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# Strong convergence for total quasi-*ϕ*-asymptotically nonexpansive semigroups in Banach spaces

- Jing Quan
^{1}Email author, - Shih-sen Chang
^{2}and - Xiongrui Wang
^{1}

**2012**:142

https://doi.org/10.1186/1687-1812-2012-142

© Quan et al.; licensee Springer. 2012

**Received:**8 May 2012**Accepted:**22 August 2012**Published:**5 September 2012

## Abstract

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-*ϕ*-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces. The main results presented in this paper extend and improve the corresponding results of many authors.

**MSC:**47H05, 47H09, 49M05.

## Keywords

- strong convergence
- total quasi-
*ϕ*-asymptotically nonexpansive semigroups - generalized projection

## 1 Introduction

*E*is a real Banach space with norm $\parallel \cdot \parallel $, ${E}^{\ast}$ is the dual space of

*E*; $\u3008\cdot ,\cdot \u3009$ is the duality pairing between

*E*and ${E}^{\ast}$;

*C*is a nonempty closed convex subset of

*E*; $\mathbb{N}$ and $\mathbb{R}$ denote the natural number set and the set of nonnegative real numbers respectively. The mapping $J:E\to {2}^{{E}^{\ast}}$ defined by

is called *the normalized duality mapping*. Let $T:C\to C$ be a nonlinear mapping; $F(T)$ denotes the set of fixed points of mapping *T*.

Alber *et al.*[1] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied the methods of approximation of fixed points. They are defined as follows.

**Definition 1.1** Let $T:C\to C$ be a mapping. *T* is said to be total asymptotically nonexpansive if there exist sequences $\{{\mu}_{n}\}$, $\{{\nu}_{n}\}$ with ${\mu}_{n},{\nu}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a strictly increasing continuous function $\psi :\mathbb{R}\to \mathbb{R}$ with $\psi (0)=0$ such that $\parallel {T}^{n}x-{T}^{n}y\parallel \le \parallel x-y\parallel +{\mu}_{n}\psi (\parallel x-y\parallel )+{\nu}_{n}$ holds for all $x,y\in C$ and all $n\in \mathbb{N}$.

*T* is said to be *total asymptotically quasi-nonexpansive* if $F(T)\ne \mathrm{\varnothing}$, there exist sequences $\{{\mu}_{n}\}$, $\{{\nu}_{n}\}$ with ${\mu}_{n},{\nu}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a strictly increasing continuous function $\psi :\mathbb{R}\to \mathbb{R}$ with $\psi (0)=0$ such that $\parallel {T}^{n}x-p\parallel \le \parallel x-p\parallel +{\mu}_{n}\psi (\parallel x-p\parallel )+{\nu}_{n}$ holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

Chidume and Ofoedu [2] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings and total asymptotically quasi-nonexpansive mappings in Banach spaces. Chidume *et al.*[3] gave a new iterative sequence and necessary and sufficient conditions for this sequence to converge to a common fixed point of finite total asymptotically nonexpansive mappings. Chang [4] established some new approximation theorems of common fixed points for a countable family of total asymptotically nonexpansive mappings in Banach spaces.

Recently, many researchers have focused on studying the convergence of iterative algorithms for quasi-*ϕ*-asymptotically nonexpansive (see [5–9]) and total quasi-*ϕ*-asymptotically nonexpansive (see [10–12]) mappings. Ye *et al.*[13] used a new hybrid projection algorithm to obtain strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. Kim [14] used hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-*ϕ*-nonexpansive mappings to prove the strong convergence theorems. Saewan [15] used the shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-*ϕ*-nonexpansive mappings.

A Banach space *E* is said to be strictly convex if $\frac{\parallel x+y\parallel}{2}<1$ for $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$; it is also said to be uniformly convex if ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$ for any two sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ in *E* such that $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty}}\frac{\parallel {x}_{n}+{y}_{n}\parallel}{2}=1$. Let $U=\{x\in E:\parallel x\parallel =1\}$ be the unit sphere of *E*, then the Banach space *E* is said to be smooth provided ${lim}_{t\to 0}\frac{\parallel x+ty\parallel -\parallel y\parallel}{t}$ exists for each $x,y\in U$. It is also said to be uniformly smooth if the limit is attained uniformly for each $x,y\in U$. It is well known that if *E* is reflexive and smooth, then the duality mapping *J* is single valued. A Banach space *E* is said to have the Kadec-Klee property if a sequence $\{{x}_{n}\}$ of *E* satisfies that ${x}_{n}\rightharpoonup x\in E$ and $\parallel {x}_{n}\parallel \to x$, then ${x}_{n}\to x$. It is known that if *E* is uniformly convex, then *E* has the Kadec-Klee property.

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

*C*is a nonempty closed convex subset of

*E*. We use $\varphi :E\times E\to {R}^{+}$ to denote the Lyapunov functional defined by

The quasi-*ϕ*-asymptotically nonexpansive and total quasi-*ϕ*-asymptotically nonexpansive mappings are defined as follows.

**Definition 1.2**A mapping $T:C\to C$ is said to be quasi-

*ϕ*-asymptotically nonexpansive, if $F(T)\ne \mathrm{\varnothing}$, there exist sequences $\{{k}_{n}\}\subset [1,+\mathrm{\infty})$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that

holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

*ϕ*-asymptotically nonexpansive, if $F(T)\ne \mathrm{\varnothing}$, there exist sequences $\{{\mu}_{n}\}$, $\{{\nu}_{n}\}$ with ${\mu}_{n},{\nu}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a strictly increasing continuous function $\psi :\mathbb{R}\to \mathbb{R}$ with $\psi (0)=0$ such that

holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

In recent years, many researchers have considered the convergence of asymptotically nonexpansive semigroups [17, 18]. The asymptotically nonexpansive semigroups are defined as follows.

**Definition 1.3**[17]

*C*into itself is said to be an asymptotically nonexpansive semigroup on

*C*, if the following conditions are satisfied:

- (a)
$T(0)x=x$ for each $x\in C$;

- (b)
$T(t+s)x=T(s)T(t)$ for any $t,s\in {R}^{+}$ and $x\in C$;

- (c)
For any $x\in C$, the mapping $t\to T(t)x$ is continuous;

- (d)There exist sequences $\{{k}_{n}\}\subset [1,+\mathrm{\infty})$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that$\parallel {T}^{n}(t)x-{T}^{n}(t)y\parallel \le {k}_{n}\parallel x-y\parallel $

holds for all $x,y\in C$, $n\in \mathbb{N}$.

We use $F(T)$ to denote the common fixed point set of the semigroup **T**, *i.e.*, $F(T)={\bigcap}_{t\ge 0}F(T(t))$.

Chang [19] used the modified Halpern-Mann type iteration algorithm for quasi-*ϕ*-asymptotically nonexpansive semigroups to prove the strong convergence in the Banach space. The quasi-*ϕ*-asymptotically nonexpansive semigroups are defined as follows.

**Definition 1.4**[19]

*C*into itself is said to be a quasi-

*ϕ*-asymptotically nonexpansive semigroup on

*C*if the conditions (a), (b), (c) in Definition 1.3 and following condition (e) are satisfied:

- (e)For all $x,y\in C$, $p\in F((T))$, $t\ge 0$, there exist sequences $\{{k}_{n}\}\subset [1,+\mathrm{\infty})$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$, such that$\varphi (p,{T}^{n}(t)x)\le {k}_{n}\varphi (p,x)$

holds for all $n\in \mathbb{N}$.

## 2 Preliminaries

This section contains some definitions and lemmas which will be used in the proofs of our main results in the following section.

**Definition 2.1**One-parameter family $\mathbf{T}:=\{T(t):t\ge 0\}$ of mappings from

*C*into itself is said to be a total quasi-

*ϕ*-asymptotically nonexpansive semigroup on

*C*if conditions (a), (b), (c) in Definition 1.3 and following condition (f) are satisfied:

- (f)If $F(T)\ne \mathrm{\varnothing}$, there exist sequences $\{{\mu}_{n}\}$, $\{{\nu}_{n}\}$ with ${\mu}_{n},{\nu}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a strictly increasing continuous function $\psi :\mathbb{R}\to \mathbb{R}$ with $\psi (0)=0$ such that$\varphi (p,{T}^{n}(t)x)\le \varphi (p,x)+{\mu}_{n}\psi (\varphi (p,x))+{\nu}_{n}$

holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

*ϕ*-asymptotically nonexpansive semigroup

**T**is said to be uniformly Lipschitzian if there exists a bounded measurable function $L:[0,\mathrm{\infty})\to (0,+\mathrm{\infty})$ such that

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-*ϕ*-nonexpansive asymptotically semigroups to prove the strong convergence in Banach spaces. The results presented in the article improve and extend the corresponding results of [5, 6, 9–12, 14, 15, 19] and many others.

In order to prove the results of this paper, we shall need the following lemmas:

**Lemma 2.1** (See [16])

*Let*

*E*

*be a smooth*,

*strictly convex and reflexive Banach space and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Then the following conclusions hold*:

- (i)
$\varphi (x,{\mathrm{\Pi}}_{C}y)+\varphi ({\mathrm{\Pi}}_{C}y,y)\le \varphi (x,y)$

*for all*$x\in C$, $y\in E$; - (ii)
*If*$x\in E$*and*$z\in C$,*then*$z={\mathrm{\Pi}}_{C}x\iff \u3008z-y,Jx-Jz\u3009\ge 0$, $\mathrm{\forall}y\in C$; - (iii)
*For*$x,y\in E$, $\varphi (x,y)=0$*if and only if*$x=y$.

**Lemma 2.2**[19]

*Let* *E* *be a uniformly convex and smooth Banach space and let*$\{{x}_{n}\}$*and*$\{{y}_{n}\}$*be two sequences of* *E*. *If*$\varphi ({x}_{n},{y}_{n})\to 0$*and either*$\{{x}_{n}\}$*or*$\{{y}_{n}\}$*is bounded*, *then*$\parallel {x}_{n}-{y}_{n}\parallel \to 0$.

**Lemma 2.3**[10]

*Let* *E* *be a real uniformly smooth and strictly convex Banach space with the Kadec*-*Klee property*, *and* *C* *be a nonempty closed convex subset of* *E*. *Let*$T:C\to C$*be a closed and total quasi*-*ϕ*-*asymptotically nonexpansive mapping defined by Definition * 1.2. *If*${\nu}_{1}=0$, *then the fixed point set*$F(T)$*of* *T* *is a closed and convex subset of* *C*.

## 3 Main results

**Theorem 3.1**

*Let*

*E*

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

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*$\mathbf{T}:=\{T(t):t\ge 0\}$

*be a total quasi*-

*ϕ*-

*asymptotically nonexpansive semigroup from*

*C*

*into itself defined by Definition*2.1.

*Suppose*$\mathbf{T}:=\{T(t):t\ge 0\}$

*is closed*,

*uniformly*

*L*-

*Lipschitz and*$F(T):={\bigcap}_{t\ge 0}F(T(t))\ne \mathrm{\varnothing}$.

*Suppose there exists*${M}^{\ast}>0$

*such that*$\psi ({\eta}_{n})\le {M}^{\ast}{\eta}_{n}$.

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

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

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

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

*satisfying the following conditions*: ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}<{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1$.

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

*be a sequence generated by*

*where*${\xi}_{n}={\mu}_{n}{M}^{\ast}{sup}_{p\in F(T)}\varphi (p,{x}_{n})$. *If*${\nu}_{1}=0$*and*$F(T)$*is bounded in* *C*, *then the iterative sequence*$\{{x}_{n}\}$*converges strongly to a common fixed point*${x}^{\ast}\in F(T)$*in* *C*.

*Proof* (I) We prove $F(T)$ and ${C}_{n}(n\in \mathbb{N})$ all are closed and convex subsets in *C*.

*C*. So $F(T)$ is closed and convex in

*C*. By the assumption we know that ${C}_{1}=C$ is closed and convex. We suppose that ${C}_{n}$ is closed and convex for some $n\ge 2$. By the definition of

*ϕ*, we have that

- (II)
We prove that $F(T)\subset {C}_{n}$.

- (III)
We prove that $\{{x}_{n}\}$ is a Cauchy sequence in

*C*.

*C*. Since

*C*is complete, without loss of generality, we can assume that ${x}_{n}\to {p}^{\ast}$ (some point in

*C*). By the assumption, we have that

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

*ϕ*-asymptotically nonexpansive semigroup with sequence ${\mu}_{n},{\nu}_{n},p\in F(T)$, we have

*E*is uniformly smooth, ${J}^{-1}$ is uniformly continuous on each bounded subset of ${E}^{\ast}$, it follows from (5) and (6) that

*J*is uniformly continuous on each bounded subset of

*E*, we have $J{x}_{n}\to J{p}^{\ast}$, and for each $t\ge 0$,

*J*is uniformly continuous, this shows that ${lim}_{n\to \mathrm{\infty}}{T}^{n}(t){x}_{n}={p}^{\ast}=0$ uniformly for $t\ge 0$. Again by the assumptions that the semigroup $\mathbf{T}:=\{T(t):t\ge 0\}$ is closed and uniformly

*L*-Lipschitzian, we have

**T**, we have that $T(t){p}^{\ast}={p}^{\ast}$,

*i.e.*, ${p}^{\ast}\in F(T(t))$. By the arbitrariness of $t\ge 0$, we have ${p}^{\ast}\in F(T)={\bigcap}_{t\ge 0}F(T(t))$.

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

In view of the definition of ${\mathrm{\Pi}}_{F(T)}{x}_{1}$, from (8), we have ${p}^{\ast}=\omega $. Therefore, ${x}_{n}\to {p}^{\ast}={\mathrm{\Pi}}_{F(T)}{x}_{1}$. This completes the proof of Theorem 3.1. □

## Declarations

### Acknowledgements

This work was supported by National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of Sichuan Provincial Education Department (No. 12ZB345 and No. 11ZA172).

## Authors’ Affiliations

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