# Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

Authors: Tanakit Thianwan
Journal: Abstract and Applied Analysis
 (2011)

---

## Abstract

We study weak convergence of the projection type Ishikawa iteration scheme
for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach
spaceE which has a Fréchet differentiable norm or its dual E* has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically
nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.


---

## Body

## 1. Introduction and Preliminaries

LetC be a nonempty closed convex subset of real normed linear space X. Let T:C→C be a mapping. A point x∈C is called a fixed point of T if and only if Tx=x. The set of all fixed points of a mapping T is denoted by F(T). A self-mapping T:C→C is said to be nonexpansive if ∥T(x)-T(y)∥≤∥x-y∥ for all x,y∈C. A self-mapping T:C→C is called asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞),kn→1 as n→∞ such that  (1.1)‖Tn(x)-Tn(y)‖≤kn‖x-y‖
for all  x,y∈C  and n≥1.  A mapping  T:C→C  is said to be uniformly L-Lipschitzian if there exists a constant  L>0  such that (1.2)‖Tn(x)-Tn(y)‖≤L‖x-y‖
for all x,y∈C  and  n≥1.  T  is uniformly Hölder continuous if there exist positive constants  L  and  α  such that (1.3)‖Tn(x)-Tn(y)‖<L‖x-y‖α
for all  x,y∈C  and  n≥1.  T  is termed as uniformly equicontinuous if, for any  ε>0,  there exists  δ>0  such that (1.4)‖Tn(x)-Tn(y)‖≤ε
whenever ∥x-y∥≤δ  for all  x,y∈C  and  n≥1  or, equivalently,  T  is uniformly equicontinuous if and only if (1.5)‖Tn(xn)-Tn(yn)‖⟶0
whenever  ∥xn-yn∥→0  as  n→∞.It is easy to see that ifT  is an asymptotically nonexpansive, then it is uniformly L-Lipschitzian with the uniform Lipschitz constant  L=sup{kn:n≥1}.Remark 1.1.
It is clear that asymptotically nonexpansiveness⇒ uniformly L-Lipschitz ⇒ uniformly Hölder continuous ⇒ uniformly equicontinuous.However, their converse fail in the presence of the following example.Example 1.2 (see [1]).
DefineT:[0,1]→[0,1]   by  Tx=(1-x3/2)2/3  for all  x∈[0,1].Fixed-point iteration process for nonexpansive self-mappings including Mann and Ishikawa iteration processes has been studied extensively by various authors [2–8]. For nonexpansive nonself-mappings, some authors (see [9–13]) have studied the strong and weak convergence theorems in Hilbert spaces or uniformly convex Banach spaces.In [14], Tan and Xu introduced a modified Ishikawa iteration process:  (1.6)xn+1=(1-bn)xn+bnT((1-γn)xn+γnTxn),n≥1,
to approximate fixed points of nonexpansive self-mappings defined on nonempty closed convex bounded subsets of a uniformly convex Banach space  X. The mapping  T  remains self-mapping of a nonempty closed convex subset  C  of a uniformly convex Banach space. If, however, the domain  C  of  T  is a proper subset of  X  (and this is the case in several applications) and  T  maps  C  into  X  then, the sequence {xn}  generated by (1.6) may not be well defined. More precisely, Tan and Xu [14] proved weak convergence of the sequences generated by (1.6) to some fixed point of  T  in a uniformly convex Banach space which satisfies Opial's condition or has a Fréchet differentiable norm.Note that eachlp(1≤p<∞)  satisfies Opial's condition, while all  Lp  do not have the property unless  p=2  and the dual of reflexive Banach spaces with a Fréchet differentiable norm has the Kadec-Klee property. It is worth mentioning that there are uniformly convex Banach spaces, which have neither a Fréchet differentiable norm nor Opial property; however, their dual does have the Kadec-Klee property (see [15, 16]).In 2005, Shahzad [11] extended Tan and Xu's result [14] to the case of nonexpansive nonself-mapping in a uniformly convex Banach space. He studied weak convergence of the modified Ishikawa type iteration process:  (1.7)xn+1=P((1-bn)xn+bnTP((1-γn)xn+γnTxn)),n≥1,
in a uniformly convex Banach space whose dual has the Kadec-Klee property. The result applies not only to  Lp  spaces with  (1≤p<∞)  but also to other spaces which do not satisfy Opial's condition or have a Fréchet differentiable norm. Meanwhile, the results of [11] generalized the results of [14].The class of asymptotically nonexpansive self-mappings is a natural generalization of the important class of nonexpansive mappings. Goebel and Kirk [17] proved that if  C  is a nonempty closed convex and bounded subset of a real uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.In 1991, the modified Mann iteration which was introduced by Schu [18] generates a sequence {xn}  in the following manner: (1.8)xn+1=(1-αn)xn+αnTnxn,n≥1,
where  {αn}  is a sequence in the interval (0,1)  and  T:C→C  is an asymptotically nonexpansive mapping. To be more precise, Schu [18] obtained the following weak convergence result for an asymptotically nonexpansive mapping in a uniformly convex Banach space which satisfies Opial's condition.Theorem 1.3 (see [18]).
LetX be a uniformly convex Banach space satisfying Opial’s condition,  ∅≠C⊂X  closed bounded and convex, and  T:C→C  asymptotically nonexpansive with sequence  {kn}⊂[1,∞)  for which  ∑n=1∞(kn-1)<∞  and  {αn}∈[0,1]  is bounded away. Let  {xn}  be a sequence generated in (1.8). Then, the sequence {xn} converges weakly to some fixed point of  T.Since then, Schu's iteration process has been widely used to approximate fixed points of asymptotically nonexpansive self-mappings in Hilbert space or Banach spaces (see [6, 14, 19, 20]).In 1994, Tan and Xu [21] obtained the following results.Theorem 1.4 (see [21]).
LetX be a uniformly convex Banach space whose norm is Fréchet differentiable, C a nonempty closed and convex subset of X, and T:C→C an asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞) such that ∑n=1∞(kn-1)<∞ such that F(T) is nonempty. Let {xn} be sequence generated in (1.8), where {αn} is a real sequence bounded away from 0 and 1. Then, the sequence {xn} converges weakly to some point in F(T).In 2001, Khan and Takahashi [22] constructed and studied the following Ishikawa iteration process:(1.9)xn+1=(1-αn)xn+αnT1nyn,yn=(1-βn)xn+βnT2nxn,n≥1,
where  T1, T2  are asymptotically nonexpansive self-mappings on C with  ∑n=1∞(kn-1)<∞ (rate of convergence) and  0≤αn, βn≤1.Note that the rate of convergence condition, namely,∑n=1∞(kn-1)<∞ has remained in extensive use to prove both weak and strong convergence theorems to approximate fixed points of asymptotically nonexpansive maps. The conditions like Opial's condition, Kadec-Klee property, or Fréchet differentiable norm have remained key to prove weak convergence theorems.In 2010, Khan and Fukhar-Ud-Din [23] established weak convergence of Ishikawa iterates of two asymptotically nonexpansive self-mappings without any condition on the rate of convergence associated with the two mappings. They got that the following new weak convergence theorem does not require any of Opial’s condition, Kadec-Klee property or Fréchet differentiable norm.Theorem 1.5 (see [23]).
LetC be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let T1,T2:C→C be asymptotically nonexpansive maps with sequences {kn},{ln}⊂[1,∞) such that limn→∞kn=1, limn→∞ln=1, respectively. Let the sequence {xn} be as in (1.9) with δ≤αn,βn≤1-δ. for some δ∈(0,1/2). If F(T1)∩F(T2)≠∅, then {xn} converges weakly to a common fixed point of T1 and T2.The concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [24] in 2003 as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.Definition 1.6 (see [24]).
LetC be a nonempty subset of a real normed linear space X. Let P:X→C  be a nonexpansive retraction of X onto C. A nonself-mapping T:C→X  is called asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞), kn→1 as n→∞ such that
(1.10)‖T(PT)n-1x-T(PT)n-1y‖≤kn‖x-y‖
for all x,y∈C and n≥1.  T is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that
(1.11)‖T(PT)n-1x-T(PT)n-1y‖≤L‖x-y‖
for all x,y∈C and n≥1.By studying the following iteration process:(1.12)x1∈C,xn+1=P((1-αn)xn+αnT(PT)n-1xn),Chidume et al. [24] got the following weak convergence theorem for asymptotically nonexpansive nonself-mapping.Theorem 1.7 (see [24]).
LetX be a real uniformly convex Banach space which has a Fréchet differentiable norm and C a nonempty closed convex subset of X. Let T:C→X be an asymptotically nonexpansive map with sequence {kn}⊂[1,∞) such that ∑n=1∞(kn2-1)<∞ and F(T)≠∅. Let {αn}⊂(0,1) be such that ϵ≤1-αn≤1-ϵ, for all n≥1 and some ϵ>0. From an arbitrary x1∈C, define the sequence {xn} by (1.12). Then, {xn} converges weakly to some fixed point of T.IfT is a self-mapping, then P becomes the identity mapping so that (1.10) and (1.11) reduce to (1.1) and (1.2), respectively. Equation (1.12) reduces to (1.8).In 2006, Wang [25] generalizes the iteration process (1.12) as follows: x1∈C,(1.13)xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1xn),n≥1,
where T1,T2:C→X are asymptotically nonexpansive nonself-mappings and {αn}, {βn} are real sequences in [0,1). He studied the strong and weak convergence of the iterative scheme (1.13) under proper conditions. Meanwhile, the results of [25] generalized the results of [24].Recently, an iterative scheme which is called the projection type Ishikawa iteration for two asymptotically nonexpansive nonself-mappings was defined and constructed by Thianwan [26]. It is given as follows:(1.14)xn+1=P((1-αn)yn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1xn),n≥1,
where {αn} and {βn} are appropriate real sequences in [0,1).In [26], Thianwan gave the following weak convergence theorem.Theorem 1.8.
LetX be a uniformly convex Banach space which satisfies Opial's condition and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [ϵ,1-ϵ] for some ϵ∈(0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, {xn} and {yn} converge weakly to a common fixed point of T1 and T2.The iterative schemes (1.14) and (1.13) are independent: neither reduces to the other. If T1=T2 and βn=0 for all n≥1, then (1.14) reduces to (1.12). It also can be reduces to Schu process (1.8).Inspired and motivated by the recent works, we prove some new weak convergence theorems of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.Now, we recall some well-known concepts and results.LetX be a Banach space with dimension X≥2. The modulus of X is the function δX:(0,2]→[0,1] defined by (1.15)δX(ϵ)=inf⁡{1-‖12(x+y)‖:‖x‖=1,‖y‖=1,ϵ=‖x-y‖}.
Banach space X is uniformly convex if and only if δX(ϵ)>0 for all ϵ∈(0,2]. It is known that a uniformly convex Banach space is reflexive and strictly convex.Recall that a Banach spaceX is said to satisfy Opial's condition [27] if xn→x weakly as n→∞ and x≠y implying that (1.16)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖.
The norm of X is said to be Fréchet differentiable if for each x∈X with ∥x∥=1 the limit  (1.17)lim⁡t→0∥x+ty∥-∥x∥t
exists and is attained uniformly for y, with ∥y∥=1. In the case of Fréchet differentiable norm, it has been obtained in [21] that(1.18)〈h,J(x)〉+12‖x‖2≤12‖x+h‖2≤〈h,J(x)〉+12‖x‖2+b(‖h‖)
for all x, h in E, where J is the normalized duality map from E to E* defined by(1.19)J(x)={x*∈E*:〈x,s*〉=‖x‖2=‖x*‖2},〈·,·〉 is the duality pairing between E and E* and b is an increasing function defined on [0,∞) such that limt↓0b(t)/t=0.A subsetC of X is said to be retract if there exists continuous mapping P:X→C such that Px=x for all x∈C. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping P:X→X is said to be a retraction if P2=P.   If a mapping P is a retraction, then Pz=z for every z∈R(P), range of P. A set C is optimal if each point outside C can be moved to be closer to all points of C. It is well known (see [28]) that(1)
ifX is a separable, strictly convex, smooth, reflexive Banach space, and if C⊂X is an optimal set with interior, then C is a nonexpansive retract of X;(2)
a subset oflp, with 1<p<∞, is a nonexpansive retract if and only if it is optimal.Note that every nonexpansive retract is optimal. In strictly convex Banach spaces, optimal sets are closed and convex. Moreover, every closed convex subset of a Hilbert space is optimal and also a nonexpansive retract.Recall that weak convergence is defined in terms of bounded linear functionals onX as follows.A sequence{xn} in a normed space X is said to be weakly convergent if there is an x∈X such that limn→∞f(xn)=f(x) for every bounded linear functional f on X. The element x is called the weak limit of {xn}, and we say that {xn} converges weakly to x. In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.A Banach spaceX is said to have the Kadec-Klee property if, for every sequence {xn} in X, xn⇀x and ∥xn∥→∥x∥ together imply ∥xn-x∥→0; for more details on Kadec-Klee property, the reader is referred to [29, 30] and the references therein.In the sequel, the following lemmas are needed to prove our main results.Lemma 1.9 (see [31]).
Letp>1, r>0 be two fixed numbers. Then, a Banach space X is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞), g(0)=0 such that
(1.20)‖λx+(1-λ)y‖p≤λ‖x‖p+(1-λ)‖y‖p-wp(λ)g(‖x-y‖)
for all x, y in Br={x∈X:∥x∥≤r}, λ∈[0,1], where
(1.21)wp(λ)=λ(1-λ)p+λp(1-λ).Lemma 1.10 (see [24]).
LetX be a uniformly convex Banach space and C a nonempty closed convex subset of X, and let T:C→X be an asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞)  and kn→1  as  n→∞. Then, I-T is demiclosed at zero; that is, if xn→x weakly and  xn-Txn→0 strongly, then x∈F(T).Lemma 1.11 (see [26]).
LetX be a uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [0,1). From an arbitrary x1∈C, define the sequence {xn} by (1.14). If q∈F(T1)∩F(T2), then limn→∞∥xn-q∥ exists.Lemma 1.12 (see [26]).
LetX be a uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [ϵ,1-ϵ]  for some ϵ∈(0,1). From an arbitrary  x1∈C, define the sequence {xn} by (1.14). Then, limn→∞∥xn-T1xn∥=limn→∞∥xn-T2xn∥=0.Lemma 1.13 (see [16]).
LetX be a real reflexive Banach space such that its dual  X*  has the Kadec-Klee property. Let {xn} be a bounded sequence in X and  x*,y*∈ωw(xn), where ωw(xn) denotes the set of all weak subsequential limits of {xn}. Suppose that limn→∞∥txn+(1-t)x*-y*∥ exists for all t∈[0,1]. Then,  x*=y*.We denote byΓ the set of strictly increasing, continuous convex functions γ:ℝ+→ℝ+ with γ(0)=0. Let C be a convex subset of the Banach space X. A mapping T:C→C is said to be type (γ) [32] if γ∈Γ and 0⩽α⩽1,  (1.22)γ(‖αTx+(1-α)Ty-T(αx+(1-α)y)‖)⩽‖x-y‖-‖Tx-Ty‖
for all x, y in C. Obviously, every type  (γ) mapping is nonexpansive. For more information about mappings of type (γ), see [33–35].Lemma 1.14 (see [36, 37]).
LetX be a uniformly convex Banach space and C a convex subset of X. Then, there exists γ∈Γ such that for each mapping S:C→C with Lipschitz constant L,
(1.23)‖αSx+(1-α)Sy-S(αx+(1-α)y)‖⩽Lγ-1(‖x-y‖-1L‖Sx-Sy‖)
for all x,y∈C and 0<α<1.


## 2. Main Results

In this section, we prove weak convergence theorems of the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.Firstly, we deal with the weak convergence of the sequence{xn} defined by (1.14) in a real uniformly convex Banach space X whose dual X* has the Kadec-Klee property. In order to prove our main results, the following lemma is needed.Lemma 2.1.
LetX be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with  P  as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [ϵ,1-ϵ] for some  ϵ∈(0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, for all u,v∈F(T1)∩F(T2), the limit limn→∞∥txn-(1-t)u-v∥ exists for all t∈[0,1].Proof.
It follows from Lemma1.11 that the sequence {xn} is bounded. Then, there exists R>0 such that {xn}⊂BR(0)∩C. Let an(t):=∥txn+(1-t)u-v∥ where t∈[0,1]. Then, limn→∞an(0)=∥u-v∥ and, by Lemma 1.11, limn→∞an(1)=limn→∞∥xn-v∥ exists. Without loss of the generality, we may assume that limn→∞∥xn-v∥=r for some positive number r. Let x∈C and t∈(0,1).  For each n≥1, define An:C→C by
(2.1)Anx=P((1-αn)yn(x)+αnT1(PT1)n-1yn(x)),
where
(2.2)yn(x)=P((1-βn)x+βnT2(PT2)n-1x).
Settingkn=1+sn and ln=1+tn. For x,z∈C, we have
(2.3)‖Anx-Anz‖=‖P((1-αn)yn(x)+αnT1(PT1)n-1yn(x))-P((1-αn)yn(z)+αnT1(PT1)n-1yn(z))‖≤‖(1-αn)(yn(x)-yn(z))+αn(T1(PT1)n-1yn(x)-T1(PT1)n-1yn(z))‖≤(1-αn)‖yn(x)-yn(z)‖+αnkn‖yn(x)-yn(z)‖≤(1-αn)‖(1-βn)(x-z)+βnT2(PT2)n-1(x-z)‖+αnkn‖(1-βn)(x-z)+βnT2(PT2)n-1(x-z)‖≤(1-αn)(1-βn)‖x-z‖+(1-αn)βnln‖x-z‖+αnkn(1-βn)‖x-z‖+αnβnknln‖x-z‖=(1-αn-βn+αnβn)‖x-z‖+(1-αn)βn(1+tn)‖x-z‖+αn(1+sn)(1-βn)‖x-z‖+αnβn(1+sn)(1+tn)‖x-z‖=‖x-z‖+βntn‖x-z‖+αnsn‖x-z‖+αnβntnsn‖x-z‖≤(1+tn+sn+tnsn)‖x-z‖.
SetSn,m:=An+m-1An+m-2⋯An, n,m≥1 and bn,m=∥Sn,m(txn+(1-t)u)-(tSn,mxn+(1-t)u)∥, where 0≤t≤1.  Also,
(2.4)‖Sn,mx-Sn,my‖≤‖An+m-1(An+m-2⋯Anx)-An+m-1(An+m-2⋯Any)‖≤(1+tn+m-1+sn+m-1+tn+m-1sn+m-1)‖An+m-2(An+m-3⋯Anx)-An+m-2(An+m-3⋯Any)‖⋮≤∏j=nn+m-1(1+tj+sj+tjsj)‖x-y‖
for all x,y∈C and Sn,mxn=xn+m, Sn,mx*=x* for all x*∈F(T1)∩F(T2).
Applying Lemma1.14 with x=xn, y=u, S=Sn,m and using the facts that limn→∞tn=limn→∞(ln-1)=0, limn→∞sn=limn→∞(kn-1)=0, and limn→∞∥xn-x*∥ exist for all x*∈F(T1)∩F(T2), we obtain limn→∞bn,m=0. Observe that
(2.5)an+m(t)=‖txn+m+(1-t)u-v‖=‖tSn,mxn+(1-t)u-Sn,mv‖=‖Sn,mv-(tSn,mxn+(1-t)u)‖=‖Sn,mv-Sn,m(txn+(1-t)u)+Sn,m(txn+(1-t)u)-(tSn,mxn+(1-t)u)‖≤‖Sn,mv-Sn,m(txn+(1-t)u)‖+bn,m=‖Sn,m(txn+(1-t)u)-Sn,mv‖+bn,m≤∏j=nn+m-1(1+tj+sj+tjsj)‖txn+(1-t)u-v‖+bn,m≤∏j=n∞(1+tj+sj+tjsj)an(t)+bn,m.
Consequently,
(2.6)limsup⁡m→∞am(t)=limsup⁡m→∞an+m(t)≤limsup⁡m→∞(bn,m+∏j=n∞(1+tj+sj+tjsj)an(t)),(2.7)limsup⁡n→∞an(t)≤liminf⁡n→∞an(t).
This implies that limn→∞an(t) exists for all  t∈[0,1]. This completes the proof.Theorem 2.2.
LetX be a real uniformly convex Banach space which has a Fréchet differentiable norm and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞,∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [ϵ,1-ϵ] for some ϵ∈(0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, {xn} converges weakly to a fixed point of T1 and T2.Proof.
Setx=p1-p2 and h=t(xn-p1) in (1.18). By using Lemmas 1.11, and 2.1 and the same proof of Lemma  4 of Osilike and Udomene [7], we can show that, for every p1,p2∈F(T1)∩F(T2),
(2.8)〈p-q,J(p1-p2)〉=0,
for all p,q∈ωw(xn). Since E is reflexive and {xn} is bounded, we from Lemma 1.13 conclude that ωw(xn)⊂F(Ti) for each i=1,2. Let p,q∈ωw(xn). It follows that p,q∈F(T1)∩F(T2); that is,
(2.9)‖p-q‖2=〈p-q,J(p-q)〉=0.
Therefore, p=q. This completes the proof.Theorem 2.3.
LetX be a real uniformly convex Banach space such that its dual X* has the Kadec-Klee property and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that ∑n=1∞(kn-1)<∞, ∑n=1∞(ln-1)<∞, respectively, and F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [ϵ,1-ϵ] for some ϵ∈(0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, {xn} converges weakly to a fixed point of T1 and T2.Proof.
It follows from Lemma1.11 that the sequence {xn} is bounded. Then, there exists a subsequence {xnj} of {xn} converging weakly to a point x*∈C. By Lemma 1.12, we have
(2.10)lim⁡n→∞‖xnj-T1xnj‖=0=lim⁡n→∞‖xnj-T2xnj‖.
Now, using Lemma 1.10, we have (I-T)x*=0; that is, Tx*=x*. Thus, x*∈F(T1)∩F(T2). It remains to show that {xn} converges weakly to x*. Suppose that {xni} is another subsequence of {xn} converging weakly to some y*. Then, y*∈C and so x*,y*∈ωw(xn)∩F(T1)∩F(T2). By Lemma 2.1,
(2.11)lim⁡n→∞‖txn-(1-t)x*-y*‖
exists for all t∈[0,1]. It follows from Lemma 1.13 that x*=y*. As a result, ωw(xn) is a singleton, and so {xn} converges weakly to a fixed point of T.In the remainder of this section, we deal with the weak convergence of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space without any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm.LetT1 and T2  be two asymptotically nonexpansive nonself-mappings of C with {kn}⊂[1,∞), limn→∞kn=1, and {ln}⊂[1,∞), limn→∞ln=1, respectively. In the sequel, we take {tn}⊂[1,∞), where tn=max{kn,ln}.We start with proving the following lemma for later use.Lemma 2.4.
LetX be a uniformly convex Banach space and C a nonempty bounded closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that kn→1, ln→1 as  n→∞,  respectively, and  F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [ϵ,1-ϵ] for some  ϵ∈(0,1). Then, for the sequence {xn} given in (1.14), we have that
(2.12)lim⁡n→∞‖xn-T1xn‖=0=lim⁡n→∞‖xn-T2xn‖.Proof.
By settingtn=max{kn,ln}, then limn→∞tn=1 if limn→∞kn=1=limn→∞ln. Let p∈F(T1)∩F(T2). Since C is bounded, there exists Br(0) such that C⊂Br(0) for some r>0. Applying Lemma 1.9 for scheme (1.14), we have
(2.13)‖yn-p‖2=‖P((1-βn)xn+βnT2(PT2)n-1xn)-p‖2≤‖(1-βn)(xn-p)+βn(T2(PT2)n-1xn-p)‖2=(1-βn)‖xn-p‖2+βnln2‖xn-p‖2-βn(1-βn)g(‖xn-T2(PT2)n-1xn‖)=(1-βn+βnln2)‖xn-p‖2-βn(1-βn)g(‖xn-T2(PT2)n-1xn‖)
and so,
(2.14)‖xn+1-p‖2=‖P((1-αn)yn+αnT1(PT1)n-1yn)-p‖2≤‖(1-αn)(yn-p)+αn(T1(PT1)n-1yn-p)‖2=(1-αn)‖yn-p‖2+αnkn2‖yn-p‖2-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)=(1-αn+αnkn2)‖yn-p‖2-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)≤(1-αn+αnkn2)((1-βn+βnln2)‖xn-p‖2-βn(1-βn)g(‖xn-T2(PT2)n-1xn‖))-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)=(1-αn+αnkn2)(1-βn+βnln2)‖xn-p‖2-(1-αn+αnkn2)βn(1-βn)g(‖xn-T2(PT2)n-1xn‖)-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)=((1-αn)(1-βn)+(1-αn)βnln2+(1-βn)αnkn2+αnkn2βnln2)‖xn-p‖2-(1-αn+αnkn2)βn(1-βn)g(‖xn-T2(PT2)n-1xn‖)-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)≤((1-αn)(1-βn)+(1-αn)βntn2+(1-βn)αntn2+αnβntn4)‖xn-p‖2-(1-αn+αnkn2)βn(1-βn)g(‖xn-T2(PT2)n-1xn‖)-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)≤((1-αn)(1-βn)tn4+(1-αn)βntn4+(1-βn)αntn4+αnβntn4)‖xn-p‖2-(1+αn(kn2-1))βn(1-βn)g(‖xn-T2(PT2)n-1xn‖)-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)≤((1-αn)(1-βn)tn4+(1-αn)βntn4+(1-βn)αntn4+αnβntn4)‖xn-p‖2-βn(1-βn)g(‖xn-T2(PT2)n-1xn‖)-αn(1-αn)g(‖yn-T1(PT1)n-1yn‖)≤‖xn-p‖2+r(tn4-1)-ε2g(‖xn-T2(PT2)n-1xn‖)-ε2g(‖yn-T1(PT1)n-1yn‖).
From (2.14), we obtain the following two important inequalities:
(2.15)‖xn+1-p‖2≤‖xn-p‖2+r(tn4-1)-ε2g(‖xn-T2(PT2)n-1xn‖),(2.16)‖xn+1-p‖2≤‖xn-p‖2+r(tn4-1)-ε2g(‖yn-T1(PT1)n-1yn‖).
Now, we prove that
(2.17)lim⁡n→∞‖xn-T2(PT2)n-1xn‖=0=lim⁡n→∞‖yn-T1(PT1)n-1yn‖.
Assume that limsupn→∞∥xn-T2(PT2)n-1xn∥>0. Then, there exists a subsequence (use the same notation for subsequence as for the sequence) of {xn} and μ>0 such that
(2.18)‖xn-T2(PT2)n-1xn‖≥μ>0.
By definition of g, we have
(2.19)g(‖xn-T2(PT2)n-1xn‖)≥g(μ)>0.
From (2.15), we have
(2.20)‖xn+1-p‖2≤‖xn-p‖2+r(tn4-1)-ε2g(μ)=‖xn-p‖2+r((tn4-1)-ε22rg(μ))-ε22g(μ).
In addition,tn4→1 and (ε2/2r)g(μ)>0; there exists n0≥1 such that (tn4-1)<(ε2/2r)g(μ) for all n≥n0. From (2.20), we obtain
(2.21)ε22g(μ)≤‖xn-p‖2-‖xn+1-p‖2
for all n≥n0.
Letm≥n0. It follows from (2.21) that
(2.22)ε22∑n=n0mg(μ)≤∑n=n0m(‖xn-p‖2-‖xn+1-p‖2)=‖xn0-p‖2.
By lettingm→∞ in (2.22), we obtain
(2.23)∞=‖xn0-p‖2<∞
which contradicts the reality. This proves that  μ=0. Thus, limsupn→∞∥xn-T2(PT2)n-1xn∥≤0. Consequently, we have
(2.24)lim⁡n→∞‖xn-T2(PT2)n-1xn‖=0.
Similarly, using (2.16), we may show that
(2.25)lim⁡n→∞‖yn-T1(PT1)n-1yn‖=0.
Using (2.24), we have
(2.26)‖xn-yn‖≤βn‖xn-T2(PT2)n-1xn‖⟶0(asn⟶∞).
From (2.25), (2.26), and the uniform equicontinuous of T1 (see Remark 1.1), we have
(2.27)‖xn-T1(PT1)n-1xn‖≤‖xn-yn‖+‖yn-T1(PT1)n-1xn‖≤‖xn-yn‖+‖yn-T1(PT1)n-1yn‖+‖T1(PT1)n-1yn-T1(PT1)n-1xn‖⟶0(asn⟶∞).
Since(2.28)‖xn-xn+1‖≤(1-αn)‖yn-xn‖+αn‖T1(PT1)n-1yn-xn‖=(1-αn)‖yn-xn‖+αn‖T1(PT1)n-1yn-T1(PT1)n-1xn+T1(PT1)n-1xn-xn‖≤‖yn-xn‖+‖T1(PT1)n-1yn-T1(PT1)n-1xn‖+‖T1(PT1)n-1xn-xn‖,
it follows from (2.26), (2.27), and the uniform equi-continuity of T1 (see Remark 1.1) that
(2.29)lim⁡n→∞‖xn-xn+1‖=0.
Sincelimn→∞∥xn-T1(PT1)n-1xn∥=0 and again from the fact that T1 is uniformly equicontinuous mapping, by Using (2.29), we have
(2.30)‖xn+1-T1(PT1)n-1xn+1‖=‖xn+1-xn+xn-T1(PT1)n-1xn+T1(PT1)n-1xn-T1(PT1)n-1xn+1‖≤‖xn+1-xn‖+‖T1(PT1)n-1xn+1-T1(PT1)n-1xn‖+‖T1(PT1)n-1xn-xn‖⟶0(asn⟶∞).
In addition,
(2.31)‖xn+1-T1(PT1)n-2xn+1‖=‖xn+1-xn+xn-T1(PT1)n-2xn+T1(PT1)n-2xn-T1(PT1)n-2xn+1‖≤‖xn+1-xn‖+‖T1(PT1)n-2xn-xn‖+‖T1(PT1)n-2xn+1-T1(PT1)n-2xn‖≤‖xn+1-xn‖+‖T1(PT1)n-2xn-xn‖+L‖xn+1-xn‖,
where L=sup{kn:n≥1}. It follows from (2.29) and (2.30) that
(2.32)lim⁡→∞‖xn+1-T1(PT1)n-2xn+1‖=0.
We denote (PT1)1-1 to be the identity maps from C onto itself. Thus, by the inequality (2.30) and (2.32), we have
(2.33)‖xn+1-T1xn+1‖=‖xn+1-T1(PT1)n-1xn+1+T1(PT1)n-1xn+1-T1xn+1‖≤‖xn+1-T1(PT1)n-1xn+1‖+‖T1(PT1)n-1xn+1-T1xn+1‖=‖xn+1-T1(PT1)n-1xn+1‖+‖T1(PT1)1-1(PT1)n-1xn+1-T1(PT1)1-1xn+1‖≤‖xn+1-T1(PT1)n-1xn+1‖+L‖(PT1)n-1xn+1-xn+1‖=‖xn+1-T1(PT1)n-1xn+1‖+L‖(PT1)(PT1)n-2xn+1-P(xn+1)‖≤‖xn+1-T1(PT1)n-1xn+1‖+L‖T1(PT1)n-2xn+1-xn+1‖⟶0(asn⟶∞),
which implies that limn→∞∥xn-T1xn∥=0. Similarly, we may show that limn→∞∥xn-T2xn∥=0. The proof is completed.Our weak convergence theorem is as follows. We do not use the rate of convergence conditions, namely,∑n=1∞(kn-1)<∞ and ∑n=1∞(ln-1)<∞ in its proof.Theorem 2.5.
LetX be a uniformly convex Banach space and C a nonempty bounded closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2:C→X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn},{ln}⊂[1,∞) such that kn→1, ln→1 as n→∞, respectively, and  F(T1)∩F(T2)≠∅. Suppose that {αn} and {βn} are real sequences in [ϵ,1-ϵ] for some ϵ∈(0,1). Then, the sequence {xn} given in (1.14) converges weakly to a common fixed point of  T1 and T2.Proof.
SinceC is a nonempty bounded closed convex subset of a uniformly convex Banach space X, there exists a subsequence {xnj} of {xn} such that xnj converges weakly to q∈ωw(xn), where ωw(xn) denotes the set of all weak subsequential limits of {xn}. This show that ωw(xn)≠∅ and, by Lemma 2.4, limn→∞∥xnj-T1xnj∥=limn→∞∥xnj-T2xnj∥=0. Since I-T1 and I-T2 are demiclosed at zero, using Lemma 1.10, we have T1q=q=T2q. Therefore, ωw(xn)⊂F(T1)∩F(T2).For any q∈ωw(xn), there exists a subsequence {xni} of {xn} such that
(2.34)xni⇀q(asi⟶∞).
It follows from (2.24) and (2.34) that
(2.35)T2(PT2)nj-1xnj=(T2(PT2)nj-1xnj-xnj)+xnj⇀q.
Now, from (1.14), (2.34), and (2.35),
(2.36)ynj=P((1-βnj)xnj+βnjT2(PT2)nj-1xnj)⇀q.
Also, from (2.25) and (2.36), we have
(2.37)T1(PT1)nj-1ynj=(T1(PT1)nj-1ynj-ynj)+ynj⇀q.
It follows from (2.36) and (2.37) that
(2.38)xnj+1=P((1-αnj)ynj+αnjT1(PT1)nj-1ynj)⇀q.
Continuing in this way, by induction, we can prove that, for any m≥0,
(2.39)xnj+m⇀q.
By induction, one can prove that ∪m=0∞{xnj+m} converges weakly to q as j→∞; in fact, {xn}n=n1∞=∪m=0∞{xnj+m}j=1∞ gives that xn⇀q as n→∞. This completes the prove.



---

*Source: 745451-2011-10-27.xml*