# A New Approximation Method for Common Fixed Points of a Finite Family of Generalized Asymptotically Quasinonexpansive Mappings in Banach Spaces

Authors: Pornsak Yatakoat; Suthep Suantai
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We introduce a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasinonexpansive
mappings. Several strong and weak convergence theorems of the proposed iteration in Banach spaces are established. The main results obtianed in this
paper generalize and refine many known results in the current literature.


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## Body

## 1. Introduction

LetC be  a convex subset of a Banach space X, and let {Ti:i=1,2,…,k} be a family of self-mappings of C. Suppose that αin∈[0,1], for all n=1,2,3,… and i=1,2,…,k.Forx1∈C, let {xn} be the sequence generated by the following algorithm:(1.1)xn+1=(1-αkn)xn+αknTkny(k-1)n,y(k-1)n=(1-α(k-1)n)xn+α(k-1)nTk-1ny(k-2)n,y(k-2)n=(1-α(k-2)n)xn+α(k-2)nTk-2ny(k-3)n,⋮y2n=(1-α2n)xn+α2nT2ny1n,y1n=(1-α1n)xn+α1nT1ny0n,
where y0n=xn for all n. The iterative process (1.1) for a finite family of mappings introduced by Khan et al.  [1], and the iterative process is the generalized form of the modified Mann (one-step) iterative process by Schu [2], the modified Ishikawa (two-step) iterative process by Tan and Xu [3], and the three-step iterative process by Xu and Noor [4].Common fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities. Many researchers [1, 5–19] are interested in studying approximation method for finding common fixed points of nonlinear mapping. Also, approximation methods for finding fixed points for nonexpansive mappings can be seen in [12–16, 20, 21].In 2003, Sun [17] studied an implicit iterative scheme initiated by Xu and Ori [22] for a finite family of asymptotically quasinonexpansive mappings. Shahzand and Udomene [18], in 2006, proved some convergence theorems for the modified Ishikawa iterative process of two asymptotically quasinonexpence mappings to a common fixed point. Nammanee et al. [23] introduced a three-step iteration scheme for asymptotically nonexpansive mappings and proved weak and strong convergence theorems of that iteration scheme under some control conditions. In 2007, Fukhar-ud-din and Khan [24] studied a new three-step iteration scheme for approximating a common fixed point of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Shahzad and Zegeye [19] introduced a new concept of generalized asymptotially nonexpansive mappings and proved some strong convergence theorems for fixed points of finite family of this class. Recently, Khan et al. [1] introduced the iterative sequence (1.1) for a finite family of asymptotically quasinonexpansive mappings in Banach spaces.Motivated by Khan et al. [1], we introduce a new iterative scheme for finding a common fixed point of a finite family of generalized asymptotically quasinonexpansive mappings as follows:Forx1∈C, let {xn} be the sequence generated by(1.2)xn+1=(1-αkn)y(k-1)n+αknTkny(k-1)n,y(k-1)n=(1-α(k-1)n)y(k-2)n+α(k-1)nTk-1ny(k-2)n,y(k-2)n=(1-α(k-2)n)y(k-3)n+α(k-2)nTk-2ny(k-3)n,⋮y2n=(1-α2n)y1n+α2nT2ny1n,y1n=(1-α1n)y0n+α1nT1ny0n,
where y0n=xn for all n.The aim of this paper is to obtain strong and weak convergence results for the iterative process (1.2) of a finite family of generalized asymptotically quasinonexpansive mappings in Banach spaces.


## 2. Preliminaries

In this section, we give some definitions and lemmas used in the main results.LetC be a nonempty subset of a real Banach space X, and let T be a self-mapping of C. The fixed point set of T is denoted by F(T)={x∈C:Tx=x}.Then letT is called(i)
nonexpansive if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈C;(ii)
quasinonexpansive if F(T)≠∅ and ∥Tx-p∥≤∥x-p∥, for all x∈C and p∈F(T);(iii)
asymptotically nonexpansive if there exists a sequence {rn} in [0,∞) with limn→∞rn=0 and ∥Tnx-Tny∥≤(1+rn)∥x-y∥, for all x,y∈C and n=1,2,3,…;(iv)
asymptotically quasinonexpansive if F(T)≠∅ and there exists a sequence {rn} in [0,∞) with limn→∞rn=0 and ∥Tnx-p∥≤(1+rn)∥x-p∥, for all x∈C, p∈F(T) and n=1,2,3,…;(v)
generalized quasinonexpansiveif F(T)≠∅ and there exists a sequence {sn} in [0,∞) with sn→0 as n→∞ such that ∥Tnx-p∥≤∥x-p∥+sn, for all x∈C, p∈F(T) and n=1,2,3,…;(vi)
generalized asymptotically quasinonexpansive [19] if F(T)≠∅ and there exist two sequences {rn} and {sn} in [0,∞) with rn→0 and sn→0 as n→∞ such that ∥Tnx-p∥≤(1+rn)∥x-p∥+sn, for all x∈C, p∈F(T) and n=1,2,3,…;(vii)
uniformlyL-Lipschitzian if there exists constant L>0 such that ∥Tnx-Tny∥≤L∥x-y∥, for all x,y∈C and n=1,2,3,…;(viii)
(L-γ)uniform Lipschitz if there are constants L>0 and γ>0 such that ∥Tnx-Tny∥≤L∥x-y∥γ, for all x,y∈C and n=1,2,3,….(ix)
semicompact if for a sequence {xn} in C with limn→∞∥xn-Txn∥=0, there exists a subsequence {xni} of {xn} such that xni→p∈C.From the definition of these mappings, it can be seen that(i)
a quasinonexpansive mapping is generalized quasinonexpansive;(ii)
an asymptotically quasinonexpansive mapping is generalized asymptotically quasinonexpansive;(iii)
a generalized quasinonexpansive mapping is generalized asymptotically quasinonexpansive;(iv)
a uniformlyL-Lipschitzian mapping is(L-1)uniform Lipschitz.The mapT:C→X is said to be demiclosed at 0 if for each sequence {xn} in C converging weakly to x∈C and T(xn) converging strongly to 0, we get Tx=0.A Banach spaceX is said to have Opial's property if for each sequence {xn} converging weakly to x∈C and x≠y, we have the condition (2.1)lim infn→∞∥xn-x∥<lim infn→∞∥xn-y∥.Condition 2 (A′′).
LetC be a subset of a normed space X. A family of self-mappings {Ti:i=1,2,…,k} of C is said to have Condition (A′′) if there exists a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(r)>0 for all r∈(0,∞) such that ∥x-Tix∥≥f(d(x,F)) for some 1≤i≤k and for all x∈C where d(x,F)=inf{∥x-p∥:p∈F=⋂i=1kF(Ti)}.The condition(A′′) defined above by the authors is the generalization of the condition (A) [25] when k=1 and condition (A′) [26] for k=2.The following lemmas are needed for proving our main results.Lemma 2.1 (cf.  [17, Lemma 2.2]).
Let the sequences{an}, {δn} and {cn} of real numbers satisfy:
(2.2)an+1≤(1+δn)an+cn,wherean≥0,δn≥0,cn≥0∀n=1,2,3,…
and ∑n=1∞,δn<∞, ∑n=1∞,cn<∞. (i)
limn→∞an exists;(ii)
ifliminfn→∞an=0, then limn→∞an=0.Lemma 2.2 (see [2, Lemma 1.3]).
LetX be a uniformly convex Banach space. Assume that 0<b≤tn≤c<1, n=1,2,3,…. Let the sequences {xn} and {yn} in X be such that lim supn→∞∥xn∥≤a, lim supn→∞∥yn∥≤a and limn→∞∥tnxn+(1-tn)yn∥=a, where a≥0. Then limn→∞∥xn-yn∥=0.


## 3. Convergence in Banach Spaces

The aim of this section is to establish the strong convergence of the iterative scheme (1.2) to converge to a common fixed point of a finite family of asymptotically quasinonexpansive mappings in a Banach space under some appropriate conditions.Lemma 3.1.
LetC be a nonempty closed convex subset of a real Banach space X, and {Ti:i=1,2,…,k} be a finite family of generalized asymtotically quasinonexpansive self-mappings of C, that is, ∥Tinx-pi∥≤(1+rin)∥x-pi∥+sin, for all x∈C with the sequence {rin},{sin}⊂[0,∞) and pi∈F(Ti), i=1,2,…,k. Suppose that F=⋂i=1kF(Ti)≠∅,x1∈C, and the iterative sequence {xn} is defined by (1.2). Let rn=max1≤i≤k{rin} and sn=max1≤i≤k{sin} Then for p∈F, we get the following:(i)
∥xn-Tinxn∥≤(2+rn)∥xn-p∥+sn,foralli=1,2,…,k;(ii)
∥y(i-1)n-Tiny(i-1)n∥≤(2+rn)∥y(i-1)n-p∥+sn,foralli=1,2,…,k;(iii)
∥ykn-p∥≤(1+rn)k∥xn-p∥+sn∑i=1k(1+rn)k-i;(iv)
∥xn+1-p∥≤(1+rn)k∥xn-p∥+sn∑i=1k(1+rn)(k-i);(v)
∥xn+1-p∥≤(1+δn)∥xn-p∥+cn,foralln∈ℕ,where cn=sn∑i=1k(1+rn)(k-i) and δn=(k1)rn+(k2)rn2+⋯+(kk)rnk.(vi)
If∑n=1∞rn<∞ and ∑n=1∞sn<∞, then limn→∞∥xn-p∥ exists.Proof.
Letp∈F. (i)
Fori=1,2,3,…,k, we have
(3.1)∥xn-Tinxn∥≤∥xn-p∥+∥Tinxn-p∥≤∥xn-p∥+(1+rn)∥xn-p∥+sin,≤(2+rn)∥xn-p∥+sn.(ii)
Similarly to part (i), we have(3.2)∥y(i-1)n-Tiny(i-1)n∥≤(2+rn)∥y(i-1)n-p∥+sn,∀i=1,2,…,k.(iii)
By part (i) and0≤αin≤1, we have
(3.3)∥y1n-p∥=∥(1-α1n)(xn-p)+α1n(T1nxn-p)∥≤(1-α1n)∥xn-p∥+α1n∥T1nxn-p∥≤(1-α1n)∥xn-p∥+α1n(1+r1n)∥xn-p∥+α1ns1n≤(1+rn)∥xn-p∥+sn,∥y2n-p∥=∥(1-α2n)(y1n-p)+α2n(T2ny1n-p)∥≤(1-α2n)∥y1n-p∥+α2n∥T2ny1n-p∥≤(1-α2n)∥y1n-p∥+α2n(1+r2n)∥y1n-p∥+α2ns2n≤(1+rn)∥y1n-p∥+sn≤(1+rn)(1+rn)∥xn-p∥+sn(1+rn)+sn≤(1+rn)2∥xn-p∥+sn((1+rn)+1),∥y3n-p∥=∥(1-α3n)(y2n-p)+α3n(T3ny2n-p)∥≤(1-α3n)∥y2n-p∥+α3n∥T3ny2n-p∥≤(1-α3n)∥y2n-p∥+α3n(1+r3n)∥y2n-p∥+α3ns3n≤(1+rn)∥y2n-p∥+sn≤(1+rn)3∥xn-p∥+sn((1+rn)2+(1+rn)+1),⋮∥ykn-p∥≤(1+rn)k∥xn-p∥+sn∑i=1k(1+rn)(k-i).(iv)
By part (ii) and part (iii), we get(3.4)∥xn+1-p∥=∥(1-αkn)(y(k-1)n-p)+αkn(Tkny(k-1)n-p)∥≤(1-αkn)∥y(k-1)n-p∥+αkn∥Tkny(k-1)n-p∥≤(1-αkn)∥y(k-1)n-p∥+αkn(1+rn)∥y(k-1)n-p∥+αknskn≤(1+rn)∥y(k-1)n-p∥+sn≤(1+rn)(1+rn)k-1∥xn-p∥+(1+rn)sn∑i=1k-1(1+rn)(k-1-i)+sn=(1+rn)k∥xn-p∥+sn∑i=1k(1+rn)(k-i).(v)
Putδn=(k1)rn+(k2)rn2+⋯+(kk)rnk and cn=sn∑i=1k(1+rn)(k-i). Then (v) is directly obtained by (iv).(vi)
By(v), we have ∥xn+1-p∥≤(1+δn)∥xn-p∥+cn for all n∈N, where cn=sn∑i=1k(1+rn)(k-i) and δn=(k1)rn+(k2)rn2+⋯+(kk)rnk. From Σn=1∞rn<∞ and Σn=1∞sn<∞, it follows that Σn=1∞δn<∞ and Σn=1∞cn<∞. By Lemma 2.1, we get limn→∞∥xn-p∥ exists.Theorem 3.2.
LetC be a nonempty closed convex subset of a real Banach space X, and {Ti:i=1,2,…,k} be a finite family of generalized asymtotically quasinonexpansive self-mappings of C, that is, ∥Tinx-pi∥≤(1+rin)∥x-pi∥+sin, for all x∈C and pi∈F(Ti),i=1,2,…,k. Suppose that F=⋂i=1kF(Ti)≠∅ is closed, x1∈C and the iterative sequence {xn} is defined by (1.2). Assume that ∑n=1∞rn<∞ and ∑n=1∞sn<∞, where rn=max1≤i≤k{rin} and sn=max1≤i≤k{sin}. Then {xn} converges strongly to a common fixed point of the family of mappings if and only if lim infn→∞d(xn,F)=0, where d(x,F)=infp∈F∥x-p∥.Proof.
The necessity is obvious and then we prove only the sufficiency. Letp∈F. Since 1+t≤et for t≥0, we obtain (1+t)k≤ekt, for k=1,2,….  Thus by Lemma 3.1(iv) and (v), for positive integers m and n, we have
(3.5)∥xn+m-p∥≤(1+r(n+m-1))k∥x(n+m-1)-p∥+c(n+m-1)≤exp{kr(n+m-1)}∥x(n+m-1)-p∥+c(n+m-1)≤⋯≤exp{k∑i=nn+m-1ri}∥xn-p∥+∑i=nn+m-1ci≤exp{k∑i=1∞ri}∥xn-p∥+∑i=n∞ci=M∥xn-p∥+∑i=n∞ci,
where M=exp{k∑i=1∞ri}.
By Lemma3.1(v), we have(3.6)∥xn+1-p∥≤(1+δn)∥xn-p∥+cn,∀p∈F,
where δn=(k1)rn+(k2)rn2+⋯+(kk)rnk and cn=sn∑i=1k(1+rn)(k-i).
It follows that(3.7)d(xn+1,F)≤(1+δn)d(xn,F).
From the given condition liminfn→∞d(xn,F)=0 and Lemma 2.1, we get
(3.8)limn→∞d(xn,F)=0.
Next, we show that{xn} is a Cauchy sequence in C. By (3.8) and ∑n=1∞cn<∞, we get that for any ϵ>0, there exists a positive integer n0 such that, for all n≥n0,(3.9)d(xn,F)<ϵ3M,∑n=n0∞cn<ϵ3.
Fromd(xn,F)<ϵ/3M, there exists p0∈F such that(3.10)∥xn0-p0∥<ϵ3M.
For any positive integerm, by (3.5), (3.9), and (3.10), we have(3.11)∥xn0+m-xn0∥≤∥xn0+m-p0∥+∥xn0-p0∥≤M∥xn0-p0∥+∑i=n0∞ci+∥xn0-p0∥<Mϵ3M+ϵ3+Mϵ3M=ϵ.
Thus, {xn} is a Cauchy sequence in X. Since X is complete, xn→q∈X. Actually, q∈C because {xn}⊂C and C is a closed subset of X. Next we show that q∈F. Since F=⋂i=1kF(Ti) is closed, by the continuity of d(x,F) with d(xn,F)→0 and xn→q as n→∞, we get d(q,F)=0 and then q∈F. Therefore, the proof is complete.Since any asymptotically quasinonexpansive mapping is generalized asymptotically quasinonexpansive, the next corollary is obtained immediately from Theorem3.2.Corollary 3.3 (see [5, Theorem 3.2]).
LetC be a nonempty closed convex subset of a real Banach space X, and {Ti:i=1,2,…,k} be a finite family of asymptotically quasinonexpansive self-mappings of C, that is, ∥Tinx-pi∥≤(1+rin)∥x-pi∥, for all x∈C and pi∈F(Ti), i=1,2,…,k. x1∈C and the iterative sequence {xn} be defined by (1.2). Then {xn} converges strongly to a common fixed point of the family of mappings if and only if lim infn→∞d(xn,F)=0, where d(x,F)=infp∈F∥x-p∥.


## 4. Convergence in Uniformly Convex Banach Spaces

In this section, strong and weak convergence results for the iterative process (1.2) on uniformly convex Banach spaces are proved without using the condition liminfn→∞d(xn,F)=0 appearing in Section 3.Theorem 4.1.
LetC be a nonempty closed convex subset of an uniformly convex real Banach space X. Let {Ti:i=1,2,…,k} be a finite family of uniformly (L-γi) Lipschitzian and generalized asymptotically quasinonexpansive self-mappings of C, that is, ∥Tinx-Tiny∥≤L∥x-y∥γi and ∥Tinx-pi∥≤(1+rin)∥x-pi∥+sin, for all x,y∈C and pi∈F(Ti), i=1,2,…,k. Suppose that {Ti:i=1,2,…,k} satisfies condition (A′′) and F=⋂i=1kF(Ti)≠∅. Let x1∈C and the iterative sequence {xn} be defined by (1.2) with {αin}i=1n⊂[a,b], where 0<a<b<1. Assume that ∑n=1∞rn<∞,∑n=1∞sn<∞ where rn=max1≤i≤k{rin} and sn=max1≤i≤k{sin}. Then {xn} converges strongly to a common fixed point of the family of mappings.Proof.
Letp∈F. By Lemma 3.1(vi), we get that limn→∞∥xn-p∥ exists. Then there is a real number c≥0 such that
(4.1)limn→∞∥xn-p∥=c.
By Lemma3.1(iii), we have(4.2)∥ykn-p∥≤(1+rn)k∥xn-p∥+sn∑i=1k(1+rn)(k-i),∀p∈F.
By takinglim sup on both sides of the above inequality, we get(4.3)lim supn→∞∥yin-p∥≤c,fori=1,2,…,k-1.
Since ∥Tiny(i-1)n-p∥≤(1+rn)∥y(i-1)n-p∥+sn and (4.3), we obtain
(4.4)lim supn→∞∥Tiny(i-1)n-p∥≤c,fori=1,2,…,k.
Since limn→∞∥xn+1-p∥=c, we have
(4.5)limn→∞∥(1-αkn)(y(k-1)n-p)+αkn(Tkny(k-1)n-p)∥=c.
Using (4.1), (4.4), and Lemma 2.2, we conclude that
(4.6)limn→∞∥y(k-1)n-Tkny(k-1)n∥=0.
We assume that
(4.7)limn→∞∥y(j-1)n-Tjny(j-1)n∥=0,forsome2≤j≤k.
It follows from (4.5) and (4.7) that
(4.8)c≤liminfn→∞∥y(j-1)n-p∥,for2≤j≤k.
By Lemma 3.1(iv), (1.2), and (4.8), we get
(4.9)limn→∞∥(1-α(j-1)n)(y(j-2)n-p)+α(j-1)n(Tj-1ny(j-2)n-p)∥=limn→∞∥y(j-1)n-p∥=c.
Using (4.3), (4.4) and Lemma 2.2, we conclude that
(4.10)limn→∞∥y(j-2)n-Tj-1ny(j-2)n∥=0.
Therefore, by mathematical induction, we obtain
(4.11)limn→∞∥y(i-1)n-Tiny(i-1)n∥=0,fori=1,2,…,k.
From (1.2), we have(4.12)∥yin-y(i-1)n∥=αin∥Tiny(i-1)n-y(j-1)n∥,fori=1,2,…,k-1.
By (4.11), we obtain that
(4.13)∥yin-y(i-1)n∥→0asn→∞,fori=1,2,…,k-1.
From
(4.14)∥xn-yin∥≤∥xn-y1n∥+∥y1n-y2n∥+⋯+∥y(i-1)n-yin∥,
for i=1,2,…,k-1, it follows by (4.13) that
(4.15)∥xn-yin∥→0asn→∞,fori=1,2,…,k-1.
From (4.11), when i=1, we get limn→∞∥xn-T1nxn∥=0. For 2≤i≤k, we have
(4.16)∥xn-Tinxn∥≤∥xn-y(i-1)n∥+∥y(i-1)n-Tiny(i-1)n∥+∥Tiny(i-1)n-Tinxn∥≤∥xn-y(i-1)n∥+∥y(i-1)n-Tiny(i-1)n∥+L∥y(i-1)n-xn∥γi.
From (4.11) and (4.15), we conclude that(4.17)limn→∞γin=0,fori=1,2,…,k,
where γin=∥xn-Tinxn∥. From (1.2), we have
(4.18)∥xn+1-xn∥≤(1-αkn)∥y(k-1)n-xn∥+αkn∥Tkny(k-1)n-xn∥≤(1-αkn)∥y(k-1)n-xn∥+αkn(∥Tkny(k-1)n-y(k-1)n∥+∥y(k-1)n-xn∥)=∥y(k-1)n-xn∥+αkn∥Tkny(k-1)n-y(k-1)n∥.
From (4.11) and (4.15),
(4.19)limn→∞∥xn+1-xn∥=0.
For i=1,2,…,k, we have
(4.20)∥xn+1-Tixn+1∥≤∥xn+1-Tin+1xn+1∥+∥Tixn+1-Tin+1xn+1∥≤γi(n+1)+L∥xn+1-Tinxn+1∥γi≤γi(n+1)+L(∥xn+1-xn∥+∥xn-Tinxn∥+∥Tinxn-Tinxn+1∥)γi≤γi(n+1)+L(∥xn+1-xn∥+γin+L∥xn-xn+1∥γi)γi.
Using (4.17) and (4.19), we obtain
(4.21)limn→∞∥xn+1-Tixn+1∥=0,fori=1,2,…,k.
Therefore, by using condition (A′′), there exists a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(r)>0 for all r∈(0,∞) such that
(4.22)limn→∞f(d(xn,F))≤limn→∞∥xn-Tjxn∥=0,
for some 1≤j≤k, that is
(4.23)limn→∞d(xn,F)=0.
By Theorem 3.2, we conclude that {xn} converges strongly to a point p∈F.Lemma 4.2.
LetC be a nonempty closed convex subset of an uniformly convex real Banach space X, and {Ti:i=1,2,…,k} be a family of (L-γi) uniform Lipschitz and generalized asymtotically quasinonexpansive self-mappings of C, that is, ∥Tinx-Tiny∥≤L∥x-y∥γi and ∥Tinx-pi∥≤(1+rin)∥x-pi∥+sin, for all x,y∈C and pi∈F(Ti), i=1,2,…,k. Suppose that F=⋂i=1kF(Ti)≠∅. Let x1∈C and the iterative sequence {xn} be defined by (1.2) with {αin}i=1n⊂[a,b], where 0<a<b<1. Assume that ∑n=1∞rn<∞, ∑n=1∞sn<∞ where rn=max1≤i≤k{rin} and sn=max1≤i≤k{sin}. Then, (i)
limn→∞∥xn-Tiny(i-1)n∥=0, for all i=1,2,,…,k;(ii)
limn→∞∥xn-Tixn∥=0,   for all i=1,2,,…,k.Proof.
(i) Letp∈F. By Lemma 3.1(vi), we obtain that limn→∞∥xn-p∥ exists and we then suppose that
(4.24)limn→∞∥xn-p∥=c.
By (4.24) and Lemma 3.1(iii), we have
(4.25)lim supn→∞∥yin-p∥≤c,fori=1,2,…,k-1.
By (1.2), we have (4.26)∥xn+1-p∥≤(1-αkn)∥y(k-1)n-p∥+αkn∥Tkny(k-1)n-p∥≤(1-αkn)∥y(k-1)n-p∥+αkn(1+rn)∥y(k-1)n-p∥+αknskn≤(1+rn)∥y(k-1)n-p∥+sn=(1+rn)∥(1-α(k-1)n)(y(k-2)n-p)+α(k-1)n(Tk-1ny(k-2)n-p)∥+sn≤(1+rn)[(1-α(k-1)n)∥y(k-2)n-p∥+α(k-1)n∥Tk-1ny(k-2)n-p∥]+sn≤(1+rn)[(1-α(k-1)n)∥y(k-2)n-p∥+α(k-1)n(1+rn)∥y(k-2)n-p∥+α(k-1)nsn]+sn≤(1+rn)2∥y(k-2)n-p∥+sn(1+rn)+sn⋮≤(1+rn)k-i∥yin-p∥+sn∑i=1k-1(1+rn)k-i,
for i=1,2,…,k-1. It follows that
(4.27)c≤liminfn→∞∥yin-p∥,fori=1,2,…,k-1.
From (4.25) and (4.27), we obtain
(4.28)limn→∞∥yin-p∥=c,fori=1,2,…,k-1,
and then
(4.29)limn→∞∥(1-αin)(y(i-1)n-p)+αin(Tiny(i-1)n-p)∥=c,
for i=1,2,…,k-1.
Since∥Tiny(i-1)n-p∥≤(1+rn)∥y(i-1)n-p∥+sn, for i=1,2,…,k-1, we have(4.30)lim supn→∞∥Tiny(i-1)n-p∥≤c,fori=1,2,…,k-1.
From (4.25), (4.29), (4.30) and Lemma 2.2, we obtain
(4.31)limn→∞∥Tiny(i-1)n-y(i-1)n∥=0,fori=1,2,…,k-1.
Now we want to show that (4.31) is also true for i=k.
Since∥Tkny(k-1)n-p∥≤(1+rn)∥y(k-1)n-p∥+sn, limn→∞rn=0 and limn→∞sn=0, it follows by (4.28),(4.32)lim supn→∞∥Tkny(k-1)n-p∥≤c.
We also have
(4.33)limn→∞∥(1-αkn)(y(k-1)n-p)+αkn(Tkny(k-1)n-p)∥=limn→∞∥xn+1-p∥=c.
Hence, by (4.25), (4.32), and Lemma 2.2, we obtain
(4.34)limn→∞∥y(k-1)n-Tkny(k-1)n∥=0.
Then, (4.31) and (4.34) give us
(4.35)limn→∞∥Tiny(i-1)n-y(i-1)n∥=0,fori=1,2,…,k.
From
(4.36)∥xn-Tiny(i-1)n∥≤∥xn-y(i-1)n∥+∥y(i-1)n-Tiny(i-1)n∥,
it implies by (4.15) and (4.35) that
(4.37)limn→∞∥xn-Tiny(i-1)n∥=0,
for i=1,2,3,…,k.
(ii) From part (i), fori=1, we have(4.38)limn→∞∥T1nxn-xn∥=0.
For i=2,3,4,…,k, we get
(4.39)∥Tinxn-xn∥≤∥Tinxn-Tiny(i-1)n∥+∥Tiny(i-1)n-xn∥≤L∥xn-y(i-1)n∥γi+∥Tiny(i-1)n-xn∥.
By part (i) and (4.15),
(4.40)limn→∞∥Tinxn-xn∥=0,fori=1,2,…,k.
For 1≤i≤k, we obtain
(4.41)∥xn-Tixn∥≤∥xn-xn+1∥+∥xn+1-Tin+1xn+1∥+∥Tin+1xn+1-Tin+1xn∥+∥Tin+1xn-Tixn∥≤∥xn-xn+1∥+∥xn+1-Tin+1xn+1∥+L∥xn+1-xn∥γi+L∥Tinxn-xn∥γi.
From (4.19) and (4.40), we then have
(4.42)limn→∞∥xn-Tixn∥=0,fori=1,2,…,k.Theorem 4.3.
Under the hypotheses of Lemma4.2, assume that Tjm is semicompact for some positive integers m and 1≤j≤k. Then {xn} converges strongly to a common fixed point of the family {Ti:i=1,2,…,k}.Proof.
Suppose thatTjm is semicompact for some positive integers m≥1 and 1≤j≤k. We have
(4.43)∥Tjmxn-xn∥≤∥Tjmxn-Tjm-1xn∥+∥Tjm-1xn-Tjm-2xn∥+⋯+∥Tj2xn-Tjxn∥+∥Tjxn-xn∥≤(m-1)L∥Tjxn-xn∥γj+∥Tjxn-xn∥.
Then, by Lemma 4.2(ii), we get ∥Tjmxn-xn∥→0 as n→∞. Since {xn} is bounded and Tjm is semicompact, there exists a subsequence {xnl} of {xn} such that xnl→q∈C as l→∞.
By continuity ofTi and Lemma 4.2(ii), we obtain (4.44)∥q-Tjq∥=liml→∞∥xnl-Tjxnl∥=0,∀j=1,2,…,k.
Therefore, q∈F and then Theorem 3.2 implies that {xn} converges strongly to a common fixed point q of the family {Ti:i=1,2,…,k}.We note that in practical Theorem4.3 is very useful in the case that one of Ti, i=1,2,3,…,k, is semicompact.Theorem 4.4.
LetC be a nonempty closed convex subset of an uniformly convex real Banach space X satisfying the Opial property, and {Ti:i=1,2,…,k} be a family of (L-γi) uniform Lipschitz and generalized asymtotically quasinonexpansive self-mappings of C, that is, ∥Tinx-Tiny∥≤L∥x-y∥γi and ∥Tinx-pi∥≤(1+rin)∥x-pi∥+sin, for all x,y∈C and pi∈F(Ti),i=1,2,…,k. Suppose that F=⋂i=1kF(Ti)≠∅. Let x1∈C and the iterative sequence {xn} be defined by (1.2) with {αin}i=1n⊂[a,b], where 0<a<b<1. Assume that ∑n=1∞rn<∞, where rn=max1≤i≤k{rin}. If I-Ti,i=1,2,…,k, is demiclosed at 0, then {xn} converges weakly to a common fixed point of the family of mappings.Proof.
Letp∈F. By Lemma 3.1(v), we get limn→∞∥xn-p∥ exists. Then we follow the proof of Theorem 3.2 by Khan et al. [1] until we can conclude that {xn} converges weakly to a common fixed point p∈F.



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*Source: 706587-2010-10-11.xml*