# Convergence Theorems on a New Iteration Process for Two Asymptotically Nonexpansive Nonself-Mappings with Errors in Banach Spaces

Authors: Murat Ozdemir; Sezgin Akbulut; Hukmi Kiziltunc
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We introduce a new two-step iterative scheme for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for this iterative scheme in a uniformly convex Banach space. The results presented extend and improve the corresponding results of Chidume et al. (2003), Wang (2006), Shahzad (2005), and Thianwan (2008).


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## Body

## 1. Introduction

LetE be a real normed space and K be a nonempty subset of E. A mapping T:K→K is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈K. A mapping T:K→K is called asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 such that ∥Tnx-Tny∥≤kn∥x-y∥ for all x,y∈K and n≥1. T is called uniformly L-Lipschitzian if there exists a real number L>0 such that ∥Tnx-Tny∥≤L∥x-y∥ for all x,y∈K and n≥1. It is easy to see that if T is an asymptotically nonexpansive, then it is uniformly L-Lipschitzian with the uniform Lipschitz constant L=sup{kn:n≥1}.Iterative techniques for nonexpansive and asymptotically nonexpansive mappings in Banach spaces including Mann type and Ishikawa type iteration processes have been studied extensively by various authors; see [1–8]. However, if the domain of T, D(T), is a proper subset of E (and this is the case in several applications), and T maps D(T) into E, then the iteration processes of Mann type and Ishikawa type studied by the authors mentioned above, and their modifications introduced may fail to be well defined.A subsetK of E is said to be a retract of E if there exists a continuous map P:E→K such that Px=x, for all x∈K. Every closed convex subset of a uniformly convex Banach space is a retract. A map P:E→K is said to be a retraction if P2=P. It follows that if a map P is a retraction, then Py=y for all y∈R(P), the range of P.The concept of asymptotically nonexpansive nonself-mappings was firstly introduced by Chidume et al. [4] as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.Definition (see [4]).
LetK be a nonempty subset of real normed linear space E. Let P:E→K be the nonexpansive retraction of E onto K. A nonself mapping T:K→E is called asymptotically nonexpansive if there exists sequence {kn}⊂[1,∞), kn→1(n→∞) such that
(1.1)∥T(PT)n-1x-T(PT)n-1y∥≤kn∥x-y∥
for all x,y∈K and n≥1. T is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that
(1.2)∥T(PT)n-1x-T(PT)n-1y∥≤L∥x-y∥
for all x,y∈K and n≥1.In [4], they study the following iterative sequence:(1.3)xn+1=P((1-αn)xn+αnT(PT)n-1xn),x1∈K,n≥1
to approximate some fixed point of T under suitable conditions. In [9], Wang generalized the iteration process (1.3) as follows:(1.4)xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-αn′)xn+αn′T2(PT2)n-1xn),x1∈K,n≥1,
where T1,T2:K→E are asymptotically nonexpansive nonself-mappings and {αn},{αn′} are sequences in [0,1]. He studied the strong and weak convergence of the iterative scheme (1.4) under proper conditions. Meanwhile, the results of [9] generalized the results of [4].In [10], Shahzad studied the following iterative sequence:(1.5)xn+1=P((1-αn)xn+αnTP[(1-βn)xn+βnTxn]),x1∈K,n≥1,
where T:K→E is a nonexpansive nonself-mapping and K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P, nonexpansive retraction.Recently, Thianwan [11] generalized the iteration process (1.5) as follows:(1.6)xn+1=P((1-αn-γn)xn+αnTP((1-βn)yn+βnTyn)+γnun),yn=P((1-αn′-γn′)xn+αn′TP((1-βn′)xn+βn′Txn)+γn′vn),x1∈K,n≥1,
where {αn}, {βn}, {γn},{αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] and {un}, {vn} are bounded sequences in K. He proved weak and strong convergence theorems for nonexpansive nonself-mappings in uniformly convex Banach spaces.The purpose of this paper, motivated by the Wang [9], Thianwan [11] and some others, is to construct an iterative scheme for approximating a fixed point of asymptotically nonexpansive nonself-mappings (provided that such a fixed point exists) and to prove some strong and weak convergence theorems for such maps.LetE be a normed space, K a nonempty convex subset of E, P:E→K the nonexpansive retraction of E onto K, and T1,T2:K→E be two asymptotically nonexpansive nonself-mappings. Then, for given x1∈K and n≥1, we define the sequence {xn} by the iterative scheme:(1.7)xn+1=P((1-αn-γn)xn+αnT1(PT1)n-1P((1-βn)yn+βnT1(PT1)n-1yn)+γnun),yn=P((1-αn′-γn′)xn+αn′T2(PT2)n-1P((1-βn′)xn+βn′T2(PT2)n-1xn)+γn′vn),
where {αn}, {βn},{γn},{αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] satisfying  αn+βn+γn=1=αn′+βn′+γn′ and {un}, {vn} are bounded sequences in K. Clearly, the iterative scheme (1.7) is generalized by the iterative schemes (1.4) and (1.6).Now, we recall the well-known concepts and results.LetE be a Banach space with dimension E≥2. The modulus of E is the function δE:(0,2]→[0,1] defined by(1.8)δE(ɛ)=inf{1-∥12(x+y)∥:∥x∥=∥y∥=1,ɛ=∥x-y∥}.
A Banach space E is uniformly convex if and only if δE(ɛ)>0 for all ɛ∈(0,2].A Banach spaceE is said to satisfy Opial's condition [12] if for any sequence {xn} in E, xn⇀x implies that(1.9)lim supn→∞∥xn-x∥<lim supn→∞∥xn-y∥
for all y∈E with y≠x, where xn⇀x denotes that {xn}converges weakly to x.The mappingT:K→E with F(T)≠∅ is said to satisfy condition (A) [13] if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0, f(t)>0 for all t∈(0,∞) such that(1.10)∥x-Tx∥≥f(d(x,F(T)))
for all x∈K, where d(x,F(T))=inf{∥x-p∥:p∈F(T)}; (see [13, page 337]) for an example of nonexpansive mappings satisfying condition (A).Two mappingsT1,T2:K→E are said to satisfy condition (A′) [14] if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0, f(t)>0 for all t∈(0,∞) such that(1.11)12(∥x-T1x∥+∥x-T2x∥)≥f(d(x,F(T)))
for all x∈K, where d(x,F(T))=inf{∥x-p∥:p∈F(T)=F(T1)∩F(T2)}.Note that condition(A′) reduces to condition (A) when T1=T2 and hence is more general than the demicompactness of T1andT2 [13]. A mapping T:K→K is called: (1) demicompact if any bounded sequence {xn} in K such that {xn-Txn} converges has a convergent subsequence, (2) semicompact (or hemicompact) if any bounded sequence {xn} in K such that {xn-Txn}→0 as n→∞ has a convergent subsequence. Every demicompact mapping is semicompact but the converse is not true in general.Senter and Dotson [13] have approximated fixed points of a nonexpansive mapping T by Mann iterates, whereas Maiti and Ghosh [14] and Tan and Xu [5] have approximated the fixed points using Ishikawa iterates under the condition (A) of Senter and Dotson [13]. Tan and Xu [5] pointed out that condition (A) is weaker than the compactness of K. Khan and Takahashi [6] have studied the two mappings case for asymptotically nonexpansive mappings under the assumption that the domain of the mappings is compact. We shall use condition (A′) instead of compactness of K to study the strong convergence of {xn} defined in (1.7).In the sequel, we need the following usefull known lemmas to prove our main results.Lemma 1.2 (see [5]).
Let{an},  {bn}, and  {δn} be sequences of nonnegative real numbers satisfying the inequality
(1.12)an+1≤(1+δn)an+bn,n≥1.
If  ∑n=1∞bn<∞ and  ∑n=1∞δn<∞, then (i)
limn→∞an exists;(ii)
In particular, if{an}  has a subsequence which converges strongly to zero, then limn→∞an=0.Lemma 1.3 (see [2]).
Suppose thatE is a uniformly convex Banach space and  0<p≤tn≤q<1 for all  n≥1. Suppose further that  {xn} and  {yn} are sequences of  Esuch that
(1.13)lim supn→∞∥xn∥≤r,lim supn→∞∥yn∥≤r,limn→∞∥tnxn+(1-tn)yn∥=r
hold for some  r≥0. Then  limn→∞∥xn-yn∥=0.Lemma 1.4 (see [4]).
LetE be a uniformly convex Banach space,  K a nonempty closed convex subset of  E, and  T:K→E be a nonexpansive mapping. Then,  (I-T) 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 fixed point of  T.


## 2. Main Results

We shall make use of the following lemmas.Lemma.
LetE be a normed space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let  T1,T2:K→E be two asymptotically nonexpansive nonself-mappings of  E with sequences  {kn},{ln}⊂[1,∞) such that  ∑n=1∞(kn-1)<∞,  ∑n=1∞(ln-1)<∞, respectively  and  F(T1)∩F(T2):={x∈K:T1x=T2x=x}≠∅. Suppose that  {un},  {vn} are bounded sequences in  K such that  ∑n=1∞γn<∞,  ∑n=1∞γn′<∞. Starting from an arbitrary  x1∈K, define the sequence  {xn} by the recursion  (1.7). Then,  limn→∞∥xn-p∥  exists for all  p∈F(T1)∩F(T2).Proof.
Letp∈F(T1)∩F(T2). Since {un} and {vn} are bounded sequences in K, we have
(2.1)r=max{supn≥1∥un-p∥,supn≥1∥vn-p∥}.
Set σn = (1-βn)yn+βnT1(PT1)n-1yn and δn= (1-βn′)xn+βn′T2(PT2)n-1xn. Firstly, we note that
(2.2)∥σn-p∥=∥(1-βn)yn+βnT1(PT1)n-1yn-p∥≤βn∥T1(PT1)n-1yn-p∥+(1-βn)∥yn-p∥≤βnkn∥yn-p∥+(1-βn)∥yn-p∥≤kn∥yn-p∥,(2.3)∥δn-p∥=∥(1-βn′)xn+βn′T2(PT2)n-1xn-p∥≤βn′∥T2(PT2)n-1xn-p∥+(1-βn′)∥xn-p∥≤βn′ln∥xn-p∥+(1-βn′)∥xn-p∥≤ln∥xn-p∥.
From (1.7) and (2.3), we have
(2.4)∥yn-p∥=∥P((1-αn′-γn′)xn+αn′T2(PT2)n-1Pδn+γn′vn)-p∥≤∥(1-αn′-γn′)xn+αn′T2(PT2)n-1Pδn+γn′vn-p∥≤αn′∥T2(PT2)n-1Pδn-p∥+(1-αn′-γn′)∥xn-p∥+γn′∥vn-p∥≤αn′ln∥δn-p∥+(1-αn′-γn′)∥xn-p∥+γn′∥vn-p∥≤αn′ln2∥xn-p∥+(1-αn′-γn′)∥xn-p∥+γn′r≤ln2∥xn-p∥+γn′r.
Substituting (2.4) into (2.2), we obtain
(2.5)∥σn-p∥≤kn∥yn-p∥≤knln2∥xn-p∥+knγn′r.
It follows from (1.7) and (2.5) that
(2.6)∥xn+1-p∥=∥P((1-αn-γn)xn+αnT1(PT1)n-1Pσn+γnun)-p∥≤∥(1-αn-γn)xn+αnT1(PT1)n-1Pσn+γnun-p∥≤αn∥T1(PT1)n-1Pσn-p∥+(1-αn-γn)∥xn-p∥+γn∥un-p∥≤αnkn∥σn-p∥+(1-αn-γn)∥xn-p∥+γn∥un-p∥≤αn(kn2ln2∥xn-p∥+kn2γn′r)+(1-αn-γn)∥xn-p∥+γnr≤kn2ln2∥xn-p∥+kn2γn′r+γnr=(1+(ln2-1)(kn2-1)+(ln2-1)+(kn2-1))∥xn-p∥+(kn2γn′+γn)r.
Note that ∑n=1∞kn-1<∞ and ∑n=1∞ln-1<∞ are equivalent to ∑n=1∞kn2-1<∞ and ∑n=1∞ln2-1<∞, respectively. Since ∑n=1∞γn<∞ and ∑n=1∞γn′<∞, we have ∑n=1∞(kn2γn′+γn)r<∞. We obtained from (2.6) and Lemma 1.2 that limn→∞∥xn-p∥ exists for all p∈F(T). This completes the proof.Lemma.
LetE be a normed space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2:K→E be nonself uniformly L1-Lipschitzian, L2-Lipschitzian, respectively. Suppose that  {un},  {vn} are bounded sequences in  K such that  ∑n=1∞γn<∞,  ∑n=1∞γn′<∞.  Starting from an arbitrary x1∈K, define the sequence {xn} by the recursion (1.7) and set Cn=∥xn-T1(PT1)n-1xn∥,Cn′=∥xn-T2(PT2)n-1xn∥ for all n≥1. If limn→∞Cn=limn→∞Cn′=0, then
(2.7)limn→∞∥xn-T1xn∥=limn→∞∥xn-T2xn∥=0.Proof.
Since{un}, {vn} are bounded, it follows from Lemma 2.1 that {un-xn} and {vn-xn} are all bounded. We set
(2.8)r1=sup{∥un-xn∥:n≥1},r2=sup{∥vn-xn∥:n≥1},r3=sup{∥un-1-xn-1∥:n≥1},r=max{ri:i=1,2,3}.
Let σn=(1-βn)yn+βnT1(PT1)n-1yn and δn=(1-βn′)xn+βn′T2(PT2)n-1xn. Then, we have
(2.9)∥σn-xn∥=∥(1-βn)yn+βnT1(PT1)n-1yn-xn∥≤βn∥T1(PT1)n-1yn-T1(PT1)n-1xn∥+βn∥T1(PT1)n-1xn-xn∥+(1-βn)∥yn-xn∥≤(L1+1)∥yn-xn∥+Cn,(2.10)∥δn-xn∥=∥(1-βn′)xn+βn′T2(PT2)n-1xn-xn∥≤βn′∥T2(PT2)n-1xn-xn∥≤Cn′.
We find the following from (1.7) and (2.10):
(2.11)∥yn-xn∥=∥P((1-αn′-γn′)xn+αn′T2(PT2)n-1Pδn+γn′vn)-xn∥≤∥(1-αn′-γn′)xn+αn′T2(PT2)n-1Pδn+γn′vn-xn∥≤αn′∥T2(PT2)n-1Pδn-T2(PT2)n-1xn∥+αn′∥T2(PT2)n-1xn-xn∥+γn′∥vn-xn∥≤L2∥δn-xn∥+Cn′+γn′r≤L2Cn′+Cn′+γn′r=(L2+1)Cn′+γn′r.
Substituting (2.11) into (2.9), we get
(2.12)∥σn-xn∥≤(L1+1)(L2+1)Cn′+(L1+1)γn′r+Cn.
It follows from (1.7) and (2.12) that
(2.13)∥xn+1-xn∥≤∥P((1-αn-γn)xn+αnT1(PT1)n-1Pσn+γnun)-xn∥≤∥T1(PT1)n-1Pσn-xn∥+γn∥un-xn∥≤∥T1(PT1)n-1Pσn-T1(PT1)n-1xn∥+∥T1(PT1)n-1xn-xn∥+γnr≤L1∥σn-xn∥+Cn+γnr≤L1((L1+1)(L2+1)Cn′+(L1+1)γn′r+Cn)+Cn+γnr=(L1+1)Cn+L1(L1+1)(L2+1)Cn′+L1(L1+1)γn′r+γnr.
Using (2.11) and (2.13), we obtain
(2.14)∥σn-1-xn∥=∥(1-βn-1)yn-1+βn-1T1(PT1)n-2yn-1-xn∥≤βn-1∥T1(PT1)n-2yn-1-T1(PT1)n-2xn-1∥+βn-1∥T1(PT1)n-2xn-1-xn-1∥+βn-1∥xn-xn-1∥+(1-βn-1)∥yn-1-xn∥≤L1∥yn-1-xn-1∥+Cn-1+∥xn-xn-1∥+∥yn-1-xn-1∥+∥xn-xn-1∥≤(L1+1)[(L2+1)Cn-1′+γn-1′r]+2[(L1+1)Cn-1+L1(L1+1)(L2+1)Cn-1′+L1(L1+1)γn-1′r+γn-1r]+Cn-1=(2L1+3)Cn-1+(2L1+1)(L1+1)(L2+1)Cn-1′+(2L1+1)(L1+1)γn-1′r+2γn-1r.
Combine (2.13) with (2.14) yields that
(2.15)∥xn-(PT1)n-1xn∥=∥xn-T1(PT1)n-2xn∥≤∥(1-αn-1-γn-1)xn-1+αn-1T1(PT1)n-2Pσn-1+γn-1un-1-T1(PT1)n-2xn∥≤αn-1∥T1(PT1)n-2Pσn-1-T1(PT1)n-2xn∥+(1-αn-1)∥xn-1-T1(PT1)n-2xn∥+γn-1∥un-1-xn-1∥≤∥T1(PT1)n-2Pσn-1-T1(PT1)n-2xn∥+∥xn-1-T1(PT1)n-2xn∥+γn-1r≤L1∥σn-1-xn∥+∥xn-1-T1(PT1)n-2xn-1∥+∥T1(PT1)n-2xn-T1(PT1)n-2xn-1∥+γn-1r≤L1[(2L1+3)Cn-1+(2L1+1)(L1+1)(L2+1)Cn-1′+(2L1+1)(L1+1)γn-1′r+2γn-1r]+Cn-1+(L1+1)Cn-1+L1(L1+1)(L2+1)Cn-1′+L1(L1+1)γn-1′r+2γn-1r=2(L1+1)2Cn-1+2L1(L1+1)2(L2+1)Cn-1′+2L1(L1+1)2γn-1′r+2(L1+1)γn-1r,
from which it follows that
(2.16)∥xn-T1xn∥=∥xn-T1(PT1)n-1xn+T1(PT1)n-1xn-T1xn∥≤∥xn-T1(PT1)n-1xn∥+∥T1(PT1)n-1xn-T1xn∥≤Cn+L1∥(PT1)n-1xn-xn∥≤Cn+2L1(L1+1)2Cn-1+2L12(L1+1)2(L2+1)Cn-1′+2L12(L1+1)2γn-1′r+2L1(L1+1)γn-1r.
It follows from limn→∞Cn=limn→∞Cn′=0 that limn→∞∥xn-T1xn∥=0. Similarly, we can show that limn→∞∥xn-T2xn∥=0. This completes the proof.Lemma.
LetE be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2:K→E be two asymptotically nonexpansive nonself-mappings of E with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn}, {βn}, {γn}, {αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] satisfyingαn+βn+γn=1=αn′+βn′+γn′, and {un}, {vn} are bounded sequences in K such that ∑n=1∞γn<∞, ∑n=1∞γn′<∞. Moreover, 0<a≤αn,αn′,βn,βn′≤b<1 for all n≥1 and some a,b∈(0,1). Starting from an arbitrary x1∈K, define the sequence {xn} by the recursion (1.7). Then,
(2.17)limn→∞∥xn-T1xn∥=limn→∞∥xn-T2xn∥=0.Proof.
Letσn=(1-βn)yn+βnT1(PT1)n-1yn and δn=(1-βn′)xn+βn′T2(PT2)n-1xn.  By Lemma 2.1, we see that limn→∞∥xn-p∥ exists. Assume that limn→∞∥xn-p∥=c. If c=0, then by the continuity of T1 and T2 the conclusion follows. Now, suppose c>0. Taking lim sup on both sides in the inequalities (2.2), (2.3), and (2.4), we have
(2.18)lim supn→∞∥σn-p∥≤c,lim supn→∞∥δn-p∥≤c,lim supn→∞∥yn-p∥≤c,
respectively. Next, we consider
(2.19)∥T1(PT1)n-1Pσn-p+γn(un-xn)∥≤∥T1(PT1)n-1Pσn-p∥+γn∥un-xn∥≤kn∥σn-p∥+γnr.
Taking lim sup on both sides in the above inequality and using (2.18), we get
(2.20)lim supn→∞∥T1(PT1)n-1Pσn-p+γn(un-xn)∥≤c.
Observe that
(2.21)∥xn-p+γn(un-xn)∥≤∥xn-p∥+γn∥un-xn∥≤∥xn-p∥+γnr,
which implies that
(2.22)lim supn→∞∥xn-p+γn(un-xn)∥≤c.lim supn→∞∥xn+1-p∥=c means that
(2.23)lim infn→∞∥αn(T1(PT1)n-1Pσn-p+γn(un-xn))+(1-αn)(xn-p+γn(un-xn))∥≥c.
On the other hand, by using (2.23) and (2.5), we have
(2.24)∥αn(T1(PT1)n-1Pσn-p+γn(un-xn))+(1-αn)(xn-p+γn(un-xn))∥≤αn∥T1(PT1)n-1Pσn-p∥+(1-αn)∥xn-p∥+γn∥un-xn∥≤αnkn∥σn-p∥+(1-αn)∥xn-p∥+γn∥un-xn∥≤αnkn(knln2∥xn-p∥+knγn′r)+(1-αn)∥xn-p∥+γnr≤kn2ln2∥xn-p∥+kn2γn′r+γnr.
Therefore, we have
(2.25)lim supn→∞∥αn(T1(PT1)n-1Pσn-p+γn(un-xn))+(1-αn)(xn-p+γn(un-xn))∥≤c.
Combining (2.23) with (2.25), we obtain
(2.26)limn→∞∥αn(T1(PT1)n-1Pσn-p+γn(un-xn))+(1-αn)(xn-p+γn(un-xn))∥=c.
Hence, applying Lemma 1.3, we find
(2.27)limn→∞∥T1(PT1)n-1Pσn-xn∥=0.
Note that
(2.28)∥xn-p∥≤∥T1(PT1)n-1Pσn-p∥+∥T1(PT1)n-1Pσn-xn∥≤kn∥σn-p∥
which yields that
(2.29)c≤liminfn→∞∥σn-p∥≤lim supn→∞∥σn-p∥≤c.
That is, limn→∞∥σn-p∥=c. This implies that
(2.30)lim infn→∞∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥≥c.
Similarly, we have
(2.31)∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥≤βn∥T1(PT1)n-1yn-p∥+(1-βn)∥(yn-p)∥≤kn∥yn-p∥,(2.32)lim supn→∞∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥≤c.
Combining (2.30) with (2.32), we obtain
(2.33)limn→∞∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥=c.
On the other hand, we have
(2.34)∥T1(PT1)n-1yn-p∥≤kn∥yn-p∥,(2.35)lim supn→∞∥T1(PT1)n-1yn-p∥≤c.
Hence, using (2.32), (2.33), (2.35), and Lemma 1.3, we find
(2.36)limn→∞∥T1(PT1)n-1yn-yn∥=0.
Note that from (2.36), we have
(2.37)∥σn-p∥=∥(1-βn)yn+βnT1(PT1)n-1yn-p∥≤(1-βn)∥yn-p∥+βn∥T1(PT1)n-1yn-p∥≤(1-βn)∥yn-p∥+βn∥T1(PT1)n-1yn-yn∥+βn∥yn-p∥=∥yn-p∥
which yields that
(2.38)c≤lim infn→∞∥yn-p∥≤lim supn→∞∥yn-p∥≤c.
That is, limn→∞∥yn-p∥=c.
Again,limn→∞∥yn-p∥=c means that
(2.39)lim infn→∞∥αn′(T2(PT2)n-1Pδn-p+γn′(vn-xn))+(1-αn′)(xn-p+γn′(vn-xn))∥≥c.
By using (2.39) and (2.3), we obtain
(2.40)∥αn′(T2(PT2)n-1Pδn-p+γn′(vn-xn))+(1-αn′)(xn-p+γn′(vn-xn))∥≤αn′∥T2(PT2)n-1Pδn-p∥+(1-αn′)∥xn-p∥+γn′∥(vn-xn)∥≤αn′ln∥δn-p∥+(1-αn′)∥xn-p∥+γn′∥(vn-xn)∥≤αn′ln2∥xn-p∥+(1-αn′)∥xn-p∥+γn′r≤ln2∥xn-p∥+γn′r.
Therefore, we have
(2.41)lim supn→∞∥αn′(T2(PT2)n-1Pδn-p+γn′(vn-xn))+(1-αn′)(xn-p+γn′(vn-xn))∥≤c.
Combining (2.39) with (2.41), we obtain
(2.42)limn→∞∥αn′(T2(PT2)n-1Pδn-p+γn′(vn-xn))+(1-αn′)(xn-p+γn′(vn-xn))∥=c.
On the other hand, we have
(2.43)∥T2(PT2)n-1Pδn-p+γn′(vn-xn)∥≤∥T2(PT2)n-1Pδn-p∥+γn′∥vn-xn∥≤ln∥δn-p∥+γn′r
which implies that
(2.44)lim supn→∞∥T2(PT2)n-1Pδn-p+γn′(vn-xn)∥≤c.
Notice that
(2.45)∥xn-p+γn′(vn-xn)∥≤∥xn-p∥+γn′∥vn-xn∥≤∥xn-p∥+γn′r,
which implies that
(2.46)lim supn→∞∥xn-p+γn′(vn-xn)∥≤c.
Using (2.42), (2.44), (2.46), and Lemma 1.3, we find
(2.47)limn→∞∥T2(PT2)n-1Pδn-xn∥=0.
Observe that
(2.48)∥xn-p∥≤∥T2(PT2)n-1Pδn-xn∥+∥T2(PT2)n-1Pδn-p∥≤ln∥δn-p∥
which yields that
(2.49)c≤lim infn→∞∥δn-p∥≤lim supn→∞∥δn-p∥≤c.
That is, limn→∞∥δn-p∥=c. This implies that
(2.50)lim infn→∞∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥≥c.
Similarly, we have
(2.51)∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥≤βn′∥T2(PT2)n-1xn-p∥+(1-βn′)∥xn-p∥≤ln∥xn-p∥,(2.52)lim supn→∞∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥≤c.
Combining (2.50) with (2.52), we obtain
(2.53)limn→∞∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥=c.
On the other hand, we have
(2.54)∥T2(PT2)n-1xn-p∥≤ln∥xn-p∥,lim supn→∞∥T2(PT2)n-1xn-p∥≤c,(2.55)lim supn→∞∥xn-p∥≤c.
Hence, using (2.53), (2.54), (2.55), and Lemma 1.3, we find
(2.56)limn→∞∥T2(PT2)n-1xn-xn∥=0.
In addition, from yn=P((1-αn′-γn′)xn+αn′T2(PT2)n-1Pδn+γn′vn) and (2.47), we have
(2.57)∥yn-xn∥=∥P((1-αn′-γn′)xn+αn′T2(PT2)n-1Pδn+γn′vn)-xn∥≤αn′∥T2(PT2)n-1Pδn-xn∥+γn′∥vn-xn∥≤∥T2(PT2)n-1Pδn-xn∥+γn′r.→0,(asn→∞).
Hence, from (2.36) and (2.57), we find
(2.58)∥T1(PT1)n-1xn-xn∥≤∥T1(PT1)n-1xn-T1(PT1)n-1yn∥+∥T1(PT1)n-1yn-yn∥+∥yn-xn∥≤kn∥yn-xn∥+∥T1(PT1)n-1yn-yn∥+∥yn-xn∥→0,(asn→∞).
That is,
(2.59)limn→∞∥T1(PT1)n-1xn-xn∥=0.
Since T1 and T2 are uniformly L1-Lipschitzian and uniformly L2-Lipschitzian, respectively, for some L1,L2≥0, it follows from (2.56), (2.59), and Lemma 2.2 that
(2.60)limn→∞∥xn-T1xn∥=limn→∞∥xn-T2xn∥=0.
This completes the proof.Theorem.
LetE be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2:K→E be two asymptotically nonexpansive nonself -mappings of E with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn}, {βn}, {γn}, {αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] satisfyingαn+βn+γn=1=αn′+βn′+γn′, and {un}, {vn} are bounded sequences in K such that ∑n=1∞γn<∞, ∑n=1∞γn′<∞. Moreover, 0<a≤αn,αn′,βn,βn′≤b<1 for all n≥1 and some a,b∈(0,1). If one of T1andT2 is completely continuous, then the sequence {xn} defined by the recursion (1.7) converges strongly to some common fixed point of T1andT2.Proof.
By Lemma2.1, {xn} is bounded. In addition, by Lemma 2.3;limn→∞∥xn-T1xn∥=limn→∞∥xn-T2xn∥=0; then {T1xn} and {T2xn} are also bounded. If T1 is completely continuous, there exists subsequence {T1xnj} of {T1xn} such that T1xnj→p as j→∞. It follows from Lemma 2.3 that limj→∞∥xnj-T1xnj∥=limj→∞∥xnj-T2xnj∥=0. So by the continuity of T1 and Lemma 1.4, we have limj→∞∥xnj-p∥=0 and p∈F(T1)∩F(T2). Furthermore, by Lemma 2.1, we get that limn→∞∥xn-p∥ exists. Thus limn→∞∥xn-p∥=0. The proof is completed.The following result gives a strong convergence theorem for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying condition(A′).Theorem.
LetE be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2:K→E be two asymptotically nonexpansive nonself-mappings of E with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn}, {βn}, {γn}, {αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] satisfying αn+βn+γn=1=αn′+βn′+γn′, and {un}, {vn} are bounded sequences in K such that ∑n=1∞γn<∞, ∑n=1∞γn′<∞. Moreover, 0<a≤αn,αn′,βn,βn′≤b<1 for all n≥1 and some a,b∈(0,1). Suppose that T1andT2 satisfy condition  (A′). Then, the sequence {xn} defined by the recursion (1.7) converges strongly to some common fixed point of T1andT2.Proof.
By Lemma2.1, we readily see that limn→∞∥xn-p∥ and so, limn→∞d(xn,F(T1)∩F(T2)) exists for all p∈F(T1)∩F(T2). Also, by Lemma 2.3, limn→∞∥T1xn-xn∥=limn→∞∥T2xn-xn∥=0. It follows from condition (A′) that
(2.61)limn→∞f(d(xn,F(T1)∩F(T2)))≤limn→∞(12(∥xn-T1xn∥+∥xn-T2xn∥))=0.
That is,
(2.62)limn→∞f(d(xn,F(T1)∩F(T2)))=0.
Since f:[0,∞)→[0,∞) is a nondecreasing function satisfying f(0)=0, f(t)>0 for all t∈(0,∞), therefore, we have
(2.63)limn→∞d(xn,F(T1)∩F(T2))=0.
Now we can take a subsequence {xnj} of {xn} and sequence {yj}⊂F such that ∥xnj-yj∥<2-j for all integers j≥1. Using the proof method of Tan and Xu [5], we have
(2.64)∥xnj+1-yj∥≤∥xnj-yj∥<2-j,
and hence
(2.65)∥yj+1-yj∥≤∥yj+1-xnj+1∥+∥xnj+1-yj∥≤2-(j+1)+2-j<2-j+1.
We get that {yj} is a Cauchy sequence in F and so it converges. Let yj→y. Since F is closed, therefore, y∈F and then xnj→y. As limn→∞∥xn-p∥ exists, xn→y∈F(T1)∩F(T2). Thereby completing the proof.Remark.
Ifγn=γn′=βn=βn′=0, then the iterative scheme (1.7) reduces to the iterative scheme (1.4) of [9]. Moreover, the condition (A′) is weaker than both the compactness of K and the semicompactness of the asymptotically nonexpansive nonself-mappings T1,T2:K→E. Also, the condition 0<a≤αn,αn′≤b<1 for all n≥1 is weaker than the condition 0<ɛ≤αn,αn′,≤1-ɛ, for all n≥1 and some ɛ∈[0,1). Hence, Theorems 2.4 and 2.5 generalize Theorems 3.3 and 3.4 in [9], respectively.In the next result, we prove the weak convergence of the iterative scheme (1.7) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying Opial's condition.Theorem.
LetE be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2:K→E be two asymptotically nonexpansive nonself-mappings of E with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn}, {βn},{γn}, {αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] satisfying αn+βn+γn=1=αn′+βn′+γn′,and {un}, {vn} are bounded sequences in K such that ∑n=1∞γn<∞, ∑n=1∞γn′<∞. Moreover, 0<a≤αn,αn′,βn,βn′≤b<1 for all n≥1 and some a,b∈(0,1). Suppose that T1andT2 satisfy Opial's condition. Then, the sequence {xn} defined by the recursion (1.7) converges weakly to some common fixed point of TandT2.Proof.
Letp∈F(T1)∩F(T2). By Lemma 2.1, we see that limn→∞∥xn-p∥ exists and {xn} bounded. Now we prove that {xn} has a unique weak subsequential limit in F(T1)∩F(T2).  Firstly, suppose that subsequences {xnk} and {xnj} of {xn} converge weakly to p1 and p2, respectively. By Lemma 2.3, we have limn→∞∥xnk-T1xnk∥=0. And Lemma 1.4 guarantees that (I-T1)p1=0, that is., T1p1=p1. Similarly, T2p1=p1. Again in the same way, we can prove that p2∈F(T1)∩F(T2).
Secondly, assumep1≠p2, then by Opial’s condition, we have
(2.66)limn→∞∥xn-p1∥=limk→∞∥xnk-p1∥<limk→∞∥xnk-p2∥=limj→∞∥xnj-p2∥<limk→∞∥xnk-p1∥=limn→∞∥xn-p1∥,
which is a contradiction, hence, p1=p2. Then, {xn} converges weakly to a common fixed point of T1 and T2. This completes the proof.Remark.
The above Theorem generalizes Theorem 3.5 of Wang [9].


## 3. Case of Two Nonself-Nonexpansive Mappings

LetT1,T2:K→E be two nonexpansive nonself-mappings. Then, the iterative scheme (1.7) is written as follows:(3.1)xn+1=P((1-αn-γn)xn+αnT1P(1-βn)yn+βnT1yn+γnun),yn=P((1-αn′-γn′)xn+αn′T2P((1-βn′)xn+βn′T2xn)+γn′vn),x1∈K,n≥1.Nothing prevents one from proving the results of the previous section for nonexpansive nonself-mappings case. Thus, one can easily prove the following.Theorem.
LetE be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2:K→E be two nonexpansive nonself-mappings of E with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn}, {βn}, {γn}, {αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] satisfying αn+βn+γn=1=αn′+βn′+γn′, and {un}, {vn} are bounded sequences in K such that ∑n=1∞γn<∞, ∑n=1∞γn′<∞. Moreover, 0<a≤αn,αn′,βn,βn′≤b<1 for all n≥1 and some a,b∈(0,1). Suppose that T1andT2 satisfy condition  (A′). Then, the sequence {xn} defined by the recursion (3.1) converges strongly to some common fixed point of T1andT2.Theorem.
LetE be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2:K→E be two nonexpansive nonself-mappings of E with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn}, {βn},{γn}, {αn′}, {βn′}, {γn′} are appropriate sequences in [0,1] satisfying αn+βn+γn=1=αn′+βn′+γn′, and {un}, {vn} are bounded sequences in K such that ∑n=1∞γn<∞, ∑n=1∞γn′<∞. Moreover, 0<a≤αn,αn′,βn,βn′≤b<1 for all n≥1 and some a,b∈(0,1). Suppose that T1andT2 satisfy Opial's condition. Then, the sequence {xn} defined by the recursion (3.1) converges weakly to some common fixed point of TandT2.Remark.
IfT1=T2=T and T is a nonexpansive nonself-mapping, then the iterative scheme (3.1) reduces to the iterative scheme (1.6) of Thianwan [11]. Then, Theorems 3.1-3.2 generalize Theorems 2.4 and 2.6 in [11], respectively.



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*Source: 307245-2010-02-24.xml*