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

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.

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

$$J(x)=\{f^{\ast }\in E^{\ast }:〈x,f^{\ast }〉={\parallel x\parallel }^2;\parallel f^{\ast }\parallel =\parallel x\parallel ,x\in E\}$$

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.

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

$$\varphi (x,y)={\parallel x\parallel }^2-2〈x,Jy〉+{\parallel y\parallel }^2,\mathrm{\forall }x,y\in E.$$

It is obvious that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2,\mathrm{\forall }x,y\in E,$$

(1)

and

$$\varphi (x,J^{-1}(\lambda Jy+(1-\lambda )Jz))\le \lambda \varphi (x,y)+(1-\lambda )\varphi (x,z).$$

(2)

Following Alber [16], the generalized projection ${\mathrm{\Pi }}_{C}x:E\to C$ is defined by

$${\mathrm{\Pi }}_{C}x=arg\underset{y\in C}{inf}\varphi (y,x),\mathrm{\forall }x\in E.$$

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 $F(T)\ne \mathrm{\varnothing }$, there exist sequences $\{k_n\}\subset [1,+\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$\varphi (p,T^{n}x)\le k_n\varphi (p,x)$$

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

A mapping $T:C\to C$ is said to be total quasi-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

$$\varphi (p,T^{n}x)\le \varphi (p,x)+{\mu }_n\psi (\varphi (p,x))+{\nu }_n$$

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

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 $\mathbf{T}:=\{T(t):t\ge 0\}$ of mappings from 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]

One-parameter family $\mathbf{T}:=\{T(t):t\ge 0\}$ 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:

• (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 $n\in \mathbb{N}$.

A total quasi-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

$$\parallel T^{(n)}(t)x-T^{(n)}(t)y\parallel \le L(t)\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,t\ge 0,n\in \mathbb{N}.$$

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, 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 $z={\mathrm{\Pi }}_{C}x\iff 〈z-y,Jx-Jz〉\ge 0$, $\mathrm{\forall }y\in C$;

• (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

$$\{\begin{array}{c}{x}_1\in E,\mathit{\text{chosen arbitrarily}};C_1=C,\hfill \\ l_{n,t}={\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n(t)x_n,\hfill \\ y_{n,t}=J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)l_{n,t}],t\ge 0,\hfill \\ C_{n+1}=\{z\in C_n:{sup}_{t\ge 0}\varphi (z,y_{n,t})\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3)

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.

It follows from Lemma 2.3 that $F(T(t))$, $t\ge 0$ is a closed and convex subset of 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

$$\begin{array}{rcl}{C}_{n+1}& =& \{z\in C_n:\underset{t\ge 0}{sup}\varphi (z,y_{n,t})\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\}\\ =& \bigcap _{t\ge 0}\{z\in C:\varphi (z,y_{n,t})\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\}\cap C_n\\ =& \bigcap _{t\ge 0}\{z\in C:2{\alpha }_n〈z,J{x}_1〉+2(1-{\alpha }_n)〈z,J{x}_n〉-2〈z,J{y}_{n,t}〉\\ \le {\alpha }_n{\parallel x_1\parallel }^2+(1-{\alpha }_n){\parallel x_n\parallel }^2-{\parallel y_{n,t}\parallel }^2\}\cap C_n.\end{array}$$

This shows that $C_{n+1}$ is closed and convex.

• (II) We prove that $F(T)\subset C_n$.

In fact $F(T)\subset C_1=C$. Suppose that $F(T)\subset C_n$, $n\ge 2$. Let

$${\omega }_{n,t}=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n(t)x_n),t\ge 0.$$

It follows from (2) that for any $u\in F(T)\subset C_n$, we have

$$\begin{array}{rcl}\varphi (u,y_{n,t})& =& \varphi (u,J^{-1}({\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{\omega }_{n,t}))\\ \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,{\omega }_{n,t}),\end{array}$$

and

$$\begin{array}{rcl}\varphi (u,{\omega }_{n,t})& =& \varphi (u,J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n(t)x_n))\\ \le & {\beta }_n\varphi (u,x_n)+(1-{\beta }_n)\varphi (u,T^n(t)x_n)\\ \le & {\beta }_n\varphi (u,x_n)+(1-{\beta }_n)[\varphi (u,x_n)+{\mu }_n\psi (\varphi (u,x_n))+{\nu }_n]\\ \le & \varphi (u,x_n)+(1-{\beta }_n)({\mu }_n{M}^{\ast }\varphi (u,x_n)+{\nu }_n).\end{array}$$

Therefore, we have

$$\begin{array}{rcl}\underset{t\ge 0}{sup}\varphi (u,y_{n,t})& \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)[\varphi (u,x_n)+(1-{\beta }_n)({\mu }_n{M}^{\ast }\varphi (u,x_n)+{\nu }_n)]\\ \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,x_n)+{\mu }_n{M}^{\ast }\underset{p\in F(T)}{sup}\varphi (p,x_n)+{\nu }_n\\ =& {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,x_n)+{\xi }_n.\end{array}$$

Where ${\xi }_n={\mu }_n{M}^{\ast }{sup}_{p\in F(T)}\varphi (p,x_n)+{\nu }_n$. This shows that $u\in C_{n+1}$, so $F(T)\subset C_{n+1}$.

• (III) We prove that $\{x_n\}$ is a Cauchy sequence in C.

Since $x_n={\mathrm{\Pi }}_{C_n}{x}_1$, from Lemma 2.1(ii), we have

$$〈x_n-y,J{x}_1-J{x}_n〉\ge 0,\mathrm{\forall }y\in C_n.$$

Again since $F(T)\subset C_n$, $n\ge 1$, we have

$$〈x_n-u,J{x}_1-J{x}_n〉\ge 0,\mathrm{\forall }u\in F(T).$$

It follows from Lemma 2.1(i) that for each $u\in F(T)$, $n\ge 1$,

$$\varphi (x_n,x_1)=\varphi ({\mathrm{\Pi }}_{C_n}{x}_1,x_1)\le \varphi (u,x_1)-\varphi (u,x_n)\le \varphi (u,x_1).$$

Therefore, $\varphi (x_n,x_1)$ 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 $\varphi (x_n,x_1)\le \varphi (x_{n+1},x_1)$. This implies that $\{\varphi (x_n,x_1)\}$ is nondecreasing. Hence, the limit ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_1)$ 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

$$\begin{array}{rcl}\varphi (x_m,x_n)& =& \varphi (x_m,{\mathrm{\Pi }}_{C_n}{x}_1)\\ \le & \varphi (x_m,x_1)-\varphi (x_n,x_1)\to 0,\text{as }m,n\to \mathrm{\infty }.\end{array}$$

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

$$\underset{n\to \mathrm{\infty }}{lim}{\xi }_n=\underset{n\to \mathrm{\infty }}{lim}[{\mu }_n{M}^{\ast }\underset{p\in F(T)}{sup}\varphi (p,x_n)+{\nu }_n]=0.$$

(4)

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

Since $x_{n+1}\in C_{n+1}$ and ${\alpha }_n\to 0$, it follows from (3) and (4) that

$$\underset{t\ge 0}{sup}\varphi (x_{n+1},y_{n,t})\le {\alpha }_n\varphi (x_{n+1},x_1)+(1-{\alpha }_n)\varphi (x_{n+1},x_n)+{\xi }_n\to 0\text{as }n\to \mathrm{\infty }.$$

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 $\mathbf{T}=\{T(t),t\ge 0\}$ is a total quasi-ϕ-asymptotically nonexpansive semigroup with sequence ${\mu }_n,{\nu }_n,p\in F(T)$, we have

$$\varphi (p,T^n(t)x)\le \varphi (p,x)+{\mu }_n\psi (\varphi (p,x))+{\nu }_n\le \varphi (p,x)+{\mu }_n{M}^{\ast }\varphi (p,x)+{\nu }_n.$$

This implies that ${\{T^n(t)x_n\}}_{t\ge 0}$ is uniformly bounded. Since for each $t\ge 0$,

$$\begin{array}{rcl}\parallel {\omega }_{n,t}\parallel & =& \parallel J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n(t)x_n)\parallel \\ \le & {\beta }_n\parallel x_n\parallel +(1-{\beta }_n)\parallel T^n(t)x_n\parallel \\ \le & max\{\parallel x_n\parallel ,\parallel T^n(t)x_n\parallel \}.\end{array}$$

This implies that $\{{\omega }_{n,t}\}$, $t\ge 0$ is also uniformly bounded. Since ${\alpha }_n\to 0$, from (3) we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{y}_{n,t}-J{\omega }_{n,t}\parallel =\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n\parallel J{x}_1-J{\omega }_{n,t}\parallel =0,t\ge 0.$$

(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

$$\underset{n\to \mathrm{\infty }}{lim}{\omega }_{n,t}=p^{\ast },\mathrm{\forall }t\ge 0.$$

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 $t\ge 0$,

$$\begin{array}{rcl}0& =& \underset{n\to \mathrm{\infty }}{lim}\parallel J{\omega }_{n,t}-J{p}^{\ast }\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n(t)x_n-J{p}^{\ast }\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\parallel {\beta }_n(J{x}_n-J{p}^{\ast })+(1-{\beta }_n)(J{T}^n(t)x_n-J{p}^{\ast })\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}(1-{\beta }_n)\parallel (J{T}^n(t)x_n-J{p}^{\ast })\parallel .\end{array}$$

By condition $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n<{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$, we have that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{T}^n(t)x_n-J{p}^{\ast }\parallel =0,\text{uniformly for }t\ge 0.$$

Since 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

(7)

By ${lim}_{n\to \mathrm{\infty }}{T}^n(t)x_n=p^{\ast }$ uniformly for $t\ge 0$, $x_n\to p^{\ast }$ and $L(t)$ is a bounded and measurable function, and from (7) we have that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel T^{n+1}(t)x_n-T^n(t)x_n\parallel =0\text{uniformly for }t\ge 0,$$

and

$$\underset{n\to \mathrm{\infty }}{lim}{T}^{n+1}(t)x_n=p^{\ast }\text{uniformly for }t\ge 0,$$

so we get

$$\underset{n\to \mathrm{\infty }}{lim}T(t)T^n(t)x_n=p^{\ast }\text{uniformly for }t\ge 0.$$

By virtue of the closeness of semigroup 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 $x_n\to p^{\ast }={\mathrm{\Pi }}_{F(T)}{x}_1$.

Let $\omega ={\mathrm{\Pi }}_{F(T)}{x}_1$. Since $\omega \in F(T)\subset C_n$ and $x_n={\mathrm{\Pi }}_{C_n}{x}_1$, we get $\varphi (x_n,x_1)\le \varphi (\omega ,x_1)$, $n\ge 1$. This implies that

$$\varphi (p^{\ast },x_1)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,x_1)\le \varphi (\omega ,x_1).$$

(8)

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. □