We introduce a general iteration scheme for a finite family
of generalized asymptotically quasi-nonexpansive mappings in Banach spaces.
The new iterative scheme includes the multistep Noor iterations with errors,
modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and
Noor, and Khan and Takahashi scheme as special cases. Our results generalize
and improve the recent ones announced by Khan et al. (2008),
H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006),
and many others.

1. Introduction

Let C be a subset of real Banach space X. Let T be a self-mapping of C and let F(T) denote the fixed points set of T, that is,F(T):={x∈C:Tx=x}. Recall that a mapping T is said to be asymptotically nonexpansive on C if there exists a sequence {bn} in [0,∞) with limn→∞bn=0 such that for each x,y∈C,

∥Tnx-Tny∥≤(1+bn)∥x-y∥,∀n≥1.

If bn=0 for all n≥1, then T is known as a nonexpansive mapping. T is called generalized asymptotically quasi-nonexpansive [1] if there exist sequences {bn},{cn} in [0,∞) with limn→∞bn=0=limn→∞cn such that

∥Tnx-p∥≤(1+bn)∥x-p∥+cn,∀n≥1,
for all x∈C and all p∈F(T). If cn=0 for all n≥1, then T is known as an asymptotically quasi-nonexpansive mapping. Tis called asymptotically nonexpansive mapping in the intermediate sense [2] provided that T is uniformly continuous and

limsupn→∞supx,y∈C(∥Tnx-Tny∥-∥x-y∥)≤0.T is said to be (L-γ)uniform Lipschitz [3] if there are constants L>0 and γ>0 such that

∥Tnx-Tny∥≤L∥x-y∥γ,∀n≥1,
for all x,y∈C. A mapping T is called semicompact if any bounded sequence {xn} in C with limn→∞∥xn-Txn∥=0, there exists a subsequence {xni} of {xn} such that {xni} converges strongly to some x* in C.

Remark 1.1.

Let T be asymptotically nonexpansive mapping in the intermediate sense. Put Gn=supx,y∈C(∥Tnx-Tny∥-∥x-y∥)∨0, ∀n≥1.

If F(T)≠Ø, we obtain that ∥Tnx-p∥≤∥x-p∥+Gn for all x∈C and all p∈F(T). Since limn→∞Gn=0, therefore T is a generalized asymptotically quasi-nonexpansive mapping.

Recall that a mapping T:C→C with F(T)≠Ø is said to satisfy condition (I) [4] if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that ∥x-Tx∥≥f(d(x,F(T))) for all x∈C, where d(x,F(T))=inf{∥x-p∥:p∈F(T)}.

Fixed-point iteration processes for asymptotically quasi-nonexpansive mapping in Banach spaces including Mann and Ishikawa iterations processes have been studied extensively by many authors; see [3, 5–11]. Many of them are used widely to study the approximate solutions of the certain problems. In 1974, Senter and Dotson [4] studied the convergence of the Mann iteration scheme defined by x1∈C,

xn+1=αnTxn+(1-αn)xn,∀n≥1,
in a uniformly convex Banach space, where {αn} is a sequence satisfying 0<a≤αn≤b<1foralln≥1 and T is a nonexpansive (or a quasi-nonexpansive) mapping. They established a relation between condition (I) and demicompactness.

Recall that a mapping T:C→C is demicompact if for every bounded sequence {xn} in C such that {xn-Txn} converges, there exists a subsequence say {xnj} of {xn} that converges strongly to some y in C. Every compact and semicompact mapping is demicompact. They actually showed that condition (I) is weaker than demicompactness for a nonexpansive mapping defined on bounded set.

Xu and Noor [12], in 2002, introduced a three-step iterative scheme as follows:

zn=anTnxn+(1-an)xn,yn=bnTnzn+(1-bn)xn,xn+1=αnTnyn+(1-αn)xn,n≥1,
where {an},{bn},{αn} are appropriate sequences in [0,1]. The theory of three-step iterative scheme is very rich, and this scheme, in the context of one or more mappings, has been extensively studied (e.g., see Khan et al. [6], Plubtieng and Wangkeeree [7], Fukhar-ud-din and Khan [5], Petrot [13], and Suantai [14]). It has been shown in [15] that three-step method performs better than two-step and one-step methods for solving variational inequalities.

In 2001, Khan and Takahashi [16] have approximated common fixed points of two asymptotically nonexpansive mappings by the modified Ishikawa iteration. Jeong and Kim [17] have approximated common fixed points of two asymptotically nonexpansive mappings. Plubtieng et al. [18], in 2006, modified Noor iterations with errors and have approximated common fixed points of three asymptotically nonexpansive mappings. Shahzad and Udomene [10] established convergence theorems for the modified Ishikawa iteration process of to asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. Plubtieng and Wangkeeree [7], in 2006, established strong convergence theorems of the modified multistep Noor iterations with errors for an asymptotically quasi-nonexpansive mapping and asymptotically nonexpansive mapping in the intermediate sense.

Very recently, Khan et al. [6], in 2008, established convergence theorems for the modified multistep Noor iterations process of finite family of asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. For rerated results with errors terms, we refer to [5–7, 17–21]. Inspired and motivated by these facts, we introduce a new iteration process for a finite family of {Ti:i=1,2,…,k} of generalized asymptotically quasi-nonexpansive mappings as follows.

Let Ti:C→C(i=1,2,…,k) be mappings and F:=⋂i=1kF(Ti). For a given x1∈C, and a fixed k∈ℕ (ℕ denote the set of all positive integers), compute the iterative sequences {xn} and {yin} by

xn+1=ykn=αknTkny(k-1)n+βknxn+γknukn,y(k-1)n=α(k-1)nTk-1ny(k-2)n+β(k-1)nxn+γ(k-1)nu(k-1)n,⋮y3n=α3nT3ny2n+β3nxn+γ3nu3n,y2n=α2nT2ny1n+β2nxn+γ2nu2n,y1n=α1nT1ny0n+β1nxn+γ1nu1n,
where y0n=xn and {u1n},{u2n},…,{ukn} are bounded sequences in C with {αin},{βin}, and {γin} are appropriate real sequences in [0,1] such that αin+βin+γin=1 for all i=1,2,…,k and all n. Our iteration includes and extends the Mann iteration (1.5), three-step iteration by Xu and Noor (1.6), the multistep Noor iterations with errors by Plubtieng and Wangkeeree [7], and the iteration defined by Khan et al. [6] simultaneously.

The purpose of this paper is to establish several strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive mappings when one mapping Ti satisfies a condition which is weaker than demicompactness and we also weak convergence theorem for a finite family of generalized asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space satisfying Opial's property. Our results generalize and improve the corresponding ones announced by Khan et al. [6], Fukhar-ud-din and Khan [5], and many others.

2. Preliminaries

In the sequel, the following lemmas are needed to prove our main results.

A mapping T with domain D(T) and range R(T) in X is said to be demiclosed at 0 if whenever {xn} is a sequence in D(T) such that {xn} converges weakly to x∈D(T) and {Txn} converging strongly to 0, we have Tx=0.

A Banach space X is said to satisfy Opial's property if for each x in X and each sequence {xn} weakly convergent to x, the following condition holds for x≠y:

liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥.
It is well known that all Hilbert spaces and lp(1<p<∞) spaces have Opial's property while Lp spaces (p≠2) have not. A family {Ti:i=1,2,…,k} of self-mappings of C with F:=⋂i=1kF(Ti)≠Ø is said to satisfy the following conditions.

Condition (A¯) [22]. If there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that 1/k∑i=1k∥x-Tix∥≥f(d(x,F)) for all x∈C, where d(x,F)=inf{∥x-p∥:p∈F}.

Condition (B¯) [22]. If there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that max1≤i≤k{∥x-Tix∥}≥f(d(x,F)) for all x∈C.

Condition (C¯) [22]. If there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that ∥x-Tlx∥≥f(d(x,F)) for all x∈C and for at least one Tl,l=1,2,…,k.

Note that (B¯) and (C¯) are equivalent, condition (B¯) reduces to condition (I) when all but one of Ti's are identities, and in addition, it also condition (A¯).

It is well known that every continuous and demicompact mapping must satisfy condition (I) (see [4]). Since every completely continuous T:C→C is continuous and demicompact so that it satisfies condition (I). Thus we will use condition (C¯) instead of the demicompactness and complete continuity of a family {Ti:i=1,2,…,k}.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B14">8</xref>, Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M155"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>]).

Let {an},{bn}, and {δn} be sequences of nonnegative real numbers satisfying the inequality
an+1≤(1+δn)an+bn,∀n=1,2,….
If ∑n=1∞δn<∞ and ∑n=1∞bn<∞, then

limn→∞an exists;

limn→∞an=0 whenever liminfn→∞an=0.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B13">7</xref>, Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M164"><mml:mrow><mml:mn>3.1</mml:mn></mml:mrow></mml:math></inline-formula>]).

Let X be a uniformly convex Banach space, {xn}, {yn}⊂X, real numbers a≥0,α,β∈(0,1), and let {αn} be a real sequence number which satisfies

0<α≤αn≤β<1,foralln≥n0 and for some n0∈ℕ;

limsupn→∞∥xn∥≤a and limsupn→∞∥yn∥≤a;

limn→∞∥αnxn+(1-αn)yn∥=a. Then limn→∞∥xn-yn∥=0.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B20">14</xref>, Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M176"><mml:mrow><mml:mn>2.7</mml:mn></mml:mrow></mml:math></inline-formula>]).

Let X be a Banach space which satisfies Opial's property and let {xn} be a sequence in X. Let u,v∈X be such that limn→∞∥xn-u∥ and limn→∞∥xn-v∥ exist. If {xnk} and {xmk} are subsequences of {xn} which converge weakly to u and v, respectively, then u=v.

3. Convergence Theorems in Banach Spaces

Our first result is the strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive mappings in a Banach space. In order to prove our main results, the following lemma is needed.

Lemma 3.1.

Let X be a Banach space and C a nonempty closed and convex subset of X, and {Ti:i=1,2,…,k} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin},{cin}⊂[0,∞) such that ∑n=1∞bin<∞ and ∑n=1∞cin<∞ for all i=1,2,…,k. Assume that F≠Ø and ∑n=1∞γin<∞ for each i=1,2,…,k. For a given x1∈C, let the sequences {xn} and {yin} be defined by (1.7). Then

there exist sequences {vn} and {ein} in [0,∞) such that ∑n=1∞vn<∞, ∑n=1∞ein<∞, and ∥yin-p∥≤(1+vn)i∥xn-p∥+ein, for all i=1,2,…,k and all p∈F;

limn→∞∥xn-p∥ exists for all p∈F;

there exist constant M>0 and {si} in [0,∞) such that ∑i=1∞si<∞ and ∥xn+m-p∥≤M∥xn-p∥+∑i=n∞si for all p∈F and n,m∈ℕ.

Proof.

(a) Let p∈F, vn=max1≤i≤k{bin} and dn=max1≤i≤k{cin} for all n.

Since ∑n=1∞bin<∞ and ∑n=1∞cin<∞, for all i=1,2,…,k, therefore ∑n=1∞vn<∞ and ∑n=1∞dn<∞. For each n≥1, we note that
∥y1n-p∥=∥α1nT1ny0n+β1nxn+γ1nu1n-p∥≤α1n∥T1nxn-p∥+β1n∥xn-p∥+γ1n∥u1n-p∥≤α1n(1+b1n)∥xn-p∥+α1nc1n+β1n∥xn-p∥+γ1n∥u1n-p∥≤α1n(1+vn)∥xn-p∥+α1ndn+β1n(1+vn)∥xn-p∥+γ1n∥u1n-p∥≤(1+vn)∥xn-p∥+e1n,
where e1n=α1ndn+γ1n∥u1n-p∥. Since {u1n} is bounded, ∑n=1∞γ1n<∞ and ∑n=1∞dn<∞, we obtain that ∑n=1∞e1n<∞. It follows from (3.1) that
∥y2n-p∥≤α2n∥T2ny1n-p∥+β2n∥xn-p∥+γ2n∥u2n-p∥≤α2n(1+vn)∥y1n-p∥+α2ndn+β2n∥xn-p∥+γ2n∥u2n-p∥≤α2n(1+vn)((1+vn)∥xn-p∥+e1n)+α2ndn+β2n(1+vn)2∥xn-p∥+γ2n∥u2n-p∥=(α2n+β2n)(1+vn)2∥xn-p∥+α2n(1+vn)e1n+α2ndn+γ2n∥u2n-p∥≤(1+vn)2∥xn-p∥+e2n,
where e2n=α2n(1+vn)e1n+α2ndn+γ2n∥u2n-p∥. Since {u2n}, {vn} are bounded, ∑n=1∞e1n<∞, ∑n=1∞dn<∞, and ∑n=1∞γ2n<∞, it follows that ∑n=1∞e2n<∞. Moreover, we see that
∥y3n-p∥≤α3n∥T3ny2n-p∥+β3n∥xn-p∥+γ3n∥u3n-p∥≤α3n(1+vn)∥y2n-p∥+α3ndn+β3n∥xn-p∥+γ3n∥u3n-p∥≤α3n(1+vn)((1+vn)2∥xn-p∥+e2n)+α3ndn+β3n(1+vn)3∥xn-p∥+γ3n∥u3n-p∥=(α3n+β3n)(1+vn)3∥xn-p∥+α3n(1+vn)e2n+α3ndn+γ3n∥u3n-p∥≤(1+vn)3∥xn-p∥+e3n,
where e3n=α3n(1+vn)e2n+α3ndn+γ3n∥u3n-p∥. Since {u3n}, {vn} are bounded, ∑n=1∞e2n<∞, ∑n=1∞dn<∞, and ∑n=1∞γ3n<∞, it follows that ∑n=1∞e3n<∞. By continuing the above method, there are nonnegative real sequences {ein} in [0,∞) such that ∑n=1∞ein<∞ and
∥yin-p∥≤(1+vn)i∥xn-p∥+ein∀i=1,2,…,k.
This completes the proof of (a).

(b) From part (a), for the case i=k, we have
∥xn+1-p∥≤(1+vn)k∥xn-p∥+ekn,∀n,p∈F.
It follows from Lemma 2.1(i) that limn→∞∥xn-p∥ exists, for all p∈F.

(c) If t≥0, then 1+t≤et and so, (1+t)k≤ekt, for k=1,2,…. Thus, from (3.5), it follows that
∥xn+m-p∥≤(1+vn+m-1)k∥xn+m-1-p∥+ek(n+m-1)≤exp{kvn+m-1}∥xn+m-1-p∥+ek(n+m-1)≤⋯≤exp{k∑i=nn+m-1vi}∥xn-p∥+∑i=nn+m-1eki≤exp{k∑i=1∞vi}∥xn-p∥+∑i=n∞eki≤M∥xn-p∥+∑i=n∞si,
where M=exp{k∑i=1∞vi} and si=eki.

Theorem 3.2.

Let X be a Banach space and C a nonempty closed and convex subset of X and {Ti:i=1,2,…,k} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin},{cin}⊂[0,∞) such that ∑n=1∞bin<∞ and ∑n=1∞cin<∞ for all i=1,2,…,k. Assume that F≠∅ is closed and ∑n=1∞γin<∞ for each i=1,2,…,k. Then the iterative sequence {xn} defined by (1.7) converges strongly to a common fixed point of the family of mappings if and only if liminfn→∞d(xn,F)=0.

Proof.

We prove only the sufficiency because the necessity is obvious. From (3.5), we have ∥xn+1-p∥≤(1+vn)k∥xn-p∥+ekn,forallnandallp∈F.

Hence, we have
d(xn+1,F)≤(1+vn)kd(xn,F)+ekn=(1+∑r=1kk(k-1)…(k-r+1)r!vnr)d(xn,F)+ekn.
Since ∑n=1∞vn<∞, it follows that ∑n=1∞∑r=1k(k(k-1)⋯(k-r+1)/r!)vnr<∞. Since ∑n=1∞ekn<∞ and liminfn→∞d(xn,F)=0, it follows from Lemma 2.1(ii) that limn→∞d(xn,F)=0. Next, we prove that {xn} is a Cauchy sequence. From Lemma 3.1(c), we have
∥xn+m-p∥≤M∥xn-p∥+∑i=n∞si,∀p∈F,n,m∈ℕ.
Since limn→∞d(xn,F)=0 and ∑i=1∞si<∞, therefore for ϵ>0, there exists n0∈ℕ such that
d(xn,F)<ϵ4M,∑i=n0∞si<ϵ4,∀n≥n0.
Therefore, there exists z1 in F such that
∥xn0-z1∥<ϵ4M.
From (3.8) to (3.10), for all n≥n0 and m≥1, we have
∥xn+m-xn∥≤∥xn+m-z1∥+∥xn-z1∥≤M∥xn0-z1∥+∑i=n0∞si+M∥xn0-z1∥+∑i=n0∞si<Mϵ4M+ϵ4+Mϵ4M+ϵ4=ϵ.
This shows that {xn} is a Cauchy sequence, hence xn→z∈C. It remains to show that z∈F. Notice that
|d(z,F)-d(xn,F)|≤∥z-xn∥,∀n.
Since limn→∞d(xn,F)=0, we obtain that z∈F.

The following corollary follows from Theorem 3.2.

Corollary 3.3.

Let X be a Banach space and C a nonempty closed and convex subset of X and {Ti:i=1,2,…,k} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin},{cin}⊂[0,∞) such that ∑n=1∞bin<∞ and ∑n=1∞cin<∞ for all i=1,2,…,k. Assume that F≠∅ is closed and ∑n=1∞γin<∞ for each i=1,2,…,k. Then the iterative sequence {xn}, defined by (1.7), converges strongly to a point p∈F if and only if there exists a subsequence {xnj} of {xn} converging to p.

Since an asymptotically quasi-nonexpansive mapping is generalized asymptotically quasi-nonexpansive mapping, so we have the following result.

Corollary 3.4.

Let X be a Banach space and C a nonempty closed and convex subset of X and {Ti:i=1,2,…,k} a finite family of asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin}⊂[0,∞) such that ∑n=1∞bin<∞ for all i=1,2,…,k. Assume that F≠Ø and ∑n=1∞γin<∞ for each i=1,2,…,k. Then the iterative sequence {xn}, defined by (1.7), converges strongly to a common fixed point of the family of mappings if and only if liminfn→∞d(xn,F)=0.

Remark 3.5.

Theorem 3.2 generalizes and extends Theorem 2.2 of Khan et al. [6], for a finite family of asymptotically quasi-nonexpansive mappings, Theorem 1 of Fukhar-ud-din and Khan [5], and Theorem 3.2 of Shahzad and Udomene [10] for two asymptotically quasi-nonexpanaive mappings to the more general class of generalized asymptotically quasi-nonexpansive mappings.

Theorem 3.6.

Let X be a Banach space and C a nonempty closed and convex subset of X and {Ti:i=1,2,…,k} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin},{cin}⊂[0,∞) such that ∑n=1∞bin<∞ and ∑n=1∞cin<∞ for all i=1,2,…,k. Suppose that F≠∅ is closed. Let x1∈C and {xn} be the sequence defined by (1.7). If ∑n=1∞γin<∞, limn→∞∥xn-Tixn∥=0 for all i=1,2,…,k and {Ti:i=1,2,…,k} satisfies condition (C¯), then {xn} converges strongly to a common fixed point of the family of mappings.

Proof.

From limn→∞∥xn-Tixn∥=0 for all i=1,2,…,k and {Ti:i=1,2,…,k} satisfying condition (C¯), there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that ∥xn-Ti0xn∥≥f(d(xn,F)) for some i0∈{1,2,…,k}, it follows that limn→∞d(xn,F)=0. From Theorem 3.2, we obtain that {xn} converges strongly to a common fixed point of the family of mappings.

4. Convergence Theorems in Uniformly Convex Banach Spaces

In this section, we establish weak and strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive and (L-γ) uniform Lipschitz mappings in a uniformly convex Banach space. In order to prove our main results, we need the following lemma.

Lemma 4.1.

Let C be a nonempty closed and convex subset of a uniformly convex Banach space X and {Ti,i=1,2,…,k} a finite family of (L-γ) uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin},{cin}⊂[0,∞) such that ∑n=1∞bin<∞ and ∑n=1∞cin<∞ for all i=1,2,…,k. Assume that F≠Ø and ∑n=1∞γin<∞ for all i=1,2,…,k. For a given x1∈C let {xn} and {yin} be the sequences defined by (1.7) with 0<η≤αin≤ρ<1, for all i=1,2,…,k and all n≥n0 and for some n0∈ℕ. Then

limn→∞∥Tjny(j-1)n-xn∥=0 for all j=1,2,…,k,

limn→∞∥Tjxn-xn∥=0 for all j=1,2,…,k,

limn→∞∥yjn-xn∥=0 for all j=1,2,…,k.

Proof.

Let p∈F, vn=max1≤i≤k{bin} and dn=max1≤i≤k{cin} for all n.

(i) From Lemma 3.1(b), we have that limn→∞∥xn-p∥ exists for all p∈F. Suppose that

limn→∞∥xn-p∥=a.
From (3.4) and (4.1), we get that
limsupn→∞∥yjn-p∥≤a,for1≤j≤k-1.
For each j∈{1,2,…,k-1} and n∈ℕ, we have
∥yjn-p∥≤αjn∥Tjny(j-1)n-p∥+βjn∥xn-p∥+γjn∥ujn-p∥≤αjn(1+vn)∥y(j-1)n-p∥+αjn(1+vn)dn+(1-αjn)(1+vn)∥xn-p∥+γjn(1+vn)∥ujn-p∥.
By using (4.3) and (1.7), for each j=1,2,…,k-1, we have
∥xn+1-p∥=∥αkn(Tkny(k-1)n-p)+βkn(xn-p)+γkn(ukn-p)∥≤αkn(1+vn)∥y(k-1)n-p∥+(1-αkn)(1+vn)∥xn-p∥+αkn(1+vn)dn+γkn∥ukn-p∥(1+vn)≤(αknα(k-1)n)(1+vn)2∥y(k-2)n-p∥+(1-αknα(k-1)n)(1+vn)2∥xn-p∥+(αkn+α(k-1)n)(1+vn)2dn+(γkn∥ukn-p∥+γ(k-1)n∥u(k-1)n-p∥)(1+vn)2⋮≤(αknα(k-1)n⋯α(j+1)n)(1+vn)(k-j)∥yjn-p∥+(1-αknα(k-1)n⋯α(j+1)n)(1+vn)(k-j)∥xn-p∥+(αkn+α(k-1)n+⋯+α(j+1)n)(1+vn)(k-j)dn+(γkn∥ukn-p∥+γ(k-1)n∥u(k-1)n-p∥+⋯+γ(j+1)n∥u(j+1)n-p∥)(1+vn)k-j.
Since 0<η≤αin≤ρ<1, for all i=1,2,…,k and all n≥n0, we have that for all n≥n0 and all j=1,2,…,k-1,
∥xn-p∥≤∥xn-p∥ηk-j-∥xn+1-p∥ηk-j(1+vn)k-j+∥yjn-p∥+ξjnηk-jdn+ϑjnηk-j,
where ξjn=αkn+α(k-1)n+⋯+α(j+1)n and ϑjn=γkn∥ukn-p∥+γ(k-1)n∥u(k-1)n-p∥+⋯+γ(j+1)n∥u(j+1)n-p∥. Since limn→∞∥xn-p∥=a and limn→∞ϑjn=limn→∞dn=limn→∞vn=0, it follows that
a≤liminfn→∞∥yjn-p∥,∀j=1,2,…,k-1.
From (4.2) and (4.6), we have
limn→∞∥yjn-p∥=a=limn→∞∥xn-p∥,∀j=1,2,…,k-1.
That is, for each j=1,2,…,k, we have
limn→∞∥αjn(Tjny(j-1)n-p+γjn(ujn-xn))+(1-αjn)(xn-p+γjn(ujn-xn))∥=a.
Since
∥Tjny(j-1)n-p+γjn(ujn-xn)∥≤∥Tjny(j-1)n-p∥+γjn∥ujn-xn∥≤(1+vn)∥y(j-1)n-p∥+dn+γjn∥ujn-xn∥,∥xn-p+γjn(ujn-xn)∥≤∥xn-p∥+γjn∥ujn-xn∥,
it follows that
limsupn→∞∥Tjny(j-1)n-p+γjn(ujn-xn)∥≤a,limsupn→∞∥xn-p+γjn(ujn-xn)∥≤a,∀j=1,2,…,k.
From (4.8) to (4.11), we can conclude from Lemma 2.2 that
limn→∞∥Tjny(j-1)n-xn∥=0,∀j=1,2,…,k.

(ii) It follows from part (i) in the case j=1 that limn→∞∥T1nxn-xn∥=0. For j=2,3,…,k, we obtain from part (i) that
∥Tjnxn-xn∥≤∥Tjnxn-Tjny(j-1)n∥+∥Tjny(j-1)n-xn∥≤L∥xn-y(j-1)n∥γ+∥Tjny(j-1)n-xn∥≤L(α(j-1)n∥Tj-1ny(j-2)n-xn∥+γ(j-1)n∥u(j-1)n-xn∥)γ+∥Tjny(j-1)n-xn∥→0asn→∞.
Therefore,
limn→∞∥Tjnxn-xn∥=0,∀j=1,2,…,k.
Since
∥xn-Tjxn∥≤∥xn+1-xn∥+∥xn+1-Tjn+1xn+1∥+∥Tjn+1xn+1-Tjn+1xn∥+∥Tjn+1xn-Tjxn∥≤αkn∥Tkny(k-1)n-xn∥+γkn∥ukn-xn∥+∥xn+1-Tjn+1xn+1∥+L(αkn∥Tkny(k-1)n-xn∥+γkn∥ukn-xn∥)γ+L∥Tjnxn-xn∥γ,
it follows from (i) and (4.14) that
limn→∞∥Tjxn-xn∥=0,∀j=1,2,…,k.

(iii) Since limn→∞γjn=0 and
∥yjn-xn∥≤αjn∥Tjny(j-1)n-xn∥+γjn∥ujn-xn∥
for all j∈{1,2,…,k}, (iii) is directly obtained by (i).

Theorem 4.2.

Let C be a nonempty closed and convex subset of a uniformly convex Banach space X satisfying the Opial's property, and {Ti,i=1,2,…,k} a finite family of (L-γ) uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin},{cin}⊂[0,∞) such that ∑n=1∞bin<∞ and ∑n=1∞cin<∞ for all i=1,2,…,k. Assume that F≠Ø and ∑n=1∞γin<∞ for all i=1,2,…,k. For a given x1∈C let {xn} be the sequence defined by (1.7) with 0<η≤αin≤ρ<1, for all i=1,2,…,k and all n≥n0 and for some n0∈ℕ. If I-Ti, i=1,2,…,k, is demiclosed at 0, then {xn} converges weakly to a common fixed point of the family {Ti:i=1,2,…,k}.

Proof.

By Lemma 4.1(ii), we have limn→∞∥Tixn-xn∥=0, for all i=1,2,…,k. Since X is uniformly convex and {xn} is bounded, without loss of generality we may assume that xn→u weakly as n→∞ for some u∈C. Since I-Ti, i=1,2,…,k, is demiclosed at 0, we have u∈F. Suppose that there are subsequences {xnk} and {xmk} of {xn} that converge weakly to u and v, respectively. Again, as above, we can prove that u,v∈F. By Lemma 3.1(b), limn→∞∥xn-u∥ and limn→∞∥xn-v∥ exist. It follows from Lemma 2.3 that u=v. Therefore {xn} converges weakly to a common fixed point of {Ti:i=1,2,…,k}.

Theorem 4.3.

Under the hypotheses of Lemma 4.1, assume that the family {Ti:i=1,2,…,k} satisfies condition (C¯). Then {xn} and {yjn} converge strongly to a common fixed point of the family of mappings for all j=1,2,…,k.

Proof.

From (3.5), we have
∥xn+1-p∥≤(1+vn)k∥xn-p∥+ekn,∀n,p∈F.
Therefore,
d(xn+1,F)≤(1+vn)kd(xn,F)+ekn=(1+∑r=1kk(k-1)⋯(k-r+1)r!vnr)d(xn,F)+ekn.
Since ∑n=1∞vn<∞, it follows that ∑n=1∞∑r=1k(k(k-1)⋯(k-r+1)/r!)vnr<∞. Since ∑n=1∞ekn<∞, we obtain from Lemma 2.1(i) that limn→∞d(xn,F) exists. By Lemma 4.1(ii), we have limn→∞∥Tixn-xn∥=0 for all i=1,2,…,k. Since {Ti:i=1,2,…,k} satisfies condition (C¯), there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that ∥xn-Ti0xn∥≥f(d(xn,F)) for some i0∈{1,2,…,k}, it follows that limn→∞d(xn,F)=0. By Theorem 3.2, we can conclude that {xn} converges strongly to a common fixed point q of the family {Ti:i=1,2,…,k}. From Lemma 4.1(iii), we have limn→∞∥yjn-xn∥=0 for all j=1,2,…,k, and we obtain that limn→∞yjn=q for all j=1,2,…,k.

Remark 4.4.

The family of generalized asymptotically quasi-nonexpansive mappings in Theorem 4.2 and 4.3 can be replaced by a family of asymptotically quasi-nonexpansive mappings. Lemma 3.1 and 4.1 generalize and improve [6, Lemma 3.1], [19, Lemmas 3.4 and 3.5], [18, Lemma 2.3], [7, Lemma 4.2], and [17, Lemma 3.3] to a finite family of (L-γ) uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Theorem 4.2 generalizes and improves [6, Theorems 3.2 and 4.2], [18, Theorem 2.9], [17, Theorem 3.1], and [21, Theorem 1] to the more general class of a finite family of (L-γ) uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Theorem 4.3 generalizes and improves [6, Theorem 3.3], [7, Theorem 4.3], [5, Theorem 2], [18, Theorem 2.4], [19, Theorem 4.2], [17, Theorem 3.2], [10, Theorem 3.4], [20, Theorem 2] and [21, Theorem 2] by using condition (C¯) instead of condition (A¯) or semicompactness or completely continuous or compactness to the more general class of a finite family of (L-γ) uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings.

Remark 4.5.

From Remark 1.1, Theorems 3.2 to 4.3 hold true for a finite family {Ti}i=1k of asymptotically nonexpansive mappings in the intermediate sense.

Acknowledgments

The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin University, and the Graduate School of Chiang Mai University, Thailand for their financial support.

ShahzadN.ZegeyeH.Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive mapsBruckR.KuczumowT.ReichS.Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial propertyQihouL.Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach spaceSenterH. F.DotsonW. G.Jr.Approximating fixed points of nonexpansive mappingsFukhar-ud-dinH.KhanS. H.Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applicationsKhanA. R.DomloA.-A.Fukhar-ud-dinH.Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spacesPlubtiengS.WangkeereeR.Strong convergence theorems for multi-step Noor iterations with errors in Banach spacesQihouL.Iterative sequences for asymptotically quasi-nonexpansive mappings with error memberQihouL.Iterative sequences for asymptotically quasi-nonexpansive mappingsShahzadN.UdomeneA.Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spacesSunZ.Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappingsXuB.NoorM. A.Fixed-point iterations for asymptotically nonexpansive mappings in Banach spacesPetrotN.Modified Noor iterative process by non-Lipschitzian mappings for nonlinear equations in Banach spacesJournal of Mathematical Analysis and Applications. In pressSuantaiS.Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappingsBnouhachemA.NoorM. A.RassiasTh. M.Three-steps iterative algorithms for mixed variational inequalitiesKhanS. H.TakahashiW.Approximating common fixed points of two asymptotically nonexpansive mappingsJeongJ. U.KimS. H.Weak and strong convergence of the Ishikawa iteration process with errors for two asymptotically nonexpansive mappingsPlubtiengS.WangkeereeR.PunpaengR.On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappingsFukhar-ud-dinH.KhanA. R.Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spacesKhanS. H.Fukhar-ud-dinH.Weak and strong convergence of a scheme with errors for two nonexpansive mappingsOsilikeM. O.AniagbosorS. C.Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappingsChidumeC. E.AliB.Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces