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).

1. Introduction

Let E 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 subset K 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 [<xref ref-type="bibr" rid="B4">4</xref>]).

Let K 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
∥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
∥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:

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:

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:

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:

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.

Let E 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:

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.

Let E be a Banach space with dimension E≥2. The modulus of E is the function δE:(0,2]→[0,1] defined by

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 space E is said to satisfy Opial's condition [12] if for any sequence {xn} in E, xn⇀x implies that

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 mapping T: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

∥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 mappings T1,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

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 [<xref ref-type="bibr" rid="B6">5</xref>]).

Let {an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying the inequality
an+1≤(1+δn)an+bn,n≥1.
If ∑n=1∞bn<∞ and ∑n=1∞δn<∞, then

limn→∞an exists;

In particular, if {an} has a subsequence which converges strongly to zero, then limn→∞an=0.

Lemma 1.3 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Suppose that E 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
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 [<xref ref-type="bibr" rid="B4">4</xref>]).

Let E 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.

Let E 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.

Let p∈F(T1)∩F(T2). Since {un} and {vn} are bounded sequences in K, we have
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
∥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∥,∥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
∥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
∥σn-p∥≤kn∥yn-p∥≤knln2∥xn-p∥+knγn′r.
It follows from (1.7) and (2.5) that
∥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.

Let E 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
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
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
∥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,∥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):
∥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
∥σn-xn∥≤(L1+1)(L2+1)Cn′+(L1+1)γn′r+Cn.
It follows from (1.7) and (2.12) that
∥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
∥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
∥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
∥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.

Let E 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,
limn→∞∥xn-T1xn∥=limn→∞∥xn-T2xn∥=0.

Proof.

Let σn=(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
lim supn→∞∥σn-p∥≤c,lim supn→∞∥δn-p∥≤c,lim supn→∞∥yn-p∥≤c,
respectively. Next, we consider
∥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
lim supn→∞∥T1(PT1)n-1Pσn-p+γn(un-xn)∥≤c.
Observe that
∥xn-p+γn(un-xn)∥≤∥xn-p∥+γn∥un-xn∥≤∥xn-p∥+γnr,
which implies that
lim supn→∞∥xn-p+γn(un-xn)∥≤c.lim supn→∞∥xn+1-p∥=c means that
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
∥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
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
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
limn→∞∥T1(PT1)n-1Pσn-xn∥=0.
Note that
∥xn-p∥≤∥T1(PT1)n-1Pσn-p∥+∥T1(PT1)n-1Pσn-xn∥≤kn∥σn-p∥
which yields that
c≤liminfn→∞∥σn-p∥≤lim supn→∞∥σn-p∥≤c.
That is, limn→∞∥σn-p∥=c. This implies that
lim infn→∞∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥≥c.
Similarly, we have
∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥≤βn∥T1(PT1)n-1yn-p∥+(1-βn)∥(yn-p)∥≤kn∥yn-p∥,lim supn→∞∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥≤c.
Combining (2.30) with (2.32), we obtain
limn→∞∥βn(T1(PT1)n-1yn-p)+(1-βn)(yn-p)∥=c.
On the other hand, we have
∥T1(PT1)n-1yn-p∥≤kn∥yn-p∥,lim supn→∞∥T1(PT1)n-1yn-p∥≤c.
Hence, using (2.32), (2.33), (2.35), and Lemma 1.3, we find
limn→∞∥T1(PT1)n-1yn-yn∥=0.
Note that from (2.36), we have
∥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
c≤lim infn→∞∥yn-p∥≤lim supn→∞∥yn-p∥≤c.
That is, limn→∞∥yn-p∥=c.

Again, limn→∞∥yn-p∥=c means that

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
∥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
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
limn→∞∥αn′(T2(PT2)n-1Pδn-p+γn′(vn-xn))+(1-αn′)(xn-p+γn′(vn-xn))∥=c.
On the other hand, we have
∥T2(PT2)n-1Pδn-p+γn′(vn-xn)∥≤∥T2(PT2)n-1Pδn-p∥+γn′∥vn-xn∥≤ln∥δn-p∥+γn′r
which implies that
lim supn→∞∥T2(PT2)n-1Pδn-p+γn′(vn-xn)∥≤c.
Notice that
∥xn-p+γn′(vn-xn)∥≤∥xn-p∥+γn′∥vn-xn∥≤∥xn-p∥+γn′r,
which implies that
lim supn→∞∥xn-p+γn′(vn-xn)∥≤c.
Using (2.42), (2.44), (2.46), and Lemma 1.3, we find
limn→∞∥T2(PT2)n-1Pδn-xn∥=0.
Observe that
∥xn-p∥≤∥T2(PT2)n-1Pδn-xn∥+∥T2(PT2)n-1Pδn-p∥≤ln∥δn-p∥
which yields that
c≤lim infn→∞∥δn-p∥≤lim supn→∞∥δn-p∥≤c.
That is, limn→∞∥δn-p∥=c. This implies that
lim infn→∞∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥≥c.
Similarly, we have
∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥≤βn′∥T2(PT2)n-1xn-p∥+(1-βn′)∥xn-p∥≤ln∥xn-p∥,lim supn→∞∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥≤c.
Combining (2.50) with (2.52), we obtain
limn→∞∥βn′(T2(PT2)n-1xn-p)+(1-βn′)(xn-p)∥=c.
On the other hand, we have
∥T2(PT2)n-1xn-p∥≤ln∥xn-p∥,lim supn→∞∥T2(PT2)n-1xn-p∥≤c,lim supn→∞∥xn-p∥≤c.
Hence, using (2.53), (2.54), (2.55), and Lemma 1.3, we find
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
∥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
∥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,
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
limn→∞∥xn-T1xn∥=limn→∞∥xn-T2xn∥=0.
This completes the proof.

Theorem.

Let E 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 Lemma 2.1, {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.

Let E 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 Lemma 2.1, 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
limn→∞f(d(xn,F(T1)∩F(T2)))≤limn→∞(12(∥xn-T1xn∥+∥xn-T2xn∥))=0.
That is,
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
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
∥xnj+1-yj∥≤∥xnj-yj∥<2-j,
and hence
∥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.

If γn=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.

Let E 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.

Let p∈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, assume p1≠p2, then by Opial’s condition, we have

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

Let T1,T2:K→E be two nonexpansive nonself-mappings. Then, the iterative scheme (1.7) is written as follows:

Nothing prevents one from proving the results of the previous section for nonexpansive nonself-mappings case. Thus, one can easily prove the following.

Theorem.

Let E 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.

Let E 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.

If T1=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.

Acknowledgment

The authors would like to thank the referees for their helpful comments.

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