)bmnM+kn(m-2)b(m-1)nM+⋯+kn2b3nM+knb2nM+b1nM.
Then we have
(12)∥xn+1-p∥≤knm∥xn-p∥+M∑i=1mkn(i-1)bin,
which leads to
(13)∥xn+1-p∥≤(1+(knm-1))∥xn-p∥+φn,n≥1,
where
(14)φn=M∑i=1mkn(i-1)bin.
Since tm-1≤mtm-1(t-1) for all t≥1, the only assumption ∑n=1∞(kn-1)<∞ is enough for the boundedness for {kn}, then kn⊂[1,D], for all n≥1, and for some D. Hence knm-1≤mDm-1(kn-1) holds for all n≥1. Therefore ∑n=1∞(knm-1)<∞ and also ∑n=1∞φn<∞. Equation (13) and Lemma 1 guarantee that the sequence {xn} is bounded and limn→∞∥xn-p∥ exists.Theorem 7.
LetX be a real uniformly convex Banach space and K a nonempty closed convex subset of X. Let Ti:K→K(i=1,2,…,m) be m asymptotically nonexpansive mappings with sequence {kn}⊂[1,∞), ∑n=1∞(kn-1)<∞ and ℱ=⋂i=1mF(Ti)≠∅. Suppose that {ain}n=1∞, {bin}n=1∞, i=1,2,…,m are appropriate sequences in [0,1] and {uin}n=1∞, i=1,2,…,m are bounded sequences in K such that ∑n=1∞bin<∞ for i=1,2,…,m. Let {xn} be given by (5). Suppose that
(15)0<liminfn→∞ain<limsupn→∞(ain+bin)<1
for i=1,…,m. Then
(16)limn→∞∥Tixn-xn∥=0,(i=1,2,…,m).Proof.
Letp∈ℱ. Then by Lemma 6, limn→∞∥xn-p∥ exists. Since {uin}n=1∞, i=1,2,…,m are bounded sequences in K, let M=supn≥1,i=1,2,…,m∥uin-p∥; moreover, it follows that {yn+m-i-p} is also bounded for each i∈{2,3,…,m}, and hence {(u(m-i+1)n-yn+i-1)} is also bounded for i∈{1,2,…,m}. By using (5), we obtain
(17)∥yn-p∥2=∥((1-amn-bmn)xn+amnTmnxn+bmnumn)-p∥2≤(1-amn-bmn)∥xn-p∥2+amn∥Tmnxn-p∥2+bmn∥umn-p∥2-(1-amn-bmn)amng(∥Tmnxn-xn∥)≤(1-amn-bmn)∥xn-p∥2+amnkn2×∥xn-p∥2+bmn∥unm-p∥2-(1-amn-bmn)amng(∥Tmnxn-xn∥)≤kn2∥xn-p∥2+bmnM2-(1-amn-bmn)×amng(∥Tmnxn-xn∥),∥yn+1-p∥2=∥((1-a(m-1)n-b(m-1)n)yn+a(m-1)n×T(m-1)nyn+b(m-1)nu(m-1)n((1-a(m-1)n-b(m-1)n)yn+a(m-1)n)-p∥2≤(1-a(m-1)n-b(m-1)n)∥yn-p∥2+a(m-1)n∥T(m-1)nyn-p∥2+b(m-1)n∥u(m-1)n-p∥2-(1-a(m-1)n-b(m-1)n)a(m-1)ng(∥T(m-1)nyn-yn∥)≤(1-a(m-1)n-b(m-1)n)∥yn-p∥2+a(m-1)nkn2∥yn-p∥2+b(m-1)nM2-(1-a(m-1)n-b(m-1)n)a(m-1)ng(∥T(m-1)nyn-yn∥)≤kn2∥yn-p∥2+b(m-1)nM2-(1-a(m-1)n-b(m-1)n)×a(m-1)ng(∥T(m-1)nyn-yn∥)≤kn2(kn2∥xn-p∥2+bmnM2-(1-amn-bmn)×amng(∥Tmnxn-xn∥)kn2∥xn-p∥2+bmnM2-(1-amn-bmn))+b(m-1)nM2-(1-a(m-1)n-b(m-1)n)a(m-1)ng(∥T(m-1)nyn-yn∥),∥yn+1-p∥2≤kn4∥xn-p∥2+kn2bmnM2+b(m-1)nM2-(1-amn-bmn)amng(∥Tmnxn-xn∥)-(1-a(m-1)n-b(m-1)n)a(m-1)ng(∥T(m-1)nyn-yn∥),∥yn+2-p∥2=∥((1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nT(m-2)nyn+1dsdsdsd+b(m-2)nu(m-2)n((1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nT(m-2)n)-p∥2≤(1-a(m-2)n-b(m-2)n)∥yn+1-p∥2+a(m-2)n∥T(m-2)nyn+1-p∥2+b(m-2)n∥u(m-2)n-p∥2-(1-a(m-2)n-b(m-2)n)a(m-2)n×g(∥T(m-2)nyn+1-yn+1∥)≤(1-a(m-2)n-b(m-2)n)∥yn+1-p∥2+a(m-2)nkn2∥yn+1-p∥2+b(m-2)nM2-(1-a(m-2)n-b(m-2)n)a(m-2)n×g(∥T(m-2)nyn+1-yn+1∥)≤kn2∥yn+1-p∥2+b(m-2)nM2-(1-a(m-2)n-b(m-2)n)a(m-2)n×g(∥T(m-2)nyn+1-yn+1∥)≤kn2(kn4∥xn-p∥2+kn2bmnM2+b(m-1)nM2-(1-amn-bmn)amng(∥Tmnxn-xn∥)-(1-a(m-1)n-b(m-1)n)a(m-1)n×g(∥T(m-1)nyn-yn∥)×(∥T(m-1)nyn-yn∥))+b(m-2)nM2-(1-a(m-2)n-b(m-2)n)a(m-2)n×g(∥T(m-2)nyn+1-yn+1∥)≤kn6∥xn-p∥2+kn4bmnM2+kn2b(m-1)nM2+b(m-2)nM2-(1-amn-bmn)amng(∥Tmnxn-xn∥)-(1-a(m-1)n-b(m-1)n)a(m-1)ng(∥T(m-1)nyn-yn∥)-(1-a(m-2)n-b(m-2)n)a(m-2)n×g(∥T(m-2)nyn+1-yn+1∥)⋮(18)∥xn+1-p∥2≤kn2m∥xn-p∥2+M2∑i=0m-1kn2ib(i+1)n-∑i=0m-2((1-a(i+1)n-b(i+1)n)a(i+1)nghghgghghg×g(∥Ti+1nyn+m-i-2-yn+m-i-2∥)(1-a(i+1)n-b(i+1)n)(1-a(i+1)n-b(i+1)n)a(i+1)n)-(1-amn-bmn)amng(∥Tmnxn-xn∥).
Note that 0≤θ2-1≤2θ(θ-1) for all θ≥1, the assumption ∑n=1∞(kn-1)<∞ implies that ∑n=1∞(kn2-1)<∞. Since {kn} is bounded, there exists D>0 such that kn∈[1,D],n≥1. Then kn2m-1≤2mD2m-1(kn-1) holds for all n≥1. Therefore, the assumption ∑n=1∞(kn-1)<∞ implies that ∑n=1∞(kn2m-1)<∞. Then
(19)(kn2m-1)∥xn-p∥2≤2mD2m-1C2(kn-1).
It follows from (18) and (19) that
(20)(1-amn-bmn)amng(∥Tmnxn-xn∥)+∑i=0m-1((1-a(i+1)n-b(i+1)n)a(i+1)ngdgdgdddd×g(∥Ti+1nyn+m-i-2-yn+m-i-2∥)((1-a(i+1)n-b(i+1)n)a(i+1)n)≤∥xn-p∥2-∥xn+1-p∥2+2mD2m-1C2(kn-1)+M2∑i=0m-1kn2ib(i+1)n.
We first obtain that
(21)(1-amn-bmn)amng(∥Tmnxn-xn∥)≤∥xn-p∥2-∥xn+1-p∥2+2mD2m-1C2(kn-1)+M2∑i=0m-1kn2ib(i+1)n.
Now if 0<liminfn→∞amn and 0<liminfn→∞amn<limsupn→∞(amn+bmn)<1, there exist a positive integer n0 and η,η′∈(0,1) such that 0<η<amn,  amn+bmn<η′<1 for all n≥n0. This implies by (21) that
(22)η(1-η′)g(∥Tmnxn-xn∥)≤∥xn-p∥2-∥xn+1-p∥2+2mD2m-1C2(kn-1)+M2∑i=0m-1kn2ib(i+1)n.
It follows from (22) that for ℓ≥n0,
(23)∑n=n0ℓg(∥Tmnxn-xn∥)≤1η(1-η′)(∑n=n0ℓ(∥xn-p∥2-∥xn+1-p∥2)gfgfgfgfgfgfgfgfgf+2mD2m-1C2∑n=n0ℓ(kn-1)fdfdfdfdfdfdfiii+M2∑n=n0ℓ∑i=0m-1kn2ib(i+1)n).
Then ∑n=n0∞g(∥Tmnxn-xn∥)<∞, and therefore limn→∞g(∥Tmnxn-xn∥)=0, and by property of g, we have limn→∞∥Tmnxn-xn∥=0. By a similar method, together with (20) and by property of g, we have
(24)limn→∞∥Tmnxn-xn∥=limn→∞∥Tm-1nyn-yn∥=limn→∞∥Tm-2nyn+1-yn+1∥⋮=limn→∞∥Tinyn+m-i-1-yn+m-i-1∥⋮=limn→∞∥T1nyn+m-2-yn+m-2∥=0
for 2≤i<m. Thus, we conclude that
(25)limsupn→∞∥Ti-1nyn+m-i-yn+m-i∥=0,
for 2≤i≤m. From (5) and for i=1,2,…,m(26)∥yn+i-yn+i-1∥=∥(1-a(m-i)n-b(m-i)n)yn+i-1dsddd+a(m-i)nT(m-i)nyn+i-1+b(m-i)nu(m-i)n-yn+i-1(1-a(m-i)n-b(m-i)n)yn+i-1∥=∥a(m-i)n(T(m-i)nyn+i-1-yn+i-1)dsdsds+b(m-i)n(u(m-i)n-yn+i-1)(T(m-i)nyn+i-1-yn+i-1)∥≤a(m-i)n∥T(m-i)nyn+i-1-yn+i-1∥+b(m-i)n∥u(m-i)n-yn+i-1∥.
This together with (25) implies that for each i=1,2,…,m-2(27)limn→∞∥yn+i-yn+i-1∥=0.
It follows from (5) that
(28)∥yn-xn∥=∥(1-amn-bmn)xn+amnTmnxn+bmnumn-xn∥≤amn∥Tmnxn-xn∥+bmn∥umn-xn∥.
Equations (24) and (28) imply that
(29)limn→∞∥yn-xn∥=0.
It follows from (5) that
(30)∥xn+1-yn+m-2∥=∥(1-a1n-b1n)yn+m-2+a1nT1nyn+m-2GFGFG+b1nu1n-yn+m-2∥≤a1n∥T1nyn+m-2-yn+m-2∥+b1n∥u1n-yn+m-2∥.
Thus, (24) and (30) guarantee that
(31)limn→∞∥xn+1-yn+m-2∥=0.
Continuing in this fashion, for each i=2,…,m we get,
(32)limn→∞∥xn+1-yn+i-2∥=0,(33)∥xn+1-xn∥=∥xn+1-yn+m-2+yn+m-2-⋯+yn+1-yn+yn-xn∥≤∥xn+1-yn+m-2∥+∥yn+m-2-yn+m-3∥+⋯+∥yn+1-yn∥+∥yn-xn∥.
Taking the limit on both sides inequality from (33), we have
(34)limn→∞∥xn+1-xn∥=0.
Since Tm is an asymptotically nonexpansive mapping with kn, we have
(35)∥xn+1-Tmnxn+1∥=∥xn+1-xn+xn-Tmnxn+Tmnxn-Tmnxn+1∥≤∥xn+1-xn∥+∥Tmnxn+1-Tmnxn∥+∥Tmnxn-xn∥≤∥xn+1-xn∥+kn∥xn+1-xn∥+∥Tmnxn-xn∥.
Taking the limit on both sides inequality (35), and by using (24), we get
(36)limn→∞∥xn+1-Tmnxn+1∥=0.
Since Tm-1 is an asymptotically nonexpansive mapping with kn, we have
(37)∥xn+1-Tm-1nxn+1∥=∥xn+1-yn+yn-Tm-1nyn+Tm-1nyn-Tm-1nxn+1∥≤∥xn+1-yn∥+∥Tm-1nxn+1-Tm-1nyn∥+∥Tm-1nyn-yn∥≤∥xn+1-yn∥+kn∥xn+1-yn∥+∥Tm-1nyn-yn∥.
Also, taking the limit on both sides inequality (37), and by using (24), we get
(38)limn→∞∥xn+1-Tm-1nxn+1∥=0.
In a similar way, one can prove that for each i=2,…,m-1(39)limn→∞∥xn-Tm-inxn∥=0.
Next, we consider
(40)∥xn-Tmxn∥≤∥xn-xn+1∥+∥Tmn+1xn+1-xn+1∥+∥Tmn+1xn+1-Tmn+1xn∥+∥Tmn+1xn-Tmxn∥≤∥xn-xn+1∥+∥Tmn+1xn+1-xn+1∥+L∥xn+1-xn∥+L∥Tmnxn-xn∥.
It follows from (34), (36), and the above inequality (40) that
(41)limn→∞∥xn-Tmxn∥=0,(42)∥xn-Tm-1xn∥≤∥xn-xn+1∥+∥Tm-1n+1xn+1-xn+1∥+∥Tm-1n+1xn+1-Tm-1n+1xn∥+∥Tm-1n+1xn-Tm-1xn∥≤∥xn-xn+1∥+∥Tm-1n+1xn+1-xn+1∥+L∥xn+1-xn∥+L∥Tm-1nxn-xn∥.
It follows from (34), (38) and (42) that
(43)limn→∞∥xn-Tm-1xn∥=0.
Continuing similar process, for each i=0,…,m-1 we get
(44)limn→∞∥xn-Tm-ixn∥=0.
The proof is completed.Theorem 8.
LetX be a real uniformly convex Banach space and K a nonempty closed convex subset of X. Let Ti:K→K(i=1,2,…,m) be m asymptotically nonexpansive mappings with sequence {kn}⊂[1,∞), ∑n=1∞(kn-1)<∞ and ℱ=⋂i=1mF(Ti)≠∅. Suppose that {ain}n=1∞, {bin}n=1∞, i=1,2,…,m are appropriate sequences in [0,1] and {uin}n=1∞, i=1,2,…,m are bounded sequences in K such that ∑n=1∞bin<∞ for i=1,2,…,m. Suppose that
(45)0<liminfn→∞ain<limsupn→∞(ain+bin)<1
for i=1,…,m. If one of {Ti} is either completely continuous or semicompact, for some i∈{1,2,…,m}, then the sequence {xn} generated by (5) converges strongly to an element of ℱ.Proof.
Assume that there existsℓ∈{1,2,…,m} such that Tℓ is semi-compact. Since {xn} is bounded and by Theorem 7, ∥xn-Tℓxn∥→0 as n→∞, there exists a subsequence {xnj} of {xn} such that xnj converges strongly to p∈K. Since limj→∞∥xnj-Tℓxnj∥=0, it follows from Lemma 4 that Tℓp=p. Also, from Theorem 7  limj→∞∥xnj-Tixnj∥=0, i=1,2,…,m. Therefore, from Lemma 4 we obtain that p∈⋂i=1mF(Ti). So {xn} converges strongly to p.
If one ofTi’s is completely continuous, say Tℓ, since {xn} is bounded, there exists a subsequence {xnj} of {xn} such that Tℓxnj converges strongly to p∈K. By Theorem 7, limj→∞∥xnj-Tℓxnj∥=0. It follows from continuity of ∥·∥ that
(46)0=limj→∞∥xnj-Tℓxnj∥=limj→∞∥xnj-p∥=0.
Using {xnj}→p as j→∞, limj→∞∥xnj-Tixnj∥=0, i=1,2,…,m and Lemma 4, we obtain that p∈ℱ=⋂i=1mF(Ti). Also using {xnj}→p as j→∞ and Lemma 6, we obtain that limn→∞∥xn-p∥=0. This completes the proof.Next, we prove a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex Banach space satisfying condition (A).Theorem 9.
LetX be a real uniformly convex Banach space and K a nonempty closed convex subset of X. Let Ti:K→K(i=1,2,…,m) be m asymptotically nonexpansive mappings with sequence {kn}⊂[1,∞), ∑n=1∞(kn-1)<∞ and satisfying the condition (A). Suppose that {ain}n=1∞, {bin}n=1∞, i=1,2,…,m are appropriate sequences in [0,1] and {uin}n=1∞, i=1,2,…,m are bounded sequences in K such that ∑n=1∞bin<∞ for i=1,2,…,m. Suppose that ℱ=⋂i=1mF(Ti)≠∅ and 0<liminfn→∞ain<limsupn→∞(ain+bin)<1 for i=1,…,m. Then the sequence {xn} generated by (5) converges strongly to an element of ℱ.Proof.
Sincelimn→∞∥xn-p∥ exists for all p∈ℱ by Lemma 6, then, for any p∈ℱ such that
(47)d(xn,ℱ)=∥xn-p∥,
we have that limn→∞∥xn-p∥ exists. It follows from (47) that limn→∞d(xn,ℱ) exists. From condition (A)
(48)0≤f(d(xn,ℱ))≤∥xn-Ti0xn∥,
where ∥xn-Ti0xn∥ is max1≤i≤m∥xn-Tixn∥. From Theorem 7  limn→∞∥xn-Ti0xn∥=0. It then follows (48) that limn→∞f(d(xn,ℱ))=0. By property of f, limn→∞d(xn,ℱ)=0. It also follows from (47) that limn→∞∥xn-p∥=0. Therefore limn→∞xn=p∈ℱ.Now, we prove the weak convergence of iteration (5) for a family of asymptotically nonexpansive mappings in a uniformly convex Banach space.Theorem 10.
LetX be a uniformly convex Banach space satisfying Opial’s condition, and let K be a nonempty closed convex subset of X. Let Ti:K→K(i=1,2,…,m) be m asymptotically nonexpansive mappings with sequence {kn}, and let the sequences {ain}n=1∞, {bin}n=1∞, and {uin}n=1∞, i=1,2,…,m be the same as in Theorem 7. Then the sequence {xn} defined by (5) converges weakly to a common fixed point of {Ti:i=1,…,m}.Proof.
It follows from Lemma6 that limn→∞∥xn-p∥ exists. Therefore, {xn-p} is a bounded sequence in X. Then by the reflexivity of X and the boundedness of {xn}, there exists a subsequence {xnj} of {xn} such that xnj⇀p weakly. By Theorem 7, limn→∞∥xn-Tixn∥=0, and I-Ti is demiclosed at 0 for i=1,2,…,m. So we obtain Tip=p for i=1,2,…,m. Finally, we prove that {xn} converges to p. Suppose p,q∈w({xn}), where w({xn}) denotes the weak limit set of {xn}. Let {xnj} and {xmj} be two subsequences of {xn} which converge weakly to p and q, respectively. Opial’s condition ensures that ω(xn) is a singleton set. It follows that p=q. Thus {xn} converges weakly to an element of ℱ. This completes the proof.



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*Source: 974317-2013-12-24.xml*