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

Fixed Point Theory and Applications20122012:142

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

• Accepted: 22 August 2012
• Published:

## 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

Throughout this article, we assume that E is a real Banach space with norm $\parallel \cdot \parallel$, ${E}^{\ast }$ is the dual space of E; $〈\cdot ,\cdot 〉$ 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
$J\left(x\right)=\left\{{f}^{\ast }\in {E}^{\ast }:〈x,{f}^{\ast }〉={\parallel x\parallel }^{2};\parallel {f}^{\ast }\parallel =\parallel x\parallel ,x\in E\right\}$

is called the normalized duality mapping. Let $T:C\to C$ be a nonlinear mapping; $F\left(T\right)$ 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 $\left\{{\mu }_{n}\right\}$, $\left\{{\nu }_{n}\right\}$ 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 $\psi \left(0\right)=0$ such that $\parallel {T}^{n}x-{T}^{n}y\parallel \le \parallel x-y\parallel +{\mu }_{n}\psi \left(\parallel x-y\parallel \right)+{\nu }_{n}$ holds for all $x,y\in C$ and all $n\in \mathbb{N}$.

T is said to be total asymptotically quasi-nonexpansive if $F\left(T\right)\ne \mathrm{\varnothing }$, there exist sequences $\left\{{\mu }_{n}\right\}$, $\left\{{\nu }_{n}\right\}$ 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 $\psi \left(0\right)=0$ such that $\parallel {T}^{n}x-p\parallel \le \parallel x-p\parallel +{\mu }_{n}\psi \left(\parallel x-p\parallel \right)+{\nu }_{n}$ holds for all $x\in C$, $p\in F\left(T\right)$ 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 [59]) and total quasi-ϕ-asymptotically nonexpansive (see [1012]) 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 $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$ 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 $U=\left\{x\in E:\parallel x\parallel =1\right\}$ 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 $\left\{{x}_{n}\right\}$ of E satisfies that ${x}_{n}⇀x\in E$ 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.

In the sequel, we assume that E is a smooth, strictly convex and reflexive Banach space and C is a nonempty closed convex subset of E. We use $\varphi :E×E\to {R}^{+}$ to denote the Lyapunov functional defined by
$\varphi \left(x,y\right)={\parallel x\parallel }^{2}-2〈x,Jy〉+{\parallel y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in E.$
It is obvious that
${\left(\parallel x\parallel -\parallel y\parallel \right)}^{2}\le \varphi \left(x,y\right)\le {\left(\parallel x\parallel +\parallel y\parallel \right)}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in E,$
(1)
and
$\varphi \left(x,{J}^{-1}\left(\lambda Jy+\left(1-\lambda \right)Jz\right)\right)\le \lambda \varphi \left(x,y\right)+\left(1-\lambda \right)\varphi \left(x,z\right).$
(2)
Following Alber [16], the generalized projection ${\mathrm{\Pi }}_{C}x:E\to C$ is defined by
${\mathrm{\Pi }}_{C}x=arg\underset{y\in C}{inf}\varphi \left(y,x\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in E.$

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 $F\left(T\right)\ne \mathrm{\varnothing }$, there exist sequences $\left\{{k}_{n}\right\}\subset \left[1,+\mathrm{\infty }\right)$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty }$ such that
$\varphi \left(p,{T}^{n}x\right)\le {k}_{n}\varphi \left(p,x\right)$

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

A mapping $T:C\to C$ is said to be total quasi-ϕ-asymptotically nonexpansive, if $F\left(T\right)\ne \mathrm{\varnothing }$, there exist sequences $\left\{{\mu }_{n}\right\}$, $\left\{{\nu }_{n}\right\}$ 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 $\psi \left(0\right)=0$ such that
$\varphi \left(p,{T}^{n}x\right)\le \varphi \left(p,x\right)+{\mu }_{n}\psi \left(\varphi \left(p,x\right)\right)+{\nu }_{n}$

holds for all $x\in C$, $p\in F\left(T\right)$ 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]

One-parameter family $\mathbf{T}:=\left\{T\left(t\right):t\ge 0\right\}$ of mappings from C into itself is said to be an asymptotically nonexpansive semigroup on C, if the following conditions are satisfied:
1. (a)

$T\left(0\right)x=x$ for each $x\in C$;

2. (b)

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

3. (c)

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

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

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

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

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]

One-parameter family $\mathbf{T}:=\left\{T\left(t\right):t\ge 0\right\}$ of mappings from 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:
1. (e)
For all $x,y\in C$, $p\in F\left(\left(T\right)\right)$, $t\ge 0$, there exist sequences $\left\{{k}_{n}\right\}\subset \left[1,+\mathrm{\infty }\right)$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty }$, such that
$\varphi \left(p,{T}^{n}\left(t\right)x\right)\le {k}_{n}\varphi \left(p,x\right)$

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 $\mathbf{T}:=\left\{T\left(t\right):t\ge 0\right\}$ 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:
1. (f)
If $F\left(T\right)\ne \mathrm{\varnothing }$, there exist sequences $\left\{{\mu }_{n}\right\}$, $\left\{{\nu }_{n}\right\}$ 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 $\psi \left(0\right)=0$ such that
$\varphi \left(p,{T}^{n}\left(t\right)x\right)\le \varphi \left(p,x\right)+{\mu }_{n}\psi \left(\varphi \left(p,x\right)\right)+{\nu }_{n}$

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

A total quasi-ϕ-asymptotically nonexpansive semigroup T is said to be uniformly Lipschitzian if there exists a bounded measurable function $L:\left[0,\mathrm{\infty }\right)\to \left(0,+\mathrm{\infty }\right)$ such that
$\parallel {T}^{\left(n\right)}\left(t\right)x-{T}^{\left(n\right)}\left(t\right)y\parallel \le L\left(t\right)\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C,t\ge 0,n\in \mathbb{N}.$

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, 912, 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:
1. (i)

$\varphi \left(x,{\mathrm{\Pi }}_{C}y\right)+\varphi \left({\mathrm{\Pi }}_{C}y,y\right)\le \varphi \left(x,y\right)$ for all $x\in C$, $y\in E$;

2. (ii)

If $x\in E$ and $z\in C$, then $z={\mathrm{\Pi }}_{C}x⇔〈z-y,Jx-Jz〉\ge 0$, $\mathrm{\forall }y\in C$;

3. (iii)

For $x,y\in E$, $\varphi \left(x,y\right)=0$ if and only if $x=y$.

Lemma 2.2[19]

Let E be a uniformly convex and smooth Banach space and let$\left\{{x}_{n}\right\}$and$\left\{{y}_{n}\right\}$be two sequences of E. If$\varphi \left({x}_{n},{y}_{n}\right)\to 0$and either$\left\{{x}_{n}\right\}$or$\left\{{y}_{n}\right\}$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$F\left(T\right)$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$\mathbf{T}:=\left\{T\left(t\right):t\ge 0\right\}$be a total quasi-ϕ-asymptotically nonexpansive semigroup from C into itself defined by Definition  2.1. Suppose$\mathbf{T}:=\left\{T\left(t\right):t\ge 0\right\}$is closed, uniformly L-Lipschitz and$F\left(T\right):={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne \mathrm{\varnothing }$. Suppose there exists${M}^{\ast }>0$such that$\psi \left({\eta }_{n}\right)\le {M}^{\ast }{\eta }_{n}$. Let${\alpha }_{n}$be a sequence in$\left[0,1\right]$and${\beta }_{n}$be a sequence in$\left(0,1\right)$satisfying the following conditions: ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$, $0<{lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}<{lim sup}_{n\to \mathrm{\infty }}{\beta }_{n}<1$. Let${x}_{n}$be a sequence generated by
$\left\{\begin{array}{c}{x}_{1}\in E,\phantom{\rule{1em}{0ex}}\mathit{\text{chosen arbitrarily}};\phantom{\rule{2em}{0ex}}{C}_{1}=C,\hfill \\ {l}_{n,t}={\beta }_{n}J{x}_{n}+\left(1-{\beta }_{n}\right)J{T}^{n}\left(t\right){x}_{n},\hfill \\ {y}_{n,t}={J}^{-1}\left[{\alpha }_{n}J{x}_{1}+\left(1-{\alpha }_{n}\right){l}_{n,t}\right],\phantom{\rule{1em}{0ex}}t\ge 0,\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:{sup}_{t\ge 0}\varphi \left(z,{y}_{n,t}\right)\le {\alpha }_{n}\varphi \left(z,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(z,{x}_{n}\right)+{\xi }_{n}\right\},\hfill \\ {x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
(3)

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

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

It follows from Lemma 2.3 that $F\left(T\left(t\right)\right)$, $t\ge 0$ is a closed and convex subset of C. So $F\left(T\right)$ 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
$\begin{array}{rcl}{C}_{n+1}& =& \left\{z\in {C}_{n}:\underset{t\ge 0}{sup}\varphi \left(z,{y}_{n,t}\right)\le {\alpha }_{n}\varphi \left(z,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(z,{x}_{n}\right)+{\xi }_{n}\right\}\\ =& \bigcap _{t\ge 0}\left\{z\in C:\varphi \left(z,{y}_{n,t}\right)\le {\alpha }_{n}\varphi \left(z,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(z,{x}_{n}\right)+{\xi }_{n}\right\}\cap {C}_{n}\\ =& \bigcap _{t\ge 0}\left\{z\in C:2{\alpha }_{n}〈z,J{x}_{1}〉+2\left(1-{\alpha }_{n}\right)〈z,J{x}_{n}〉-2〈z,J{y}_{n,t}〉\\ \le {\alpha }_{n}{\parallel {x}_{1}\parallel }^{2}+\left(1-{\alpha }_{n}\right){\parallel {x}_{n}\parallel }^{2}-{\parallel {y}_{n,t}\parallel }^{2}\right\}\cap {C}_{n}.\end{array}$
This shows that ${C}_{n+1}$ is closed and convex.
1. (II)

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

In fact $F\left(T\right)\subset {C}_{1}=C$. Suppose that $F\left(T\right)\subset {C}_{n}$, $n\ge 2$. Let
${\omega }_{n,t}={J}^{-1}\left({\beta }_{n}J{x}_{n}+\left(1-{\beta }_{n}\right)J{T}^{n}\left(t\right){x}_{n}\right),\phantom{\rule{1em}{0ex}}t\ge 0.$
It follows from (2) that for any $u\in F\left(T\right)\subset {C}_{n}$, we have
$\begin{array}{rcl}\varphi \left(u,{y}_{n,t}\right)& =& \varphi \left(u,{J}^{-1}\left({\alpha }_{n}J{x}_{1}+\left(1-{\alpha }_{n}\right)J{\omega }_{n,t}\right)\right)\\ \le & {\alpha }_{n}\varphi \left(u,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(u,{\omega }_{n,t}\right),\end{array}$
and
$\begin{array}{rcl}\varphi \left(u,{\omega }_{n,t}\right)& =& \varphi \left(u,{J}^{-1}\left({\beta }_{n}J{x}_{n}+\left(1-{\beta }_{n}\right)J{T}^{n}\left(t\right){x}_{n}\right)\right)\\ \le & {\beta }_{n}\varphi \left(u,{x}_{n}\right)+\left(1-{\beta }_{n}\right)\varphi \left(u,{T}^{n}\left(t\right){x}_{n}\right)\\ \le & {\beta }_{n}\varphi \left(u,{x}_{n}\right)+\left(1-{\beta }_{n}\right)\left[\varphi \left(u,{x}_{n}\right)+{\mu }_{n}\psi \left(\varphi \left(u,{x}_{n}\right)\right)+{\nu }_{n}\right]\\ \le & \varphi \left(u,{x}_{n}\right)+\left(1-{\beta }_{n}\right)\left({\mu }_{n}{M}^{\ast }\varphi \left(u,{x}_{n}\right)+{\nu }_{n}\right).\end{array}$
Therefore, we have
$\begin{array}{rcl}\underset{t\ge 0}{sup}\varphi \left(u,{y}_{n,t}\right)& \le & {\alpha }_{n}\varphi \left(u,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\left[\varphi \left(u,{x}_{n}\right)+\left(1-{\beta }_{n}\right)\left({\mu }_{n}{M}^{\ast }\varphi \left(u,{x}_{n}\right)+{\nu }_{n}\right)\right]\\ \le & {\alpha }_{n}\varphi \left(u,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(u,{x}_{n}\right)+{\mu }_{n}{M}^{\ast }\underset{p\in F\left(T\right)}{sup}\varphi \left(p,{x}_{n}\right)+{\nu }_{n}\\ =& {\alpha }_{n}\varphi \left(u,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(u,{x}_{n}\right)+{\xi }_{n}.\end{array}$
Where ${\xi }_{n}={\mu }_{n}{M}^{\ast }{sup}_{p\in F\left(T\right)}\varphi \left(p,{x}_{n}\right)+{\nu }_{n}$. This shows that $u\in {C}_{n+1}$, so $F\left(T\right)\subset {C}_{n+1}$.
1. (III)

We prove that $\left\{{x}_{n}\right\}$ is a Cauchy sequence in C.

Since ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$, from Lemma 2.1(ii), we have
$〈{x}_{n}-y,J{x}_{1}-J{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in {C}_{n}.$
Again since $F\left(T\right)\subset {C}_{n}$, $n\ge 1$, we have
$〈{x}_{n}-u,J{x}_{1}-J{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u\in F\left(T\right).$
It follows from Lemma 2.1(i) that for each $u\in F\left(T\right)$, $n\ge 1$,
$\varphi \left({x}_{n},{x}_{1}\right)=\varphi \left({\mathrm{\Pi }}_{{C}_{n}}{x}_{1},{x}_{1}\right)\le \varphi \left(u,{x}_{1}\right)-\varphi \left(u,{x}_{n}\right)\le \varphi \left(u,{x}_{1}\right).$
Therefore, $\varphi \left({x}_{n},{x}_{1}\right)$ is bounded. By virtue of (1), ${x}_{n}$ is also bounded. Since ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$ and ${x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}$, we have $\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left({x}_{n+1},{x}_{1}\right)$. This implies that $\left\{\varphi \left({x}_{n},{x}_{1}\right)\right\}$ is nondecreasing. Hence, the limit ${lim}_{n\to \mathrm{\infty }}\varphi \left({x}_{n},{x}_{1}\right)$ exists. By the construction of ${C}_{n}$, for any positive integer $m\ge n$, we have ${C}_{m}\subset {C}_{n}$ and ${x}_{m}={\mathrm{\Pi }}_{{C}_{1}}{x}_{1}\in {C}_{n}$. This shows that
It follows from Lemma 2.2 that ${lim}_{n,m\to \mathrm{\infty }}\parallel {x}_{m}-{x}_{n}\parallel =0$. Hence ${x}_{n}$ is a Cauchy sequence in 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
$\underset{n\to \mathrm{\infty }}{lim}{\xi }_{n}=\underset{n\to \mathrm{\infty }}{lim}\left[{\mu }_{n}{M}^{\ast }\underset{p\in F\left(T\right)}{sup}\varphi \left(p,{x}_{n}\right)+{\nu }_{n}\right]=0.$
(4)
1. (IV)

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

Since ${x}_{n+1}\in {C}_{n+1}$ and ${\alpha }_{n}\to 0$, it follows from (3) and (4) that
Since ${x}_{n}\to {p}^{\ast }$, by Lemma 2.2, for each $t\ge 0$, we have
$\underset{n\to \mathrm{\infty }}{lim}{y}_{n,t}={p}^{\ast }.$
(5)
Since ${x}_{n}$ is bounded, and $\mathbf{T}=\left\{T\left(t\right),t\ge 0\right\}$ is a total quasi-ϕ-asymptotically nonexpansive semigroup with sequence ${\mu }_{n},{\nu }_{n},p\in F\left(T\right)$, we have
$\varphi \left(p,{T}^{n}\left(t\right)x\right)\le \varphi \left(p,x\right)+{\mu }_{n}\psi \left(\varphi \left(p,x\right)\right)+{\nu }_{n}\le \varphi \left(p,x\right)+{\mu }_{n}{M}^{\ast }\varphi \left(p,x\right)+{\nu }_{n}.$
This implies that ${\left\{{T}^{n}\left(t\right){x}_{n}\right\}}_{t\ge 0}$ is uniformly bounded. Since for each $t\ge 0$,
$\begin{array}{rcl}\parallel {\omega }_{n,t}\parallel & =& \parallel {J}^{-1}\left({\beta }_{n}J{x}_{n}+\left(1-{\beta }_{n}\right)J{T}^{n}\left(t\right){x}_{n}\right)\parallel \\ \le & {\beta }_{n}\parallel {x}_{n}\parallel +\left(1-{\beta }_{n}\right)\parallel {T}^{n}\left(t\right){x}_{n}\parallel \\ \le & max\left\{\parallel {x}_{n}\parallel ,\parallel {T}^{n}\left(t\right){x}_{n}\parallel \right\}.\end{array}$
This implies that $\left\{{\omega }_{n,t}\right\}$, $t\ge 0$ is also uniformly bounded. Since ${\alpha }_{n}\to 0$, from (3) we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel J{y}_{n,t}-J{\omega }_{n,t}\parallel =\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}\parallel J{x}_{1}-J{\omega }_{n,t}\parallel =0,\phantom{\rule{1em}{0ex}}t\ge 0.$
(6)
Since E is uniformly smooth, ${J}^{-1}$ is uniformly continuous on each bounded subset of ${E}^{\ast }$, it follows from (5) and (6) that
$\underset{n\to \mathrm{\infty }}{lim}{\omega }_{n,t}={p}^{\ast },\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\ge 0.$
Since ${x}_{n}\to {p}^{\ast }$ and J is uniformly continuous on each bounded subset of E, we have $J{x}_{n}\to J{p}^{\ast }$, and for each $t\ge 0$,
$\begin{array}{rcl}0& =& \underset{n\to \mathrm{\infty }}{lim}\parallel J{\omega }_{n,t}-J{p}^{\ast }\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {\beta }_{n}J{x}_{n}+\left(1-{\beta }_{n}\right)J{T}^{n}\left(t\right){x}_{n}-J{p}^{\ast }\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\parallel {\beta }_{n}\left(J{x}_{n}-J{p}^{\ast }\right)+\left(1-{\beta }_{n}\right)\left(J{T}^{n}\left(t\right){x}_{n}-J{p}^{\ast }\right)\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\left(1-{\beta }_{n}\right)\parallel \left(J{T}^{n}\left(t\right){x}_{n}-J{p}^{\ast }\right)\parallel .\end{array}$
By condition $0<{lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}<{lim sup}_{n\to \mathrm{\infty }}{\beta }_{n}<1$, we have that
Since J is uniformly continuous, this shows that ${lim}_{n\to \mathrm{\infty }}{T}^{n}\left(t\right){x}_{n}={p}^{\ast }=0$ uniformly for $t\ge 0$. Again by the assumptions that the semigroup $\mathbf{T}:=\left\{T\left(t\right):t\ge 0\right\}$ is closed and uniformly L-Lipschitzian, we have
(7)
By ${lim}_{n\to \mathrm{\infty }}{T}^{n}\left(t\right){x}_{n}={p}^{\ast }$ uniformly for $t\ge 0$, ${x}_{n}\to {p}^{\ast }$ and $L\left(t\right)$ is a bounded and measurable function, and from (7) we have that
and
so we get
By virtue of the closeness of semigroup T, we have that $T\left(t\right){p}^{\ast }={p}^{\ast }$, i.e., ${p}^{\ast }\in F\left(T\left(t\right)\right)$. By the arbitrariness of $t\ge 0$, we have ${p}^{\ast }\in F\left(T\right)={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)$.
1. (V)

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

Let $\omega ={\mathrm{\Pi }}_{F\left(T\right)}{x}_{1}$. Since $\omega \in F\left(T\right)\subset {C}_{n}$ and ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$, we get $\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left(\omega ,{x}_{1}\right)$, $n\ge 1$. This implies that
$\varphi \left({p}^{\ast },{x}_{1}\right)=\underset{n\to \mathrm{\infty }}{lim}\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left(\omega ,{x}_{1}\right).$
(8)

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

(1)
Department of Mathematics, Yibin University, Yibin, Sichuan, 644000, China
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, 650221, China

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