# Strong convergence theorems for a countable family of totally quasi-*ϕ*-asymptotically nonexpansive nonself mappings in Banach spaces with applications

- Li Yi
^{1}Email author

**2012**:268

https://doi.org/10.1186/1029-242X-2012-268

© Yi; licensee Springer 2012

**Received: **21 May 2012

**Accepted: **5 November 2012

**Published: **22 November 2012

## Abstract

In this paper, we introduce a class of totally quasi-*ϕ*-asymptotically nonexpansive nonself mappings and study the strong convergence under a limit condition only in the framework of Banach spaces. Meanwhile, our results are applied to study the approximation problem of a solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang *et al.* (Appl. Math. Comput. 218:7864-7870, 2012).

**MSC:**47J05, 47H09, 49J25.

## Keywords

*ϕ*-asymptotically nonexpansive nonself mappingtotally quasi-

*ϕ*-asymptotically nonexpansive nonself mappingiterative sequencenonexpansive retraction

## 1 Introduction

*X*is a real Banach space with the dual ${X}^{\ast}$,

*D*is a nonempty closed convex subset of

*X*. We also denote by

*J*the normalized duality mapping from

*X*to ${2}^{{X}^{\ast}}$ which is defined by

where $\u3008\cdot ,\cdot \u3009$ denotes the generalized duality pairing.

Let *D* be a nonempty closed subset of a real Banach space *X*. A mapping $T:D\to D$ is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ for all $x,y\in D$. An element $p\in D$ is called a fixed point of $T:D\to D$ if $p=T(p)$. The set of fixed points of *T* is represented by $F(T)$.

A Banach space *X* is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel \le 1$ for all $x,y\in X$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. A Banach space is said to be uniformly convex if ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$ for any two sequences $\{{x}_{n}\},\{{y}_{n}\}\subset X$ with $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =0$.

*X*is said to be Gâteaux differentiable if for each $x,y\in S(x)$, the limit

exists, where $S(x)=\{x:\parallel x\parallel =1,x\in X\}$. In this case, *X* is said to be smooth. The norm of a Banach space *X* is said to be Fréchet differentiable if for each $x\in S(x)$, the limit (1.1) is attained uniformly for $y\in S(x)$ and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for $x,y\in S(x)$. In this case, *X* is said to be uniformly smooth.

A subset *D* of *X* is said to be a retract of *X* if there exists a continuous mapping $P:X\to D$ such that $Px=x$ for all $x\in X$. It is well known that every nonempty closed convex subset of a uniformly convex Banach space *X* is a retract of *X*. A mapping $P:X\to D$ is said to be a retraction if ${P}^{2}=P$. It follows that if a mapping *P* is a retraction, then $Py=y$ for all *y* in the range of *P*. A mapping $P:X\to D$ is said to be a nonexpansive retraction if it is nonexpansive and it is a retraction from *X* to *D*.

*X*is a smooth, strictly convex and reflexive Banach space and

*D*is a nonempty closed convex subset of

*X*. In the sequel, we always use $\varphi :X\times X\to {R}^{+}$ to denote the Lyapunov functional defined by

for all $\lambda \in [0,1]$ and $x,y,z\in X$.

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.

In the sequel, we denote the strong convergence and weak convergence of the sequence $\{{x}_{n}\}$ by ${x}_{n}\to x$ and ${x}_{n}\rightharpoonup x$, respectively.

**Lemma 1.1** (see [1])

*Let*

*X*

*be a smooth*,

*strictly convex and reflexive Banach space and*

*D*

*be a nonempty closed convex subset of*

*X*.

*Then the following conclusions hold*:

- (a)
$\varphi (x,y)=0$

*if and only if*$x=y$; - (b)
$\varphi (x,{\mathrm{\Pi}}_{D}y)+\varphi ({\mathrm{\Pi}}_{D}y,y)\le \varphi (x,y)$, $\mathrm{\forall}x,y\in D$;

- (c)
*if*$x\in X$*and*$z\in D$,*then*$z={\mathrm{\Pi}}_{D}x$*if and only if*$\u3008z-y,Jx-Jz\u3009\ge 0$, $\mathrm{\forall}y\in D$.

**Remark 1.1** (see [2])

Let ${\mathrm{\Pi}}_{D}$ be the generalized projection from a smooth, reflexive and strictly convex Banach space *X* onto a nonempty closed convex subset *D* of *X*. Then ${\mathrm{\Pi}}_{D}$ is a closed and quasi-*ϕ*-nonexpansive from *X* onto *D*.

**Remark 1.2** (see [2])

If *H* is a real Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel}^{2}$, and ${\mathrm{\Pi}}_{D}$ is the metric projection of *H* onto *D*.

**Definition 1.1**Let $P:X\to D$ be a nonexpansive retraction.

- (1)A nonself mapping $T:D\to X$ is said to be quasi-
*ϕ*-nonexpansive if $F(T)\ne \mathrm{\Phi}$, and$\varphi (p,T{(PT)}^{n-1}x)\le \varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),\mathrm{\forall}n\ge 1;$(1.7) - (2)A nonself mapping $T:D\to X$ is said to be quasi-
*ϕ*-asymptotically nonexpansive if $F(T)\ne \mathrm{\Phi}$, and there exists a real sequence ${k}_{n}\subset [1,+\mathrm{\infty})$, ${k}_{n}\to 1$ (as $n\to \mathrm{\infty}$), such that$\varphi (p,T{(PT)}^{n-1}x)\le {k}_{n}\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),\mathrm{\forall}n\ge 1;$(1.8) - (3)A nonself mapping $T:D\to X$ is said to be totally quasi-
*ϕ*-asymptotically nonexpansive if $F(T)\ne \mathrm{\Phi}$, and there exist nonnegative real sequences $\{{v}_{n}\}$, $\{{\mu}_{n}\}$ with ${v}_{n},{\mu}_{n}\to 0$ (as $n\to \mathrm{\infty}$) and a strictly increasing continuous function $\zeta :{R}^{+}\to {R}^{+}$ with $\zeta (0)=0$ such that$\varphi (p,T{(PT)}^{n-1}x)\le \varphi (p,x)+{v}_{n}\zeta [\varphi (p,x)]+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,\mathrm{\forall}n\ge 1,p\in F(T).$(1.9)

**Remark 1.3** From the definitions, it is obvious that a quasi-*ϕ*-nonexpansive nonself mapping is a quasi-*ϕ*-asymptotically nonexpansive nonself mapping, and a quasi-*ϕ*-asymptotically nonexpansive nonself mapping is a totally quasi-*ϕ*-asymptotically nonexpansive nonself mapping, but the converse is not true.

Next, we present an example of a quasi-*ϕ*-nonexpansive nonself mapping.

**Example 1.1** (see [2])

Let *H* be a real Hilbert space, *D* be a nonempty closed and convex subset of *H* and $f:D\times D\to R$ be a bifunction satisfying the conditions: (A1) $f(x,x)=0$, $\mathrm{\forall}x\in D$; (A2) $f(x,y)+f(y,x)\le 0$, $\mathrm{\forall}x,y\in D$; (A3) for each $x,y,z\in D$, ${lim}_{t\to 0}f(tz+(1-t)x,y)\le f(x,y)$; (A4) for each given $x\in D$, the function $y\mapsto f(x,y)$ is convex and lower semicontinuous. The so-called equilibrium problem for *f* is to find an ${x}^{\ast}\in D$ such that $f({x}^{\ast},y)\ge 0$, $\mathrm{\forall}y\in D$. The set of its solutions is denoted by $EP(f)$.

then (1) ${T}_{r}$ is single-valued, and so $z={T}_{r}(x)$; (2) ${T}_{r}$ is a relatively nonexpansive nonself mapping, therefore it is a closed quasi-*ϕ*-nonexpansive nonself mapping; (3) $F({T}_{r})=EP(f)$ and $F({T}_{r})$ is a nonempty and closed convex subset of *D*; (4) ${T}_{r}:D\to D$ is nonexpansive. Since $F({T}_{r})$ is nonempty, and so it is a quasi-*ϕ*-nonexpansive nonself mapping from *D* to *H*, where $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $x,y\in H$.

Now, we give an example of a totally quasi-*ϕ*-asymptotically nonexpansive nonself mapping.

**Example 1.2** (see [2])

*D*be a unit ball in a real Hilbert space ${l}^{2}$, and let $T:D\to {l}^{2}$ be a nonself mapping defined by

where $\{{a}_{i}\}$ is a sequence in $(0,1)$ such that ${\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}$.

- (i)
$\parallel Tx-Ty\parallel \le 2\parallel x-y\parallel $, $\mathrm{\forall}x,y\in D$;

- (ii)
$\parallel {T}^{n}x-{T}^{n}y\parallel \le 2{\prod}_{j=2}^{n}{a}_{j}$, $\mathrm{\forall}x,y\in D$, $n\ge 2$.

*D*is a unit ball in a real Hilbert space ${l}^{2}$, it follows from Remark 1.2 that $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $\mathrm{\forall}x,y\in D$. The above inequality can be written as

where *P* is the nonexpansive retraction. This shows that the mapping *T* defined as above is a totally quasi-*ϕ*-asymptotically nonexpansive nonself mapping.

**Lemma 1.2** (see [4])

*Let* *X* *be a uniformly convex and smooth Banach space*, *and let* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *be two sequences of* *X* *such that* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *are bounded*; *if* $\varphi ({x}_{n},{y}_{n})\to 0$, *then* $\parallel {x}_{n}-{y}_{n}\parallel \to 0$.

**Lemma 1.3** *Let* *X* *be a smooth*, *strictly convex and reflexive Banach space and* *D* *be a nonempty closed convex subset of* *X*. *Let* $T:D\to X$ *be a totally quasi*-*ϕ*-*asymptotically nonexpansive nonself mapping with* ${\mu}_{1}=0$, *then* $F(T)$ *is a closed and convex subset of* *D*.

*Proof*Let $\{{x}_{n}\}$ be a sequence in $F(T)$ such that ${x}_{n}\to p$. Since

*T*is a totally quasi-

*ϕ*-asymptotically nonexpansive nonself mapping, we have

By Lemma 1.2, we obtain $Tp=p$. So, we have $p\in F(T)$. This implies $F(T)$ is closed.

*ϕ*, let $\{{u}_{n}\}$ be a sequence generated by ${u}_{1}=Tw$, ${u}_{2}=T(PT)w$, ${u}_{3}=T{(PT)}^{2}w,\dots ,{u}_{n}=T{(PT)}^{n-1}w=TP{u}_{n-1}$, we have

Hence, we have ${u}_{n}\to w$. This implies that ${u}_{n+1}\to w$. Since *TP* is closed and ${u}_{n+1}=T{(PT)}^{n}w=TP{u}_{n}$, we have $TPw=w$. Since $w\in C$, and so $Tw=w$, *i.e.*, $w\in F(T)$. This implies $F(T)$ is convex. This completes the proof of Lemma 1.3. □

**Definition 1.2**(1) (see [5]) A countable family of nonself mappings $\{{T}_{i}\}:D\to X$ is said to be uniformly quasi-

*ϕ*-asymptotically nonexpansive if ${\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\Phi}$, and there exist nonnegative real sequences ${k}_{n}\subset [1,+\mathrm{\infty})$, ${k}_{n}\to 1$, such that for each $i\ge 1$,

- (2)A countable family of nonself mappings $\{{T}_{i}\}:D\to X$ is said to be uniformly totally quasi-
*ϕ*-asymptotically nonexpansive if ${\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\Phi}$, and there exist nonnegative real sequences $\{{v}_{n}\}$, $\{{\mu}_{n}\}$ with ${v}_{n},{\mu}_{n}\to 0$ (as $n\to \mathrm{\infty}$) and a strictly increasing continuous function $\zeta :{R}^{+}\to {R}^{+}$ with $\zeta (0)=0$ such that for each $i\ge 1$,$\varphi (p,{T}_{i}{(P{T}_{i})}^{n-1}x)\le \varphi (p,x)+{v}_{n}\zeta [\varphi (p,x)]+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,\mathrm{\forall}n\ge 1,p\in F(T).$(1.14)

*L*-Lipschitz continuous if there exists a constant $L>0$ such that

Considering the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-*ϕ*-nonexpansive and quasi-*ϕ*-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [4–19]).

The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm for a family of totally quasi-*ϕ*-asymptotically nonexpansive nonself mappings to have the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang *et al.* [5, 6, 20, 21], Su *et al.* [16], Kiziltunc *et al.* [10], Yildirim *et al.* [11], Yang *et al.* [22], Wang [18, 19], Pathak *et al.* [14], Thianwan [17], Qin *et al.* [15], Hao *et al.* [9], Guo *et al.* [7], Nilsrakoo *et al.* [13] and others.

## 2 Main results

**Theorem 2.1**

*Let*

*X*

*be a real uniformly smooth and uniformly convex Banach space*,

*D*

*be a nonempty closed convex subset of*

*X*.

*Let*$\{{T}_{i}\}:D\to X$

*be a family of uniformly totally quasi*-

*ϕ*-

*asymptotically nonexpansive nonself mappings with sequences*$\{{v}_{n}\}$, $\{{\mu}_{n}\}$,

*with*${v}_{n},{\mu}_{n}\to 0$ (

*as*$n\to \mathrm{\infty}$),

*and a strictly increasing continuous function*$\zeta :{R}^{+}\to {R}^{+}$

*with*$\zeta (0)=0$

*such that for each*$i\ge 1$, $\{{T}_{i}\}:D\to X$

*is uniformly*${L}_{i}$-

*Lipschitz continuous*.

*Let*$\{{\alpha}_{n}\}$

*be a sequence in*$[0,1]$

*and*$\{{\beta}_{n}\}$

*be a sequence in*$(0,1)$

*satisfying the following conditions*:

- (i)
${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$;

- (ii)
$0<{lim}_{n\to \mathrm{\infty}}inf{\beta}_{n}\le {lim}_{n\to \mathrm{\infty}}sup{\beta}_{n}<1$.

*Let*${x}_{n}$

*be a sequence generated by*

*where* ${\xi}_{n}={v}_{n}{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}$, $\mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$, ${\mathrm{\Pi}}_{{D}_{n+1}}$*is the generalized projection of* *X* *onto* ${D}_{n+1}$. *If* ℱ *is nonempty*, *then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

*Proof* (I) First, we prove that ℱ and ${D}_{n}$ are closed and convex subsets in *D*.

*D*. Therefore, ℱ is a closed and convex subset in

*D*. By the assumption that ${D}_{1}=D$ is closed and convex, suppose that ${D}_{n}$ is closed and convex for some $n\ge 1$. In view of the definition of

*ϕ*, we have

- (II)
Next, we prove that $\mathcal{F}\subset {D}_{n}$ for all $n\ge 1$.

In fact, it is obvious that $\mathcal{F}\subset {D}_{1}$. Suppose that $\mathcal{F}\subset {D}_{n}$.

- (III)
Now, we prove that $\{{x}_{n}\}$ converges strongly to some point ${p}^{\ast}$.

Therefore, $\{\varphi ({x}_{n},{x}_{1})\}$ is bounded, and so is $\{{x}_{n}\}$. Since ${x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1}$ and ${x}_{n+1}={\mathrm{\Pi}}_{{D}_{n+1}}{x}_{1}\in {D}_{n+1}\subset {D}_{n}$, we have $\varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1})$. This implies that $\{\varphi ({x}_{n},{x}_{1})\}$ is nondecreasing. Hence, ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})$ exists.

It follows from Lemma 1.2 that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{m}-{x}_{n}\parallel =0$. Hence, $\{{x}_{n}\}$ is a Cauchy sequence in *D*. Since *D* is complete, without loss of generality, we can assume that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast}$ (some point in *D*).

- (IV)
Now, we prove that ${p}^{\ast}\in \mathcal{F}$.

*ϕ*-asymptotically nonexpansive nonself mappings, we have

This implies that $\{{T}_{i}{(P{T}_{i})}^{n-1}{x}_{n}\}$ is uniformly bounded.

this implies that $\{{w}_{n,i}\}$ is also uniformly bounded.

for each $i\ge 1$.

*J*is uniformly continuous on each bounded subset of

*X*, we have

*J*is uniformly continuous, this shows that

for each $i\ge 1$.

We get ${lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}{(P{T}_{i})}^{n}{x}_{n}-{T}_{i}{(P{T}_{i})}^{n-1}{x}_{n}\parallel =0$. Since ${lim}_{n\to \mathrm{\infty}}{T}_{i}{(P{T}_{i})}^{n-1}{x}_{n}={P}^{\ast}$ and ${lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast}$, we have ${lim}_{n\to \mathrm{\infty}}{T}_{i}P{T}_{i}{(P{T}_{i})}^{n-1}{x}_{n}={p}^{\ast}$.

- (V)
Finally, we prove that ${p}^{\ast}={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$ and so ${x}_{n}\to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}={p}^{\ast}$.

which yields that ${p}^{\ast}=w={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$. Therefore, ${x}_{n}\to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$. The proof of Theorem 3.1 is completed. □

By Remark 1.3, the following corollary is obtained.

**Corollary 2.1** *Let* *X*, *D*, $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ *be the same as in Theorem * 2.1. *Let* $\{{T}_{i}\}:D\to X$ *be a family of uniformly quasi*-*ϕ*-*asymptotically nonexpansive nonself mappings with the sequence* ${k}_{n}\subset [1,+\mathrm{\infty})$, ${k}_{n}\to 1$, *such that for each* $i\ge 1$, $\{{T}_{i}\}:D\to X$ *is uniformly* ${L}_{i}$-*Lipschitz continuous*.

*Let*${x}_{n}$

*be a sequence generated by*

*where* ${\xi}_{n}=({k}_{n}-1){sup}_{p\in \mathcal{F}}\varphi (p,{x}_{n})$, $\mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$, ${\mathrm{\Pi}}_{{D}_{n+1}}$ *is the generalized projection of* *X* *onto* ${D}_{n+1}$. *If* ℱ *is nonempty*, *then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

## 3 Application

In this section we utilize Corollary 2.1 to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result.

**Theorem 3.1**

*Let*

*H*

*be a real Hilbert space*,

*D*

*be a nonempty closed and convex subset of H*. $\{{\alpha}_{n}\}$, $({\beta}_{n})$

*be the same as in Theorem*2.1.

*Let*$\{{f}_{i}\}:D\times D\to R$

*be a countable family of bifunctions satisfying conditions*(A1)-(A4)

*as given in Example*1.1.

*Let*$\{{T}_{r,i}:D\to D\subset H\}$

*be the family of mappings defined by*(1.9),

*i*.

*e*.,

*Let*$\{{x}_{n}\}$

*be the sequence generated by*

*If* $\mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{r,i})\ne \mathrm{\Phi}$, *then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$, *which is a common solution of the system of equilibrium problems for* *f*.

*Proof*In Example 1.1, we have pointed out that ${u}_{n,i}={T}_{r,i}({x}_{n})$, $F({T}_{r,i})=EP({f}_{i})$ is nonempty and convex for all $i\ge 1$, ${T}_{r,i}$ is a countable family of quasi-

*ϕ*-nonexpansive nonself mappings. Since $F({T}_{r,i})$ is nonempty, so ${T}_{r,i}$ is a countable family of quasi-

*ϕ*-nonexpansive mappings and for all $i\ge 1$, ${T}_{r,i}$ is a uniformly 1-Lipschitzian mapping. Hence, (3.1) can be rewritten as follows:

Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □

## Declarations

### Acknowledgements

The author is very grateful to both reviewers for careful reading of this paper and for their comments.

## Authors’ Affiliations

## References

- Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*. Edited by: Kartosator AG. Dekker, New York; 1996:15–50.Google Scholar - Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*Math. Stud.*1994, 63(1/4):123–145.MathSciNetGoogle Scholar - Cioranescu I:
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems*. Kluwer Academic, Dordrecht; 1990.View ArticleGoogle Scholar - Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space.
*SIAM J. Optim.*2002, 13: 938–945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar - Chang SS, Wang L, Tang YK, Wang B, Qin LJ: Strong convergence theorems for a countable family of quasi-
*ϕ*-asymptotically nonexpansive nonself mappings.*Appl. Math. Comput.*2012, 218: 7864–7870. 10.1016/j.amc.2012.02.002MathSciNetView ArticleGoogle Scholar - Chang SS, Lee HWJ, Chan CK, Yang L: Approximation theorems for total quasi-
*ϕ*-asymptotically nonexpansive mappings with applications.*Appl. Math. Comput.*2011, 218: 2921–2931. 10.1016/j.amc.2011.08.036MathSciNetView ArticleGoogle Scholar - Guo W: Weak convergence theorems for asymptotically nonexpansive nonself-mappings.
*Appl. Math. Lett.*2011, 24: 2181–2185. 10.1016/j.aml.2011.06.022MathSciNetView ArticleGoogle Scholar - Halpren B: Fixed points of nonexpansive maps.
*Bull. Am. Math. Soc.*1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View ArticleGoogle Scholar - Hao Y, Cho SY, Qin X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings.
*Fixed Point Theory Appl.*2010., 2010: Article ID 218573Google Scholar - Kiziltunc H, Temir S: Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces.
*Comput. Math. Appl.*2011, 61(9):2480–2489. 10.1016/j.camwa.2011.02.030MathSciNetView ArticleGoogle Scholar - Yildirim I, Özdemir M: A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings.
*Nonlinear Anal., Theory Methods Appl.*2009, 71(3–4):991–999. 10.1016/j.na.2008.11.017View ArticleGoogle Scholar - Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.
*J. Math. Anal. Appl.*2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleGoogle Scholar - Nilsrakoo W, Sajung S: Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces.
*Appl. Math. Comput.*2011, 217(14):6577–6586. 10.1016/j.amc.2011.01.040MathSciNetView ArticleGoogle Scholar - Pathak HK, Cho YJ, Kang SM: Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings.
*Nonlinear Anal., Theory Methods Appl.*2009, 70(5):1929–1938. 10.1016/j.na.2008.02.092MathSciNetView ArticleGoogle Scholar - Qin XL, Cho YJ, Kang SM, Zhou HY: Convergence of a modified Halpern-type iterative algorithm for quasi-
*ϕ*-nonexpansive mappings.*Appl. Math. Lett.*2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015MathSciNetView ArticleGoogle Scholar - Su YF, Xu HK, Zhang X: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications.
*Nonlinear Anal.*2010, 73: 3890–3906. 10.1016/j.na.2010.08.021MathSciNetView ArticleGoogle Scholar - Thianwan S: Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space.
*J. Comput. Appl. Math.*2009, 224(2):688–695. 10.1016/j.cam.2008.05.051MathSciNetView ArticleGoogle Scholar - Wang L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings.
*J. Math. Anal. Appl.*2006, 323(1):550–557. 10.1016/j.jmaa.2005.10.062MathSciNetView ArticleGoogle Scholar - Wang L: Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings.
*Comput. Math. Appl.*2007, 53(7):1012–1019. 10.1016/j.camwa.2007.01.001MathSciNetView ArticleGoogle Scholar - Chang SS, Chan CK, Lee HWJ: Modified block iterative algorithm for quasi-
*ϕ*-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces.*Appl. Math. Comput.*2011, 217: 7520–7530. 10.1016/j.amc.2011.02.060MathSciNetView ArticleGoogle Scholar - Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications.
*Nonlinear Anal. TMA*2010, 73: 2260–2270. 10.1016/j.na.2010.06.006MathSciNetView ArticleGoogle Scholar - Yang L, Xie X: Weak and strong convergence theorems of three step iteration process with errors for nonself-asymptotically nonexpansive mappings.
*Math. Comput. Model.*2010, 52(5–6):772–780. 10.1016/j.mcm.2010.05.006MathSciNetView ArticleGoogle Scholar

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