# Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

Authors: Esref Turkmen; Safeer Hussain Khan; Murat Ozdemir
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

Suppose thatK is nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction and F:=F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅. Let T1,T2:K→E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences {kn(i)}⊂[1,∞) satisfying ∑n=1∞(kn(i)-1)<∞(i=1,2), respectively. For any given x1∈K, suppose that {xn} is a sequence generated iteratively by xn+1=(1-αn)(PT1)nyn+αn(PT2)nyn, yn=(1-βn)xn+βn(PT1)nxn, n∈ℕ, where {αn} and {βn} are sequences in [a,1-a] for some a∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of T1 and T2 are obtained.


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## Body

## 1. Introduction

LetE be a real Banach space with K, its nonempty subset. Let T:K→K be a mapping. A point x∈K is called a fixed point ofT if and only if Tx=x. In this paper, ℕ stands for the set of natural numbers. We will also denote by F(T) the set of fixed points of T, that is, F(T)={x∈K:Tx=x} and by F:=F(T1)∩F(T2), the set of common fixed points of two mappings T1 and T2. T is called asymptotically nonexpansive if for a sequence  {kn}⊂[1,∞) with limn→∞kn=1, ∥Tnx-Tny∥≤kn∥x-y∥ for all x,y∈K and all n∈ℕ. T is called uniformly L-Lipschitzian if for some L>0, ∥Tnx-Tny∥≤L∥x-y∥ for all n∈ℕ and all x,y∈K. T is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈K. Let P:E→K be a nonexpansive retraction of E into K. A nonself-mapping T:K→E is called asymptotically nonexpansive (according to Chidume et al. [1]) if for a sequence  {kn}⊂[1,∞) with limn→∞kn=1, we have ∥T(PT)n-1x-T(PT)n-1y∥≤kn∥x-y∥ for all x,y∈K and n∈ℕ. T is called uniformly L-Lipschitzian if for some L>0, ∥T(PT)n-1x-T(PT)n-1y∥≤L∥x-y∥ for all n∈ℕ and all x,y∈K.In what follows, we fixx1∈K as a starting point of the process under consideration, and take {αn}, {βn} sequences in (0,1).Agarwal et al. [2] recently introduced the iteration process
(1.1)xn+1=(1-αn)Tnxn+αnTnyn,yn=(1-βn)xn+βnTnxn,n∈N.
They showed that their process is independent of Mann and Ishikawa and converges faster than both of these. See Proposition  3.1 [2].Obviously the above process deals with one self-mapping only. The case of two mappings in iteration processes has also remained under study since Das and Debata [3] gave and studied a two mappings scheme. Also see, for example, Takahashi and Tamura [4] and Khan and Takahashi [5]. Note that two mappings case, that is, approximating the common fixed points, has its own importance as it has a direct link with the minimization problem, see, for example, Takahashi [6].Being an important generalization of the class of nonexpansive self-mappings, the class of asymptotically nonexpansive self-mappings was introduced by Goebel and Kirk [7] whereas the concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [1] in 2003 as a generalization of asymptotically nonexpansive self-mappings. Actually they studied the iteration process
(1.2)xn+1=P((1-αn)xn+αnT(PT)n-1xn),n∈N.Nonself asymptotically nonexpansive mappings have been studied by many authors [8–11]. Wang [10] studied the process
(1.3)xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1xn),n∈N.Very recently, Thianwan [12] considered a new iterative scheme (called projection type Ishikawa iteration) as follows:
(1.4)xn+1=P((1-αn)yn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1xn),n∈N.As a matter of fact, ifT is a self-mapping, then P is an identity mapping. In addition, if T:K→E is asymptotically nonexpansive and P:E→K is a nonexpansive retraction, then PT:K→K is asymptotically nonexpansive. Indeed, for all x,y∈K and n∈ℕ, it follows that
(1.5)‖(PT)nx-(PT)ny‖=‖PT(PT)n-1x-PT(PT)n-1y‖≤‖T(PT)n-1x-T(PT)n-1y‖≤kn‖x-y‖.
The converse, however, may not be true. Therefore, Zhou et al. [13] introduced the following generalized definition recently.Definition 1.1 (see [13]).
LetK be a nonempty subset of real normed linear space E. Let P:E→K be the nonexpansive retraction of E into K.(i)
A nonself-mappingT:K→E is called asymptotically nonexpansive with respect to P if there exists sequences {kn}∈[1,∞) with kn→1 as n→∞ such that
(1.6)‖(PT)nx-(PT)ny‖≤kn‖x-y‖,∀x,y∈K,n∈N.(ii)
A nonself-mappingT:K→E is said to be uniformly L-Lipschitzian with respect to P if there exists a constant L≥0 such that
(1.7)‖(PT)nx-(PT)ny‖≤L‖x-y‖,∀x,y∈K,n∈N.Futhermore, by studying the following iterative process(1.8)x1∈K,xn+1=αnxn+βn(PT1)nxn+γn(PT2)nxn,n∈N,
where {αn}, {βn}, and {γn} are three sequences in [a,1-a] for some a∈(0,1), satisfying αn+βn+γn=1, Zhou et al. [13] obtained some strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings with respect to P in uniformly convex Banach spaces. As a consequence, the main results of Chidume et al. [1] were deduced.Incorporating the ideas of Agarwal et al. [2], Thianwan [12], and Zhou et al. [13], a new two-step iterative scheme for two nonself asymptotically nonexpansive mappings is introduced and studied in this paper. Our process reads as follows.LetK be a nonempty closed convex subset of a real normed linear space E with retraction P. Let T1,T2:K→E be two nonself asymptotically nonexpansive mappings with respect to P:
(1.9)x1∈K,xn+1=(1-αn)(PT1)nyn+αn(PT2)nyn,yn=(1-βn)xn+βn(PT1)nxn,n∈N,
where {αn} and {βn} are sequences in [0,1]. Following the method of Agarwal et al. [2], it is not difficult to see that our process is able to compute common fixed points at a rate better than (1.3) and (1.4).Under suitable conditions, the sequence{xn} defined by (1.9) can also be generalized to iterative sequence with errors. Thus all the results proved in this paper can also be proved for the iterative process with errors. In this case our main iterative process (1.9) looks like
(1.10)x1∈K,xn+1=αn(PT1)nyn+βn(PT2)nyn+γnun,yn=αn′xn+βn′(PT1)nxn+γn′vn,n∈N,
where {αn}, {βn}, {γn}, {αn′}, {βn′}, {γn′} are real sequences in [0,1] satisfying αn+βn+γn=1=αn′+βn′+γn′ and {un}, {vn} are bounded sequences in K. Observe that the iterative process (1.10) with errors reduces to the iterative process (1.9) when γn=γn′=0.


## 2. Preliminaries

For the sake of convenience, we restate the following concepts and results.LetE be a Banach space with its dimension greater than or equal to 2. The modulus of E is the function δE(ε):(0,2]→[0,1] defined by
(2.1)δE(ε)=inf⁡{1-‖12(x+y)‖:‖x‖=1,‖y‖=1,ε=‖x-y‖}.A Banach spaceE is uniformly convex if and only if δE(ε)>0 for all ε∈(0,2].LetE be a Banach space and S(E)={x∈E:∥x∥=1}. The space E is said to be smooth if
(2.2)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈S(E).A subsetK of E is said to be a retract if there exists a continuous mapping P:E→K such that Px=x for all x∈K. A mapping P:E→E is said to be a retraction if P2=P. Let C and K be subsets of a Banach space E. A mapping P from C into K is called sunny if P(Px+t(x-Px))=Px for x∈C with Px+t(x-Px)∈C and t≥0.Note that, ifP is a retraction, then Pz=z for every z∈R(P) (the range of P). It is well-known that every closed convex subset of a uniformly convex Banach space is a retract.For anyx∈K, the inward set IK(x) is defined as follows:
(2.3)IK(x)={y∈E:y=x+λ(z-x),z∈K,λ≥0}.
A mapping T:K→E is said to satisfy the inward condition if Tx∈IK(x) for all x∈K. T is said to be weakly inward if Tx∈clIK(x) for each x∈K, where clIK(x) is the closure of IK(x).A Banach spaceE is said to satisfy Opial's condition if, for any sequence  {xn} in E, xn⇀x implies that
(2.4)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖
for all y∈E with y≠x, where xn⇀x means that {xn} converges weakly to x.Recall that the mappingT:K→K with F(T)≠∅ is 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 ∥x-Tx∥≥f(d(x,F(T))) for all x∈K, where d(x,F(T))=inf{∥x-p∥:p∈F(T)}. Khan and Fukhar-ud-din [15] modified condition (A) for two mappings as follows: Two mappings T1, T2:K→K are said to satisfy condition (A′) [15] if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0, f(t)>0 for all t∈(0,∞) such that
(2.5)12(‖x-T1x‖+‖x-T2x‖)≥f(d(x,F))
for all x∈K, where d(x,F)=inf{∥x-p∥:p∈F:=F(T1)∩F(T2)}.Note that condition(A′) reduces to condition (A) when T1=T2. It is also well-known that condition (A) is weaker than demicompactness or semicompactness, see [14].A mappingT with domain D(T) and range R(T) in E is said to be demiclosed at p if whenever {xn} is a sequence in D(T) such that {xn} converges weakly to x*∈D(T) and {Txn} converges strongly to p, then Tx*=p.We need the following lemmas for our main results.Lemma 2.1 (see [16]).
If{rn}, {tn} are two sequences of nonnegative real numbers such that
(2.6)rn+1≤(1+tn)rn,n∈N
and ∑n=1∞tn<∞, then limn→∞rn exists.Lemma 2.2 (see [17]).
Suppose thatE is a uniformly convex Banach space and  0<p≤tn≤q<1 for all  n∈ℕ. Also, suppose that  {xn} and  {yn} are sequences of  E  such that
(2.7)limsup⁡n→∞‖xn‖≤r,limsup⁡n→∞‖yn‖≤r,lim⁡n→∞‖(1-tn)xn+tnyn‖=r
hold for some  r≥0. Then  limn→∞∥xn-yn∥=0.Lemma 2.3 (see [18]).
LetE be real smooth Banach space, let K be nonempty closed convex subset of E with P as a sunny nonexpansive retraction, and let T:K→E be a mapping satisfying weakly inward condition. Then F(PT)=F(T).Lemma 2.4 (see [1]).
LetE be a uniformly convex Banach space and let C be a nonempty closed convex subset of E. Let T be a nonself asymptotically nonexpansive mapping. Then I-T  is demiclosed with respect to zero, that is, xn⇀x and xn-Txn→0 imply that Tx=x.


## 3. Main Results

### 3.1. Convergence Theorems in Real Banach Spaces

In this section, we prove the strong convergence of the iteration scheme (1.9) to a common fixed point of nonself asymptotically nonexpansive mappings T1 and T2 with respect to P in real Banach spaces. Let T1, T2:K→E be two nonself asymptotically nonexpansive mappings with respect to P with sequences {kn(i)}⊂[1,∞) satisfying ∑n=1∞(kn(i)-1)<∞(i=1,2), respectively. Put kn=max{kn(1),kn(2)}, then obviously ∑n=1∞(kn-1)<∞. From now on we will take this sequence  {kn} for both T1 and T2.We first prove the following lemmas.Lemma 3.1.
LetE be a real normed linear space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let  T1,T2:K→E be two nonself asymptotically nonexpansive mappings with respect to  P with  sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9) and F≠∅. Then,  (i)
limn→∞∥xn-p∥ exists for all  p∈F;(ii)
there exists a constantM>0 such that ∥xn+m-p∥≤M∥xn-p∥  for all  m,n∈ℕ and p∈F.Proof.
(i) Letp∈F. From (1.9), we have
(3.1)‖yn-p‖=‖(1-βn)xn+βn(PT1)nxn-p‖≤(1-βn)‖xn-p‖+βn‖(PT1)nxn-p‖≤(1-βn)‖xn-p‖+βnkn‖xn-p‖=(1+βn(kn-1))‖xn-p‖≤(1+(kn-1))‖xn-p‖=kn‖xn-p‖.
By (3.1) and (1.9), we obtain
(3.2)‖xn+1-p‖=‖(1-αn)(PT1)nyn-αn(PT2)nyn-p‖=‖(1-αn)((PT1)nyn-p)-αn((PT2)nyn-p)‖≤(1-αn)kn‖yn-p‖+αnkn‖yn-p‖=kn‖yn-p‖≤kn2‖xn-p‖=(1+(kn2-1))‖xn-p‖.
Note that ∑n=1∞(kn-1)<∞ is equivalent to ∑n=1∞(kn2-1)<∞. Thus, by (3.2) and Lemma 2.1, limn→∞∥xn-p∥ exists for all p∈F(T).
(ii) From (3.2), we have
(3.3)‖xn+1-p‖≤kn2‖xn-p‖.
It is well known that 1+x≤ex for all x≥0. Using it for the above inequality, we have
(3.4)‖xn+m-p‖≤(1+(kn+m-12-1))‖xn+m-1-p‖≤ekn+m-12-1‖xn+m-1-p‖≤ekn+m-12-1[(1+(kn+m-22-1))‖xn+m-2-p‖]≤ekn+m-12-1+kn+m-22-1‖xn+m-2-p‖⋮≤e∑j=nn+m-1(kj2-1)‖xn-p‖≤M‖xn-p‖,
where M=e∑j=nn+m-1(kj2-1). That is, ∥xn+m-p∥≤M∥xn-p∥  for allm,n∈ℕ and p∈F.Theorem 3.2.
LetE be a real Banach space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let  T1,T2:K→E be two nonself asymptotically nonexpansive mappings with respect to  P with  sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9) and  F≠∅.  Then, {xn} converges strongly to a common fixed point of  T1 and T2 if and only if liminfn→∞d(xn,F)=0, where d(xn,F)=inf{∥x-p∥:p∈F}.Proof.
The necessity of the conditions is obvious. Thus, we need only prove the sufficiency. Suppose thatliminfn→∞d(xn,F)=0. From (3.2), we have
(3.5)d(xn+1,F)≤(1+(kn2-1))d(xn,F).
As ∑n=1∞(kn-1)<∞, therefore limn→∞d(xn,F) exists by Lemma 2.1. But by hypothesis liminfn→∞d(xn,F)=0, therefore we must have limn→∞d(xn,F)=0.
Next we show that{xn} is a Cauchy sequence. Let ε>0. Since limn→∞d(xn,F)=0, therefore there exists a constant n0 such that for all n≥n0, we have
(3.6)d(xn,F)<ε4M,
where M>0 is the constant in Lemma 3.1(ii). So we can find p′∈F such that
(3.7)‖xn0-p′‖<ε2M.
Using Lemma 3.1(ii), we have for all n≥n0 and m∈ℕ that
(3.8)‖xn+m-xn‖≤‖xn+m-p′‖+‖xn-p′‖≤M‖xn0-p′‖+M‖xn0-p′‖=2M‖xn0-p′‖<ε.
Hence {xn} is a Cauchy sequence in a closed subset K of a Banach space E, therefore it must converge to a point in K. Let limn→∞xn=q. Now, limn→∞d(xn,F)=0 gives that d(q,F)=0. Since the set of fixed points of asymptotically nonexpansive mappings is closed, we have q∈F. This completes the proof of the theorem.On the lines similar to this theorem, we can also prove the following theorem which addresses the error terms.Theorem 3.3.
LetE be a real Banach space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1, T2:K→E be two asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by (1.10) with ∑n=1∞γn<∞, ∑n=1∞γn′<∞ and F≠∅, Then, {xn} converges strongly to a common fixed point of  T1 and T2 if and only if  liminfn→∞d(xn,F)=0, where d(xn,F)=inf{∥x-p∥:p∈F}.


### 3.2. Convergence Theorems in Real Uniformly Convex Banach Spaces

In this section, we prove the strong and weak convergence of the sequence defined by the iteration scheme (1.9) to a common fixed point of nonself asymptotically nonexpansive mappings T1 and T2 with respect to P in real uniformly convex and smooth Banach space. We first prove the following lemma.Lemma 3.4.
LetK be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T1, T2:K→E be two nonself asymptotically nonexpansive mappings with respect to  P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by(1.9), where {αn} and {βn} are sequences in [a,1-a] for some a∈(0,1). If  F≠∅, then
(3.9)lim⁡n→∞‖xn-(PT1)xn‖=lim⁡n→∞‖xn-(PT2)xn‖=0.Proof.
By Lemma3.1(i), limn→∞∥xn-p∥ exists. Assume that limn→∞∥xn-p∥=c. If c=0, the conclusion is obvious. Suppose c>0. Taking limsup on both sides in that inequality (3.1), we have
(3.10)limsup⁡n→∞‖yn-p‖≤limsup⁡n→∞‖xn-p‖=lim⁡n→∞‖xn-p‖=c.
Thus ∥(PT1)nyn-p∥≤kn∥yn-p∥ for all n∈ℕ implies that
(3.11)limsup⁡n→∞‖(PT1)nyn-p‖≤c.
Similarly,
(3.12)limsup⁡n→∞‖(PT2)nyn-p‖≤c.
Further,
(3.13)c=lim⁡n→∞‖xn+1-p‖=lim⁡n→∞‖(1-αn)(PT1)nyn+αn(PT2)nyn-p‖=lim⁡n→∞‖(1-αn)((PT1)nyn-p)+αn((PT2)nyn-p)‖≤lim⁡n→∞[(1-αn)‖limsup⁡n→∞((PT1)nyn-p)‖+αn‖limsup⁡n→∞((PT2)nyn-p)‖]≤lim⁡n→∞[(1-αn)c+αnc]=c
gives that
(3.14)lim⁡n→∞‖(1-αn)((PT1)nyn-p)+αn((PT2)nyn-p)‖=c.
Hence, using (3.11), (3.12), (3.14), and Lemma 2.2, we obtain
(3.15)lim⁡n→∞‖(PT2)nyn-(PT1)nyn‖=0.
Noting that
(3.16)‖xn+1-p‖=‖(1-αn)(PT1)nyn+αn(PT2)nyn-p‖≤‖(PT1)nyn-p‖+αn‖(PT2)nyn-(PT1)nyn‖≤kn‖yn-p‖+αn‖(PT2)nyn-(PT1)nyn‖
which yields that
(3.17)c≤liminf⁡n→∞‖yn-p‖.
By (3.10) and (3.17), we obtain
(3.18)lim⁡n→∞‖yn-p‖=c.
Moreover, ∥(PT1)nxn-p∥≤kn∥xn-p∥ for all n∈ℕ implies that
(3.19)limsup⁡n→∞‖(PT1)nxn-p‖≤c.
Hence
(3.20)c=lim⁡n→∞‖yn-p‖=lim⁡n→∞‖(1-βn)xn+βn(PT1)nxn-p‖=lim⁡n→∞‖(1-βn)(xn-p)+βn((PT1)nxn-p)‖≤lim⁡n→∞[(1-βn)‖limsup⁡n→∞(xn-p)‖+βn‖limsup⁡n→∞((PT1)nxn-p)‖]≤lim⁡n→∞[(1-βn)c+βnc]=c
gives that
(3.21)lim⁡n→∞‖(1-βn)(xn-p)+βn((PT1)nxn-p)‖=c.
Again by Lemma 2.2, we obtain
(3.22)lim⁡n→∞‖(PT1)nxn-xn‖=0.
In addition, from yn=(1-βn)xn+βn(PT1)nxn, we have
(3.23)‖yn-xn‖=βn‖(PT1)nxn-xn‖.
Hence by (3.22),
(3.24)lim⁡n→∞‖yn-xn‖=0.
Also
(3.25)‖(PT2)nyn-xn‖≤‖(PT2)nyn-(PT1)nyn‖+‖(PT1)nyn-(PT1)nxn‖+‖(PT1)nxn-xn‖≤‖(PT2)nyn-(PT1)nyn‖+kn‖yn-xn‖+‖(PT1)nxn-xn‖
implies by (3.15), (3.22), and (3.24) that
(3.26)lim⁡n→∞‖(PT2)nyn-xn‖=0.
Using (3.24) and (3.26), we obtain
(3.27)‖(PT2)nxn-xn‖≤‖(PT2)nxn-(PT2)nyn‖+‖(PT2)nyn-xn‖≤kn‖xn-yn‖+‖(PT2)nyn-xn‖,
so that
(3.28)lim⁡n→∞‖(PT2)nxn-xn‖=0.
Then
(3.29)‖(PT1)nyn-xn‖≤‖(PT1)nyn-(PT1)nxn‖+‖(PT1)nxn-xn‖≤kn‖yn-xn‖+‖(PT1)nxn-xn‖
gives
(3.30)lim⁡n→∞‖(PT1)nyn-xn‖=0.
From (3.15), and (3.30), we have
(3.31)‖xn+1-xn‖=‖(1-αn)(PT1)nyn+αn(PT2)nyn-xn‖≤‖(PT1)nyn-xn‖+αn‖(PT2)nyn-(PT1)nyn‖⟶0asn⟶∞.
Thus from ∥xn+1-yn∥≤∥xn+1-xn∥+∥xn-yn∥, we get
(3.32)lim⁡n→∞‖xn+1-yn‖=0.
From (3.30), (3.31) and
(3.33)‖xn+1-(PT1)nyn‖≤‖xn+1-xn‖+‖xn-(PT1)nyn‖,
we have
(3.34)lim⁡n→∞‖xn+1-(PT1)nyn‖=0.
Now we make use of the fact that every nonself asymptotically nonexpansive mapping with respect to P must be uniformly L-Lipschitzian with respect to P combined with (3.22), (3.32), and (3.34), where L=supn∈ℕ{kn}≥1, to reach at
(3.35)‖xn-(PT1)xn‖≤‖xn-(PT1)nxn‖+‖(PT1)nxn-(PT1)nyn-1‖+‖(PT1)nyn-1-(PT1)xn‖≤‖xn-(PT1)nxn‖+kn‖xn-yn-1‖+L‖(PT1)n-1yn-1-xn‖.
Thus
(3.36)lim⁡n→∞‖xn-(PT1)xn‖=0.
From (3.24), (3.26), and (3.31), we have
(3.37)‖xn+1-(PT2)nxn‖≤‖xn+1-xn‖+‖xn-(PT2)nyn‖+‖(PT2)nyn-(PT2)nxn‖≤‖xn+1-xn‖+‖xn-(PT2)nyn‖+kn‖yn-xn‖⟶0asn→∞,
and so
(3.38)lim⁡n→∞‖xn+1-(PT2)nxn‖=0.
Again making use of the fact that every nonself asymptotically nonexpansive mapping with respect to P must be uniformly L-Lipschitzian with respect to P and (3.28), (3.31) and (3.38), we have
(3.39)‖xn+1-(PT2)xn+1‖≤‖xn+1-(PT2)n+1xn+1‖+‖(PT2)n+1xn+1-(PT2)n+1xn‖+‖(PT2)n+1xn-(PT2)xn+1‖≤‖xn+1-(PT2)n+1xn+1‖+kn+1‖xn+1-xn‖+L‖(PT2)nxn-xn+1‖.
This gives,
(3.40)lim⁡n→∞‖xn-(PT2)xn‖=0.
This completes the proof of the lemma.Theorem 3.5.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9),  where {αn} and {βn} are two sequences in [a,1-a] for some a∈(0,1). If one of T1 and T2 is completely continuous and F≠∅, then{xn} converges strongly to a common fixed point of T1 and T2.Proof.
By Lemma3.1(i), limn→∞∥xn-p∥ exists for any p∈F. It is sufficient to show that {xn} has a subsequence which converges strongly to a common fixed point of T1 and T2. By Lemma 3.4, limn→∞∥xn-(PT1)xn∥=limn→∞∥xn-(PT2)xn∥=0. Suppose that T1 is completely continuous. Noting that P is nonexpansive, we conclude that there exists subsequence  {PT1xnj} of {PT1xn} such that PT1xnj→p. Thus ∥xnj-p∥≤∥xnj-PT1xnj∥+∥PT1xnj-p∥ implies xnj→p as j→∞. Again limj→∞∥xnj-(PT1)xnj∥=0 yields by continuity of P and T1 that p=PT1p. Similarly p=PT2p. By Lemma 2.3, p=T1p=T2p. Since F is closed, so p∈F. Thus {xn} converges strongly to a common fixed point p of T1 and T2. This completes the proof.Theorem 3.6.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9),  where {αn} and {βn} are two sequences in [a,1-a] for some a∈(0,1). If T1 and T2 satisfy condition (A′) and F≠∅, then {xn} converges strongly to a common fixed point of T1 and T2.Proof.
By Lemma3.1(i), limn→∞∥xn-p∥ exists, and so, limn→∞d(xn,F) exists for all  p∈F. Also, by Lemma 3.4  limn→∞∥xn-(PT1)xn∥=limn→∞∥xn-(PT2)xn∥=0. It follows from condition (A′) and  Lemma 2.3 that
(3.41)lim⁡n→∞f(d(xn,F))≤lim⁡n→∞(12(‖xn-(PT1)xn‖+‖xn-(PT2)xn‖))=0.
That is,
(3.42)lim⁡n→∞f(d(xn,F))=0.
Since f:[0,∞)→[0,∞) is a nondecreasing function satisfying f(0)=0, f(t)>0 for all t∈(0,∞), therefore we have
(3.43)lim⁡n→∞d(xn,F)=0.
From Theorem 3.2, we obtain that {xn} is a Cauchy sequence in K. Since K is a closed subset of a complete space, there exists a q∈K such that xn→q as n→∞. Then, limn→∞d(xn,F)=0 yields that d(q,F)=0. Further, it follows from the closedness of F that q∈F. This completes the proof.Theorem 3.7.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E satisfying Opial's condition with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by (1.9), where {αn} and {βn} are two sequences in [a,1-a] for some a∈(0,1). If F≠∅, then {xn} converges weakly to a common fixed point of T1 and T2.Proof.
Letp∈F. By Lemma 3.1(i), limn→∞∥xn-p∥ exists, and {xn} is bounded. Note that PT1 and PT2 are self-mappings from K into itself.  We now prove that {xn} has a unique weak subsequential limit in F. Suppose that subsequences {xnk} and {xnj} of {xn} converge weakly to p1 and p2, respectively. By Lemma 3.4, we have limn→∞∥xnk-(PTi)xnk∥=0, (i=1,2). Lemma 2.4 guarantees that (I-PT1)p1=0, that is, (PT1)p1=p1. Similarly, (PT2)p1=p1. Again in the same way, we can prove that p2∈F. Lemma 2.3 now assures that p1,p2∈F. For uniqueness, assume that p1≠p2, then by Opial’s condition, we have
(3.44)lim⁡n→∞‖xn-p1‖=lim⁡k→∞‖xnk-p1‖<lim⁡k→∞‖xnk-p2‖=lim⁡j→∞‖xnj-p2‖<lim⁡j→∞‖xnj-p1‖=lim⁡n→∞‖xn-p1‖,
which is a contradiction and hence p1=p2. As a result, {xn} converges weakly to a common fixed point of T1 and T2.In a way similar to the above, we can also prove the results involving error terms as follows.Theorem 3.8.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is the sequence defined by (1.10) satisfying the following conditions:(i)
∑n=1∞γn<∞,  ∑n=1∞γn′<∞;(ii)
{αn} and {αn′} are two sequences in [a,1-a] for some a∈(0,1).If one of T1 and T2 is completely continuous and F≠∅, then {xn} converges strongly to  a common fixed point of T1 and T2.Theorem 3.9.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is the sequence defined by (1.10) satisfying the following conditions:(i)
∑n=1∞γn<∞,  ∑n=1∞γn′<∞;(ii)
{αn} and {αn′} are two sequences in [a,1-a] for some a∈(0,1). If T1 and T2 satisfy condition (A′) and F≠∅, then {xn} converges strongly to a common fixed point of T1 and T2.Theorem 3.10.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E satisfying Opial's condition with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is the sequence defined by (1.10) satisfying the following conditions:(i)
∑n=1∞γn<∞,  ∑n=1∞γn′<∞;(ii)
{αn} and {αn′} are two sequences in [a,1-a] for some a∈(0,1).If F≠∅, then {xn} converges weakly to a common fixed point of T1 and T2.


## 3.1. Convergence Theorems in Real Banach Spaces

In this section, we prove the strong convergence of the iteration scheme (1.9) to a common fixed point of nonself asymptotically nonexpansive mappings T1 and T2 with respect to P in real Banach spaces. Let T1, T2:K→E be two nonself asymptotically nonexpansive mappings with respect to P with sequences {kn(i)}⊂[1,∞) satisfying ∑n=1∞(kn(i)-1)<∞(i=1,2), respectively. Put kn=max{kn(1),kn(2)}, then obviously ∑n=1∞(kn-1)<∞. From now on we will take this sequence  {kn} for both T1 and T2.We first prove the following lemmas.Lemma 3.1.
LetE be a real normed linear space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let  T1,T2:K→E be two nonself asymptotically nonexpansive mappings with respect to  P with  sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9) and F≠∅. Then,  (i)
limn→∞∥xn-p∥ exists for all  p∈F;(ii)
there exists a constantM>0 such that ∥xn+m-p∥≤M∥xn-p∥  for all  m,n∈ℕ and p∈F.Proof.
(i) Letp∈F. From (1.9), we have
(3.1)‖yn-p‖=‖(1-βn)xn+βn(PT1)nxn-p‖≤(1-βn)‖xn-p‖+βn‖(PT1)nxn-p‖≤(1-βn)‖xn-p‖+βnkn‖xn-p‖=(1+βn(kn-1))‖xn-p‖≤(1+(kn-1))‖xn-p‖=kn‖xn-p‖.
By (3.1) and (1.9), we obtain
(3.2)‖xn+1-p‖=‖(1-αn)(PT1)nyn-αn(PT2)nyn-p‖=‖(1-αn)((PT1)nyn-p)-αn((PT2)nyn-p)‖≤(1-αn)kn‖yn-p‖+αnkn‖yn-p‖=kn‖yn-p‖≤kn2‖xn-p‖=(1+(kn2-1))‖xn-p‖.
Note that ∑n=1∞(kn-1)<∞ is equivalent to ∑n=1∞(kn2-1)<∞. Thus, by (3.2) and Lemma 2.1, limn→∞∥xn-p∥ exists for all p∈F(T).
(ii) From (3.2), we have
(3.3)‖xn+1-p‖≤kn2‖xn-p‖.
It is well known that 1+x≤ex for all x≥0. Using it for the above inequality, we have
(3.4)‖xn+m-p‖≤(1+(kn+m-12-1))‖xn+m-1-p‖≤ekn+m-12-1‖xn+m-1-p‖≤ekn+m-12-1[(1+(kn+m-22-1))‖xn+m-2-p‖]≤ekn+m-12-1+kn+m-22-1‖xn+m-2-p‖⋮≤e∑j=nn+m-1(kj2-1)‖xn-p‖≤M‖xn-p‖,
where M=e∑j=nn+m-1(kj2-1). That is, ∥xn+m-p∥≤M∥xn-p∥  for allm,n∈ℕ and p∈F.Theorem 3.2.
LetE be a real Banach space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let  T1,T2:K→E be two nonself asymptotically nonexpansive mappings with respect to  P with  sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9) and  F≠∅.  Then, {xn} converges strongly to a common fixed point of  T1 and T2 if and only if liminfn→∞d(xn,F)=0, where d(xn,F)=inf{∥x-p∥:p∈F}.Proof.
The necessity of the conditions is obvious. Thus, we need only prove the sufficiency. Suppose thatliminfn→∞d(xn,F)=0. From (3.2), we have
(3.5)d(xn+1,F)≤(1+(kn2-1))d(xn,F).
As ∑n=1∞(kn-1)<∞, therefore limn→∞d(xn,F) exists by Lemma 2.1. But by hypothesis liminfn→∞d(xn,F)=0, therefore we must have limn→∞d(xn,F)=0.
Next we show that{xn} is a Cauchy sequence. Let ε>0. Since limn→∞d(xn,F)=0, therefore there exists a constant n0 such that for all n≥n0, we have
(3.6)d(xn,F)<ε4M,
where M>0 is the constant in Lemma 3.1(ii). So we can find p′∈F such that
(3.7)‖xn0-p′‖<ε2M.
Using Lemma 3.1(ii), we have for all n≥n0 and m∈ℕ that
(3.8)‖xn+m-xn‖≤‖xn+m-p′‖+‖xn-p′‖≤M‖xn0-p′‖+M‖xn0-p′‖=2M‖xn0-p′‖<ε.
Hence {xn} is a Cauchy sequence in a closed subset K of a Banach space E, therefore it must converge to a point in K. Let limn→∞xn=q. Now, limn→∞d(xn,F)=0 gives that d(q,F)=0. Since the set of fixed points of asymptotically nonexpansive mappings is closed, we have q∈F. This completes the proof of the theorem.On the lines similar to this theorem, we can also prove the following theorem which addresses the error terms.Theorem 3.3.
LetE be a real Banach space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1, T2:K→E be two asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by (1.10) with ∑n=1∞γn<∞, ∑n=1∞γn′<∞ and F≠∅, Then, {xn} converges strongly to a common fixed point of  T1 and T2 if and only if  liminfn→∞d(xn,F)=0, where d(xn,F)=inf{∥x-p∥:p∈F}.


## 3.2. Convergence Theorems in Real Uniformly Convex Banach Spaces

In this section, we prove the strong and weak convergence of the sequence defined by the iteration scheme (1.9) to a common fixed point of nonself asymptotically nonexpansive mappings T1 and T2 with respect to P in real uniformly convex and smooth Banach space. We first prove the following lemma.Lemma 3.4.
LetK be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T1, T2:K→E be two nonself asymptotically nonexpansive mappings with respect to  P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by(1.9), where {αn} and {βn} are sequences in [a,1-a] for some a∈(0,1). If  F≠∅, then
(3.9)lim⁡n→∞‖xn-(PT1)xn‖=lim⁡n→∞‖xn-(PT2)xn‖=0.Proof.
By Lemma3.1(i), limn→∞∥xn-p∥ exists. Assume that limn→∞∥xn-p∥=c. If c=0, the conclusion is obvious. Suppose c>0. Taking limsup on both sides in that inequality (3.1), we have
(3.10)limsup⁡n→∞‖yn-p‖≤limsup⁡n→∞‖xn-p‖=lim⁡n→∞‖xn-p‖=c.
Thus ∥(PT1)nyn-p∥≤kn∥yn-p∥ for all n∈ℕ implies that
(3.11)limsup⁡n→∞‖(PT1)nyn-p‖≤c.
Similarly,
(3.12)limsup⁡n→∞‖(PT2)nyn-p‖≤c.
Further,
(3.13)c=lim⁡n→∞‖xn+1-p‖=lim⁡n→∞‖(1-αn)(PT1)nyn+αn(PT2)nyn-p‖=lim⁡n→∞‖(1-αn)((PT1)nyn-p)+αn((PT2)nyn-p)‖≤lim⁡n→∞[(1-αn)‖limsup⁡n→∞((PT1)nyn-p)‖+αn‖limsup⁡n→∞((PT2)nyn-p)‖]≤lim⁡n→∞[(1-αn)c+αnc]=c
gives that
(3.14)lim⁡n→∞‖(1-αn)((PT1)nyn-p)+αn((PT2)nyn-p)‖=c.
Hence, using (3.11), (3.12), (3.14), and Lemma 2.2, we obtain
(3.15)lim⁡n→∞‖(PT2)nyn-(PT1)nyn‖=0.
Noting that
(3.16)‖xn+1-p‖=‖(1-αn)(PT1)nyn+αn(PT2)nyn-p‖≤‖(PT1)nyn-p‖+αn‖(PT2)nyn-(PT1)nyn‖≤kn‖yn-p‖+αn‖(PT2)nyn-(PT1)nyn‖
which yields that
(3.17)c≤liminf⁡n→∞‖yn-p‖.
By (3.10) and (3.17), we obtain
(3.18)lim⁡n→∞‖yn-p‖=c.
Moreover, ∥(PT1)nxn-p∥≤kn∥xn-p∥ for all n∈ℕ implies that
(3.19)limsup⁡n→∞‖(PT1)nxn-p‖≤c.
Hence
(3.20)c=lim⁡n→∞‖yn-p‖=lim⁡n→∞‖(1-βn)xn+βn(PT1)nxn-p‖=lim⁡n→∞‖(1-βn)(xn-p)+βn((PT1)nxn-p)‖≤lim⁡n→∞[(1-βn)‖limsup⁡n→∞(xn-p)‖+βn‖limsup⁡n→∞((PT1)nxn-p)‖]≤lim⁡n→∞[(1-βn)c+βnc]=c
gives that
(3.21)lim⁡n→∞‖(1-βn)(xn-p)+βn((PT1)nxn-p)‖=c.
Again by Lemma 2.2, we obtain
(3.22)lim⁡n→∞‖(PT1)nxn-xn‖=0.
In addition, from yn=(1-βn)xn+βn(PT1)nxn, we have
(3.23)‖yn-xn‖=βn‖(PT1)nxn-xn‖.
Hence by (3.22),
(3.24)lim⁡n→∞‖yn-xn‖=0.
Also
(3.25)‖(PT2)nyn-xn‖≤‖(PT2)nyn-(PT1)nyn‖+‖(PT1)nyn-(PT1)nxn‖+‖(PT1)nxn-xn‖≤‖(PT2)nyn-(PT1)nyn‖+kn‖yn-xn‖+‖(PT1)nxn-xn‖
implies by (3.15), (3.22), and (3.24) that
(3.26)lim⁡n→∞‖(PT2)nyn-xn‖=0.
Using (3.24) and (3.26), we obtain
(3.27)‖(PT2)nxn-xn‖≤‖(PT2)nxn-(PT2)nyn‖+‖(PT2)nyn-xn‖≤kn‖xn-yn‖+‖(PT2)nyn-xn‖,
so that
(3.28)lim⁡n→∞‖(PT2)nxn-xn‖=0.
Then
(3.29)‖(PT1)nyn-xn‖≤‖(PT1)nyn-(PT1)nxn‖+‖(PT1)nxn-xn‖≤kn‖yn-xn‖+‖(PT1)nxn-xn‖
gives
(3.30)lim⁡n→∞‖(PT1)nyn-xn‖=0.
From (3.15), and (3.30), we have
(3.31)‖xn+1-xn‖=‖(1-αn)(PT1)nyn+αn(PT2)nyn-xn‖≤‖(PT1)nyn-xn‖+αn‖(PT2)nyn-(PT1)nyn‖⟶0asn⟶∞.
Thus from ∥xn+1-yn∥≤∥xn+1-xn∥+∥xn-yn∥, we get
(3.32)lim⁡n→∞‖xn+1-yn‖=0.
From (3.30), (3.31) and
(3.33)‖xn+1-(PT1)nyn‖≤‖xn+1-xn‖+‖xn-(PT1)nyn‖,
we have
(3.34)lim⁡n→∞‖xn+1-(PT1)nyn‖=0.
Now we make use of the fact that every nonself asymptotically nonexpansive mapping with respect to P must be uniformly L-Lipschitzian with respect to P combined with (3.22), (3.32), and (3.34), where L=supn∈ℕ{kn}≥1, to reach at
(3.35)‖xn-(PT1)xn‖≤‖xn-(PT1)nxn‖+‖(PT1)nxn-(PT1)nyn-1‖+‖(PT1)nyn-1-(PT1)xn‖≤‖xn-(PT1)nxn‖+kn‖xn-yn-1‖+L‖(PT1)n-1yn-1-xn‖.
Thus
(3.36)lim⁡n→∞‖xn-(PT1)xn‖=0.
From (3.24), (3.26), and (3.31), we have
(3.37)‖xn+1-(PT2)nxn‖≤‖xn+1-xn‖+‖xn-(PT2)nyn‖+‖(PT2)nyn-(PT2)nxn‖≤‖xn+1-xn‖+‖xn-(PT2)nyn‖+kn‖yn-xn‖⟶0asn→∞,
and so
(3.38)lim⁡n→∞‖xn+1-(PT2)nxn‖=0.
Again making use of the fact that every nonself asymptotically nonexpansive mapping with respect to P must be uniformly L-Lipschitzian with respect to P and (3.28), (3.31) and (3.38), we have
(3.39)‖xn+1-(PT2)xn+1‖≤‖xn+1-(PT2)n+1xn+1‖+‖(PT2)n+1xn+1-(PT2)n+1xn‖+‖(PT2)n+1xn-(PT2)xn+1‖≤‖xn+1-(PT2)n+1xn+1‖+kn+1‖xn+1-xn‖+L‖(PT2)nxn-xn+1‖.
This gives,
(3.40)lim⁡n→∞‖xn-(PT2)xn‖=0.
This completes the proof of the lemma.Theorem 3.5.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9),  where {αn} and {βn} are two sequences in [a,1-a] for some a∈(0,1). If one of T1 and T2 is completely continuous and F≠∅, then{xn} converges strongly to a common fixed point of T1 and T2.Proof.
By Lemma3.1(i), limn→∞∥xn-p∥ exists for any p∈F. It is sufficient to show that {xn} has a subsequence which converges strongly to a common fixed point of T1 and T2. By Lemma 3.4, limn→∞∥xn-(PT1)xn∥=limn→∞∥xn-(PT2)xn∥=0. Suppose that T1 is completely continuous. Noting that P is nonexpansive, we conclude that there exists subsequence  {PT1xnj} of {PT1xn} such that PT1xnj→p. Thus ∥xnj-p∥≤∥xnj-PT1xnj∥+∥PT1xnj-p∥ implies xnj→p as j→∞. Again limj→∞∥xnj-(PT1)xnj∥=0 yields by continuity of P and T1 that p=PT1p. Similarly p=PT2p. By Lemma 2.3, p=T1p=T2p. Since F is closed, so p∈F. Thus {xn} converges strongly to a common fixed point p of T1 and T2. This completes the proof.Theorem 3.6.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by  (1.9),  where {αn} and {βn} are two sequences in [a,1-a] for some a∈(0,1). If T1 and T2 satisfy condition (A′) and F≠∅, then {xn} converges strongly to a common fixed point of T1 and T2.Proof.
By Lemma3.1(i), limn→∞∥xn-p∥ exists, and so, limn→∞d(xn,F) exists for all  p∈F. Also, by Lemma 3.4  limn→∞∥xn-(PT1)xn∥=limn→∞∥xn-(PT2)xn∥=0. It follows from condition (A′) and  Lemma 2.3 that
(3.41)lim⁡n→∞f(d(xn,F))≤lim⁡n→∞(12(‖xn-(PT1)xn‖+‖xn-(PT2)xn‖))=0.
That is,
(3.42)lim⁡n→∞f(d(xn,F))=0.
Since f:[0,∞)→[0,∞) is a nondecreasing function satisfying f(0)=0, f(t)>0 for all t∈(0,∞), therefore we have
(3.43)lim⁡n→∞d(xn,F)=0.
From Theorem 3.2, we obtain that {xn} is a Cauchy sequence in K. Since K is a closed subset of a complete space, there exists a q∈K such that xn→q as n→∞. Then, limn→∞d(xn,F)=0 yields that d(q,F)=0. Further, it follows from the closedness of F that q∈F. This completes the proof.Theorem 3.7.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E satisfying Opial's condition with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is defined by (1.9), where {αn} and {βn} are two sequences in [a,1-a] for some a∈(0,1). If F≠∅, then {xn} converges weakly to a common fixed point of T1 and T2.Proof.
Letp∈F. By Lemma 3.1(i), limn→∞∥xn-p∥ exists, and {xn} is bounded. Note that PT1 and PT2 are self-mappings from K into itself.  We now prove that {xn} has a unique weak subsequential limit in F. Suppose that subsequences {xnk} and {xnj} of {xn} converge weakly to p1 and p2, respectively. By Lemma 3.4, we have limn→∞∥xnk-(PTi)xnk∥=0, (i=1,2). Lemma 2.4 guarantees that (I-PT1)p1=0, that is, (PT1)p1=p1. Similarly, (PT2)p1=p1. Again in the same way, we can prove that p2∈F. Lemma 2.3 now assures that p1,p2∈F. For uniqueness, assume that p1≠p2, then by Opial’s condition, we have
(3.44)lim⁡n→∞‖xn-p1‖=lim⁡k→∞‖xnk-p1‖<lim⁡k→∞‖xnk-p2‖=lim⁡j→∞‖xnj-p2‖<lim⁡j→∞‖xnj-p1‖=lim⁡n→∞‖xn-p1‖,
which is a contradiction and hence p1=p2. As a result, {xn} converges weakly to a common fixed point of T1 and T2.In a way similar to the above, we can also prove the results involving error terms as follows.Theorem 3.8.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is the sequence defined by (1.10) satisfying the following conditions:(i)
∑n=1∞γn<∞,  ∑n=1∞γn′<∞;(ii)
{αn} and {αn′} are two sequences in [a,1-a] for some a∈(0,1).If one of T1 and T2 is completely continuous and F≠∅, then {xn} converges strongly to  a common fixed point of T1 and T2.Theorem 3.9.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is the sequence defined by (1.10) satisfying the following conditions:(i)
∑n=1∞γn<∞,  ∑n=1∞γn′<∞;(ii)
{αn} and {αn′} are two sequences in [a,1-a] for some a∈(0,1). If T1 and T2 satisfy condition (A′) and F≠∅, then {xn} converges strongly to a common fixed point of T1 and T2.Theorem 3.10.
LetK be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E satisfying Opial's condition with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and nonself asymptotically nonexpansive mappings with respect to P with sequence  {kn}⊂[1,∞) satisfying  ∑n=1∞(kn-1)<∞. Suppose that {xn} is the sequence defined by (1.10) satisfying the following conditions:(i)
∑n=1∞γn<∞,  ∑n=1∞γn′<∞;(ii)
{αn} and {αn′} are two sequences in [a,1-a] for some a∈(0,1).If F≠∅, then {xn} converges weakly to a common fixed point of T1 and T2.



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*Source: 487864-2011-04-27.xml*