Total quasi-ϕ-asymptotically nonexpansive semigroups and strong convergence theorems in Banach spaces

Abstract

The purpose of this article is to modify the Halpern-Mann-type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding recent results announced by many authors.

MSC:47J05, 47H09, 49J25.

1 Introduction

Throughout this paper, we assume that E is a real Banach space with the dual $E^{\ast }$, C is a nonempty closed convex subset of E, and $J:E\to 2^{E^{\ast }}$ is the normalized duality mapping defined by

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

Let $T:C\to E$ be a nonlinear mapping; we denote by $F(T)$ the set of fixed points of T.

Recall that a mapping $T:C\to C$ is said to be nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

$T:C\to C$ is said to be quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and

$$\parallel Tx-p\parallel \le \parallel x-p\parallel ,\mathrm{\forall }x\in C,p\in F(T).$$

$T:C\to C$ is said to be asymptotically nonexpansive if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that

$$\parallel T^{n}x-T^{n}y\parallel \le k_n\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,n\ge 1.$$

$T:C\to C$ is said to be quasi-asymptotically nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that

$$\parallel T^{n}x-p\parallel \le k_n\parallel x-p\parallel ,\mathrm{\forall }x\in C,p\in F(T),n\ge 1.$$

A one-parameter family $\mathcal{T}:=\{T(t):t\ge 0\}$ of mappings from C into C is said to be a nonexpansive semigroup if the following conditions are satisfied:

• (i) $T(0)x=x$ for all $x\in C$;

• (ii) $T(s+t)=T(s)T(t)$, $\mathrm{\forall }s,t\ge 0$;

• (iii) for each $x\in C$, the mapping $t\mapsto T(t)x$ is continuous;

• (iv) $\parallel T(t)x-T(t)y\parallel \le \parallel x-y\parallel$, $\mathrm{\forall }x,y\in C$.

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

A one-parameter family $\mathcal{T}:=\{T(t):t\ge 0\}$ of mappings from C into C is said to be a quasi-nonexpansive semigroup if $F(\mathcal{T})\ne \mathrm{\varnothing }$ and the above conditions (i)-(iii) and the following condition (v) are satisfied:

• (v) $\parallel T(t)x-p\parallel \le \parallel x-p\parallel$, $\mathrm{\forall }x\in C$, $p\in F(\mathcal{T})$, $t\ge 0$.

A one-parameter family $\mathcal{T}:=\{T(t):t\ge 0\}$ of mappings from C into C is said to be an asymptotically nonexpansive semigroup if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that the above conditions (i)-(iii) and the following condition (vi) are satisfied:

• (vi) $\parallel T^n(t)x-T^n(t)y\parallel \le k_n\parallel x-y\parallel$, $\mathrm{\forall }x,y\in C$, $n\ge 1$, $t\ge 0$.

A one-parameter family $\mathcal{T}:=\{T(t):t\ge 0\}$ of mappings from C into C is said to be a quasi-asymptotically nonexpansive semigroup if $F(\mathcal{T})\ne \mathrm{\varnothing }$ and there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that the above conditions (i)-(iii) and the following condition (vii) are satisfied:

• (vii) $\parallel T^n(t)x-p\parallel \le k_n\parallel x-p\parallel$, $\mathrm{\forall }x\in C$, $p\in F(\mathcal{T})$, $t\ge 0$, $n\ge 1$.

As is well known, the construction of fixed points of nonexpansive mappings (asymptotically nonexpansive mappings) and of common fixed points of nonexpansive semi-groups (asymptotically nonexpansive semigroups) is an important problem in the theory of nonexpansive mappings and its applications; in particular, in image recovery, convex feasibility problem, and signal processing problem (see, for example, [1–3]).

Iterative approximation of a fixed point for nonexpansive mappings, asymptotically nonexpansive mappings, nonexpansive semigroups, and asymptotically nonexpansive semigroups in Hilbert or Banach spaces has been studied extensively by many authors (see, for example, [4–31] and the references therein).

The purpose of this article is to introduce the concept of total quasi-ϕ-asymptotically nonexpansive semigroups; to modify the Halpern and Mann-type iteration algorithm [13, 14] for total quasi-ϕ-asymptotically nonexpansive semigroups; and to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of Kim [32], Suzuki [4], Xu [5], Chang et al. [6–8, 22, 23, 30, 33], Cho et al. [10], Thong [11], Buong [12], Mann [13], Halpern [14], Qin et al. [15], Nakajo et al. [18] and others.

2 Preliminaries

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. In what follows, we always use $\varphi :E\times E\to {\mathcal{R}}^{+}$ to denote the Lyapunov functional defined by

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

(2.1)

It is obvious from the definition of ϕ that

(2.2)

(2.3)

and

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

(2.4)

Following Alber [34], the generalized projection ${\mathrm{\Pi }}_C:E\to C$ is defined by

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

Lemma 2.1 ([34])

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:

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

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

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

Remark 2.2 If E is a real Hilbert space H, then $\varphi (x,y)={\parallel x-y\parallel }^2$ and ${\mathrm{\Pi }}_C=P_C$ (the metric projection of H onto C).

Definition 2.3 A mapping $T:C\to C$ is said to be closed if, for any sequence $\{x_n\}\subset C$ with $x_n\to x$ and $T{x}_n\to y$, $Tx=y$.

Definition 2.4 (1) A mapping $T:C\to C$ is said to be quasi-ϕ-nonexpansive, if $F(T)\ne \mathrm{\varnothing }$ and

$$\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,p\in F(T).$$

• (2) A mapping $T:C\to C$ is said to be $(\{k_n\})$-quasi-ϕ-asymptotically nonexpansive, if $F(T)\ne \mathrm{\varnothing }$ and there exists a real sequence $\{k_n\}\subset [1,\mathrm{\infty })$, $k_n\to 1$ such that

  $$\varphi (p,T^{n}x)\le k_n\varphi (p,x),\mathrm{\forall }n\ge 1,x\in C,p\in F(T).$$

• (3) A mapping $T:C\to C$ is said to be $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and there exist nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$) and a strictly increasing continuous function $\zeta :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that

  $$\varphi (p,T^{n}x)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n,\mathrm{\forall }n\ge 1,x\in C,p\in F(T).$$

  (2.5)

Remark 2.5 ([22])

From the definitions, it is obvious that a quasi-ϕ-nonexpansive mapping is a $(\{k_n=1\})$-quasi-ϕ-asymptotically nonexpansive mapping and a $(\{k_n\})$-quasi-ϕ-asymptotically nonexpansive mapping is a $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive mapping with ${\nu }_n=k_n-1$, ${\mu }_n=0$, $\zeta (t)=t$, $\mathrm{\forall }t\ge 0$. However, the converse is not true.

Example 2.6 ([23])

Let E be a uniformly smooth and strictly convex Banach space and $A:E\to E^{\ast }$ be a maximal monotone mapping such that $A^{-1}0\ne \mathrm{\varnothing }$, then $J_r={(J+rA)}^{-1}J$ is closed and quasi-ϕ-nonexpansive from E onto $D(A)$.

Example 2.7 ([30])

• (1) Let C be a unit ball in a real Hilbert space $l^2$ and let $T:C\to C$ be a mapping defined by

  $$T:(x_1,x_2,\dots )\to (0,x_1^2,a_2{x}_2,a_3{x}_3,\dots ),(x_1,x_2,\dots )\in l^2,$$

  (2.6)

where $\{a_i\}$ is a sequence in $(0,1)$ such that ${\mathrm{\Pi }}_{i=2}^{\mathrm{\infty }}{a}_i=\frac{1}{2}$. It is proved in [30] that T is (single-valued) total quasi-ϕ-asymptotically nonexpansive.

• (2) Let $I=[0,1]$, $X=C(I)$ (the Banach space of continuous functions defined on I with the uniform convergence norm ${\parallel f\parallel }_C={sup}_{t\in I}|f(t)|$), $D=\{f\in X:f(x)\ge 0,\mathrm{\forall }x\in I\}$ and a, b be two constants in $(0,1)$ with $a<b$. Let $T:D\to 2^D$ be a multi-valued mapping defined by

  $$T(f)=\{\begin{array}{cc}\{g\in D:a\le f(x)-g(x)\le b,\mathrm{\forall }x\in I\},\hfill & \text{if }f(x)>1,\mathrm{\forall }x\in I;\hfill \\ \{0\},\hfill & \text{otherwise}.\hfill \end{array}$$

  (2.7)

It is proved that $T:C\to 2^C$ is a multi-valued total quasi-ϕ-asymptotically nonexpansive mapping.

Example 2.8 Let ${\mathrm{\Pi }}_C$ be the generalized projection from a smooth, reflexive, and strictly convex Banach space E onto a nonempty closed convex subset C of E, then ${\mathrm{\Pi }}_C$ is a closed and quasi-ϕ-nonexpansive from E onto C.

Lemma 2.9 Let E be a smooth, reflexive, and strictly convex real Banach space such that both E and $E^{\ast }$ have the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let $T:C\to C$ be a closed and $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive mapping, then $F(T)$ is a closed convex subset of C.

Proof Let $\{x_n\}$ be any sequence in $F(T)$ such that $x_n\to x^{\ast }$. Now, we prove that $x^{\ast }\in F(T)$. In fact, since $T{x}_n=x_n\to x^{\ast }$ and T is closed, we have $x^{\ast }=T{x}^{\ast }$, i.e., $F(T)$ is a closed subset in C.

Next, we prove that $F(T)$ is convex. In fact, let $x,y\in F(T)$ and $p=tx+(1-t)y$, where $t\in (0,1)$. Now, we prove that $p=Tp$. Indeed, since T is $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive, for any $n\ge 1$, we have

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

(2.8)

and

$$\varphi (y,T^{n}p)\le \varphi (y,p)+{\nu }_n\zeta (\varphi (y,p))+{\mu }_n.$$

(2.9)

On the other hand, it follows from (2.4) that

$$\varphi (x,T^{n}p)=\varphi (x,p)+\varphi (p,T^{n}p)+2〈x-pJp-J{T}^{n}p〉$$

(2.10)

and

$$\varphi (y,T^{n}p)=\varphi (y,p)+\varphi (p,T^{n}p)+2〈y-pJp-J{T}^{n}p〉.$$

(2.11)

It follows from (2.8)-(2.11) that

$$\begin{array}{rcl}\varphi (p,T^{n}p)& =& 2〈p-xJp-J{T}^{n}p〉+\varphi (x,T^{n}p)-\varphi (x,p)\\ \le & 2〈p-xJp-J{T}^{n}p〉+{\nu }_n\zeta (\varphi (x,p))+{\mu }_n\end{array}$$

(2.12)

and

$$\begin{array}{rcl}\varphi (p,T^{n}p)& =& 2〈p-yJp-J{T}^{n}p〉+\varphi (y,T^{n}p)-\varphi (y,p)\\ \le & 2〈p-yJp-J{T}^{n}p〉+{\nu }_n\zeta (\varphi (y,p))+{\mu }_n.\end{array}$$

(2.13)

Multiplying t and $(1-t)$ on both sides of (2.12) and (2.13), respectively and then adding up these two inequalities, we have that

$$\varphi (p,T^{n}p)\le t{\nu }_n\zeta (\varphi (x,p))+(1-t){\nu }_n\zeta (\varphi (y,p))+{\mu }_n.$$

Letting $n\to \mathrm{\infty }$, we have that $\varphi (p,T^{n}p)\to 0$. Hence, it follows from (2.2) that

$$\parallel T^{n}p\parallel \to \parallel p\parallel ,$$

(2.14)

and so

$$\parallel J\left(T^{n}p\right)\parallel \to \parallel Jp\parallel .$$

(2.15)

E is reflexive and so is $E^{\ast }$. Without loss of generality, we can assume that $J(T^{n}p)\rightharpoonup x^{\ast }$ (some point in $E^{\ast }$). In view of the reflexivity of E, we have $J(E)=E^{\ast }$. This shows that there exists an element $x\in E$ such that $Jx=x^{\ast }$. Hence, we have

$$\begin{array}{rcl}\varphi (p,T^{n}p)& =& {\parallel p\parallel }^2-2〈p,J\left(T^{n}p\right)〉+{\parallel T^{n}p\parallel }^2\\ =& {\parallel p\parallel }^2-2〈p,J\left(T^{n}p\right)〉+{\parallel J\left(T^{n}p\right)\parallel }^2.\end{array}$$

Taking ${lim}_{n\to \mathrm{\infty }}$ on both sides of the equality above, we obtain that

$$\begin{array}{rcl}0& =& {\parallel p\parallel }^2-2〈p,Jx〉+{\parallel Jp\parallel }^2\\ =& {\parallel p\parallel }^2-2〈p,Jx〉+{\parallel p\parallel }^2\\ =& 2\{{\parallel p\parallel }^2-〈p,Jx〉\}\\ =& 2〈p,Jp-Jx〉.\end{array}$$

This implies that $Jp-Jx=0$. Therefore, we have $J(T^{n}p)\rightharpoonup Jp$. Since $E^{\ast }$ has the Kadec-Klee property, this together with (2.15) shows that $J(T^{n}p)\to Jp$. Since E is reflexive and strictly convex, $J^{-1}$ is norm-weak-continuous, $T^{n}p\rightharpoonup p$. Again, since E has the Kadec-Klee property, this together with (2.14) shows that $T^{n}p\to p$ (as $n\to \mathrm{\infty }$). Therefore, $T{T}^{n}p=T^{n+1}p\to p$. By virtue of the closeness of T, it follows that $p=Tp$, i.e., $p\in F(T)$. The convexity of $F(T)$ is proved.

This completes the proof of Lemma 2.9. □

Definition 2.10 (I) Let E be a real Banach space, C be a nonempty closed convex subset of E. $\mathcal{T}:=\{T(t):t\ge 0\}$ be a one-parameter family of mappings from C into C. $\mathcal{T}$ is said to be

• (1) a quasi-ϕ-nonexpansive semigroup if $\mathcal{F}={\bigcap }_{t\ge 0}F(T(t))\ne \mathrm{\varnothing }$ and the following conditions are satisfied:

• (i) $T(0)x=x$ for all $x\in C$;

• (ii) $T(s+t)=T(s)T(t)$ for all $s,t\ge 0$;

• (iii) for each $x\in C$, the mapping $t\mapsto T(t)x$ is continuous;

• (iv) $\varphi (p,T(t)x)\le \varphi (p,x)$, $\mathrm{\forall }t\ge 0$, $p\in \mathcal{F}$, $x\in C$.

• (2) $\mathcal{T}$ is said to be a $(\{k_n\})$-quasi-ϕ-asymptotically nonexpansive semigroup if the set $\mathcal{F}={\bigcap }_{t\ge 0}F(T(t))$ is nonempty, and there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that the conditions (i)-(iii) and the following condition (v) are satisfied:

• (v) $\varphi (p,T^n(t)x)\le k_n\varphi (p,x)$, $\mathrm{\forall }t\ge 0$, $p\in \mathcal{F}$, $n\ge 1$, $x\in C$.

• (3) $\mathcal{T}$ is said to be a $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive semigroup if the set $\mathcal{F}={\bigcap }_{t\ge 0}F(T(t))$ is nonempty, and there exists nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ (as $n\to \mathrm{\infty }$) and a strictly increasing continuous function $\zeta :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\zeta (0)=0$ such that the conditions (i)-(iii) and the following condition (vi) are satisfied:

• (vi) $\varphi (p,T^n(t)x)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n$, $\mathrm{\forall }n\ge 1$, $x\in C$, $p\in F(T)$.

• (II) A total quasi-ϕ-asymptotically nonexpansive semigroup $\mathcal{T}$ is said to be uniformly Lipschitzian if there exists a bounded measurable function $L:[0,\mathrm{\infty })\to (0,\mathrm{\infty })$ such that

  $$\parallel T^n(t)x-T^n(t)y\parallel \le L(t)\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,\mathrm{\forall }n\ge 1,t\ge 0.$$

3 Main results

Theorem 3.1 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and $E^{\ast }$ have the Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let $\mathcal{T}:=\{T(t):t\ge 0\}$ be a closed, uniformly L-Lipschitz and $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive semigroup. 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) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$.

Let $\{x_n\}$ be a sequence generated by

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

(3.1)

where $\mathcal{F}:={\bigcap }_{t\ge 0}^{\mathrm{\infty }}F(T(t))$, ${\xi }_n={\nu }_n{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,x_n))+{\mu }_n$, ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$. If $\mathcal{F}$ is bounded in C, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_1$.

Proof (I) First, we prove that $\mathcal{F}$ and $C_n$, $n\ge 1$ all are closed and convex subsets in C.

In fact, it follows from Lemma 2.9 that $F(T(t))$, $t\ge 0$ is a closed and convex subset of C. Therefore, $\mathcal{F}$ is closed and convex in C.

Again, by the assumption that $C_1=C$ is closed and convex, suppose that $C_n$ is closed and convex for some $n\ge 2$. In view of the definition of ϕ, we have that

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

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

• (II) Now, we prove that $\mathcal{F}\subset C_n$, $\mathrm{\forall }n\ge 1$.

In fact, it is obvious that $\mathcal{F}\subset C_1=C$. Suppose that $\mathcal{F}\subset C_n$ for some $n\ge 2$. Letting

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

it follows from (2.3) that for any $u\in \mathcal{F}\subset C_n$, we have

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

(3.2)

and

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

(3.3)

Therefore, we have

$$\begin{array}{rcl}\underset{t\ge 0}{sup}\varphi (u,y_{n,t})& \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\{\varphi (u,x_n)+{\nu }_n\zeta (\varphi (u,x_n))+{\mu }_n\}\\ \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,x_n)+{\nu }_n\underset{p\in \mathcal{F}}{sup}\zeta (\varphi (p,x_n))+{\mu }_n\\ =& {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,x_n)+{\xi }_n,\end{array}$$

where ${\xi }_n={\nu }_n{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,x_n))+{\mu }_n$. This shows that $u\in C_{n+1}$, and so $\mathcal{F}\subset C_{n+1}$. The conclusion is proved.

• (III) Next, we prove that $\{x_n\}$ is bounded and $\{\varphi (x_n,x_1)\}$ is convergent.

In fact, since $x_n={\mathrm{\Pi }}_{C_n}{x}_1$, from Lemma 2.1(b), we have

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

Again, since $\mathcal{F}\subset C_n$, $\mathrm{\forall }n\ge 1$, we have

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

It follows from Lemma 2.1(a) that for each $u\in \mathcal{F}$ and for each $n\ge 1$,

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

(3.4)

Therefore, $\{\varphi (x_n,x_1)\}$ is bounded. By virtue of (2.2), $\{x_n\}$ is also bounded.

Again, since $x_n={\mathrm{\Pi }}_{C_n}{x}_1$, $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1$, and $x_{n+1}\in C_{n+1}\subset C_n$, $\mathrm{\forall }n\ge 1$, we have

$$\varphi (x_n,x_1)\le \varphi (x_{n+1},x_1),\mathrm{\forall }n\ge 1.$$

This implies that $\{\varphi (x_n,x_1)\}$ is nondecreasing and bounded. Hence, ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_1)$ exists. The conclusions are proved.

• (IV) Next, we prove that $x_n\to p^{\ast }$ (some point in C).

In fact, since $\{x_n\}$ is bounded and the space E is reflexive, we may assume that there exists a subsequence of $\{x_{n_i}\}$ such that $x_{n_i}\rightharpoonup p^{\ast }$. Since $C_n$, $\mathrm{\forall }n\ge 1$ is closed and convex, we see that $p^{\ast }\in C_n$, $\mathrm{\forall }n\ge 1$. This implies that $\varphi (x_{n_i},x_1)\le \varphi (p^{\ast },x_1)$, $\mathrm{\forall }{n}_i$. On the other hand, it follows from the weakly lower semicontinuity of the norm that

$$\begin{array}{rcl}\varphi (p^{\ast },x_1)& =& {\parallel p^{\ast }\parallel }^2-2〈p^{\ast },J{x}_1〉+{\parallel x_1\parallel }^2\\ \le & \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\{{\parallel x_{n_i}\parallel }^2-2〈x_{n_i},J{x}_1〉+{\parallel x_1\parallel }^2\}\\ =& \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (x_{n_i},x_1)\\ \le & \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (x_{n_i},x_1)\\ \le & \varphi (p^{\ast },x_1),\end{array}$$

which implies that $\varphi (x_{n_i},x_1)\to \varphi (p^{\ast },x_1)$ (as $n_i\to \mathrm{\infty }$). Hence, $\parallel x_{n_i}\parallel \to \parallel p^{\ast }\parallel$ (as $n_i\to \mathrm{\infty }$). In view of the Kadec-Klee property of E, we see that $x_{n_i}\to p^{\ast }$ (as $n_i\to \mathrm{\infty }$).

If there exists another subsequence $\{x_{n_j}\}\subset \{x_n\}$ such that $x_{n_j}\to q^{\ast }\in C$, we have

$$\begin{array}{rcl}\varphi (p^{\ast },q^{\ast })& =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},x_{n_j})\\ =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},{\mathrm{\Pi }}_{C_{n_j}}{x}_1)\\ \le & \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},x_1)-\varphi ({\mathrm{\Pi }}_{C_{n_j}}{x}_1,x_1)\\ =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},x_1)-\varphi (x_{n_j},x_1)=0,\end{array}$$

i.e., $p^{\ast }=q^{\ast }$. This shows that $x_n\to p^{\ast }$. Therefore, we have

$$\underset{n\to \mathrm{\infty }}{lim}{\xi }_n=\underset{n\to \mathrm{\infty }}{lim}\{{\nu }_n\underset{p\in \mathcal{F}}{sup}\zeta (\varphi (p,x_n))+{\mu }_n\}=0.$$

(3.5)

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

In fact, since $x_{n+1}\in C_{n+1}$, $x_n\to p^{\ast }$ and ${\alpha }_n\to 0$, it follows from (3.1) and (3.5) that

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

(3.6)

This implies that for each $t\ge 0$,

$$\underset{n\to \mathrm{\infty }}{lim}{(\parallel y_{n,t}\parallel -\parallel x_{n+1}\parallel )}^2=0.$$

(3.7)

Therefore,

$$\parallel y_{n,t}\parallel \to \parallel p^{\ast }\parallel ,\text{uniformly in }t\ge 0,$$

(3.8)

and so

$$\parallel J(y_{n,t})\parallel \to \parallel J{p}^{\ast }\parallel ,\text{uniformly in }t\ge 0.$$

(3.9)

This shows that $\{J(y_{n,t})\}$ is uniformly bounded. E is reflexive and so is $E^{\ast }$. Without loss of generality, we can assume that $J(y_{n,t})\rightharpoonup y^{\ast }$ (some point in $E^{\ast }$). Since E is reflexive, $J(E)=E^{\ast }$. Hence, there exists $y\in E$ such that $Jy=y^{\ast }$. This implies that $J(y_{n,t})\rightharpoonup Jy$. Since

$$\begin{array}{rcl}\varphi (x_{n+1},y_{n,t})& =& {\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{y}_{n,t}〉+{\parallel y_{n,t}\parallel }^2\\ =& {\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{y}_{n,t}〉+{\parallel J{y}_{n,t}\parallel }^2.\end{array}$$

Letting $n\to \mathrm{\infty }$, from (3.6), we have

$$\begin{array}{rcl}0& =& {\parallel p^{\ast }\parallel }^2-2〈p^{\ast },Jy〉+{\parallel J{p}^{\ast }\parallel }^2\\ =& {\parallel p^{\ast }\parallel }^2-2〈p^{\ast },Jy〉+{\parallel p^{\ast }\parallel }^2\\ =& 2〈p^{\ast },J{p}^{\ast }-Jy〉,\end{array}$$

which shows that $J{p}^{\ast }=Jy$, and so

$$J(y_{n,t})\rightharpoonup J{p}^{\ast }.$$

(3.10)

This together with (3.9) and the Kadec-Klee property of $E^{\ast }$ shows that $J(y_{n,t})\to J{p}^{\ast }$. Since $J^{-1}$ is norm-weak-continuous, we have

$$y_{n,t}\rightharpoonup p^{\ast }.$$

(3.11)

It follows from (3.8), (3.11) and the Kadec-Klee property of E, we have

$$y_{n,t}\to p^{\ast },\text{uniformly in }t\ge 0.$$

(3.12)

On the other hand, since $\{x_n\}$ is bounded and $\mathcal{T}:=\{T(t):t\ge 0\}$ is a $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive semigroup, for any given $p\in \mathcal{F}$, we have

$$\varphi (p,T^n(t)x_n)\le \varphi (p,x_n)+{\nu }_n\zeta (\varphi (p,x_n))+{\mu }_n,\mathrm{\forall }t\ge 0,n\ge 1.$$

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

$$\begin{array}{rcl}\parallel w_{n,t}\parallel & =& \parallel J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n(t)x_n)\parallel \\ \le & {\beta }_n\parallel x_n\parallel +(1-{\beta }_n)\parallel T^n(t)x_n\parallel \\ \le & max\{\parallel x_n\parallel ,\parallel T^n(t)x_n\parallel \},t\ge 0,\end{array}$$

it implies that ${\{w_{n,t}\}}_{t\ge 0}$ is also uniformly bounded.

Since ${\alpha }_n\to 0$, from (3.1), we have

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

(3.13)

It follows from (3.12) that $J{w}_{n,t}\to J{p}^{\ast }$ (as $n\to \mathrm{\infty }$), uniformly in $t\ge 0$. Therefore, we have

$$w_{n,t}\rightharpoonup p^{\ast }.$$

(3.14)

Since

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}|\parallel w_{n,t}\parallel -\parallel p^{\ast }\parallel |& =& \underset{n\to \mathrm{\infty }}{lim}|\parallel J(w_{n,t})\parallel -\parallel J\left(p^{\ast }\right)\parallel |\\ \le & \underset{n\to \mathrm{\infty }}{lim}\parallel J(w_{n,t})-J\left(p^{\ast }\right)\parallel =0.\end{array}$$

This together with (3.14) shows that

$$w_{n,t}\to p^{\ast }(\text{as }n\to \mathrm{\infty }),\text{ uniformly in }t\ge 0.$$

(3.15)

Since $x_n\to p^{\ast }$, we have $J{x}_n\to J{p}^{\ast }$, and so for each $t\ge 0$,

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

By condition (ii), we have that

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

(3.16)

Since $J^{-1}$ is norm-weakly-continuous, this implies that

$$T^n(t)x_n\rightharpoonup p^{\ast },\text{for each }t\ge 0.$$

(3.17)

It follows from (3.16) that for each $t\ge 0$,

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}|\parallel T^n(t)x_n\parallel -\parallel p^{\ast }\parallel |& =& \underset{n\to \mathrm{\infty }}{lim}|\parallel J(T^n(t)x_n)\parallel -\parallel J\left(p^{\ast }\right)\parallel |\\ \le & \parallel J(T^n(t)x_n)-J\left(p^{\ast }\right)\parallel =0.\end{array}$$

This together with (3.17) and the Kadec-Klee property of E shows that

$$T^n(t)x_n\to p^{\ast }(\text{as }n\to \mathrm{\infty })\text{ uniformly in }t\ge 0.$$

Again, by the assumptions that the semigroup $\mathcal{T}:=\{T(t):t\ge 0\}$ is closed and uniformly L-Lipschitzian, we have

$$\begin{array}{rcl}\parallel T^{n+1}(t)x_n-T^n(t)x_n\parallel & \le & \parallel T^{n+1}(t)x_n-T^{n+1}(t)x_{n+1}\parallel +\parallel T^{n+1}(t)x_{n+1}-x_{n+1}\parallel \\ +\parallel x_{n+1}-x_n\parallel +\parallel x_n-T^n(t)x_n\parallel \\ \le & (L(t)+1)\parallel x_{n+1}-x_n\parallel +\parallel T^{n+1}(t)x_{n+1}-x_{n+1}\parallel \\ +\parallel x_n-T^n(t)x_n\parallel .\end{array}$$

(3.18)

Since ${lim}_{n\to \mathrm{\infty }}{T}^n(t)x_n=p^{\ast }$ uniformly in $t\ge 0$, $x_n\to p^{\ast }$ and $L(t):[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a bounded and measurable function, these together with (3.9) imply that

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

and so

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

i.e.,

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

In view of the closeness of the semigroup $\mathcal{T}$, it yields that $T(t)p^{\ast }=p^{\ast }$, i.e., $p^{\ast }\in F(T(t))$. By the arbitrariness of $t\ge 0$, we have $p^{\ast }\in \mathcal{F}:={\bigcap }_{t\ge 0}F(T(t))$.

• (VI) Finally, we prove that $x_n\to p^{\ast }={\mathrm{\Pi }}_{\mathcal{F}}{x}_1$.

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

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

(3.19)

In view of the definition of ${\mathrm{\Pi }}_{\mathcal{F}}{x}_1$, from (3.10) we have $p^{\ast }=w$. Therefore, $x_n\to p^{\ast }={\mathrm{\Pi }}_{\mathcal{F}}{x}_1$. This completes the proof of Theorem 3.1. □

From Theorem 3.1, we can obtain the following.

Theorem 3.2 Let E, C, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ be the same as in Theorem  3.1. Let $\mathcal{T}:=\{T(t):t\ge 0\}$ be a closed, uniformly L-Lipschitz and $(\{k_n\})$-quasi-ϕ-asymptotically nonexpansive semigroup with $\{k_n\}\subset [1,\mathrm{\infty })$, $k_n\to 1$. Let $\{x_n\}$ be a sequence generated by

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

(3.20)

where $\mathcal{F}:={\bigcap }_{t\ge 0}^{\mathrm{\infty }}F(T(t))$, ${\xi }_n=(k_n-1){sup}_{p\in \mathcal{F}}\varphi (p,x_n)$, ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$. If $\mathcal{F}$ is bounded in C, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_1$.

Proof It follows from Definition 2.10 that if $\mathcal{T}:=\{T(t):t\ge 0\}$ is a closed, uniformly L-Lipschitz and $(\{k_n\})$-quasi-ϕ-asymptotically nonexpansive semigroup, then it must be a closed, uniformly L-Lipschitz $(\{{\nu }_n\},\{{\mu }_n\},\zeta )$-total quasi-ϕ-asymptotically nonexpansive semigroup with ${\nu }_n=k_n-1$, ${\mu }_n=0$, $\mathrm{\forall }n\ge 1$ and $\zeta (t)=t$, $t\ge 0$. Therefore, all the conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 can be obtained from Theorem 3.1 immediately. □

Theorem 3.3 Let E, C, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ be the same as in Theorem  3.1. Let $\mathcal{T}:=\{T(t):t\ge 0\}$ be a closed, quasi-ϕ-nonexpansive semigroup such that the set $\mathcal{F}:={\bigcap }_{t\ge 0}F(T(t))$ is nonempty. Let $\{x_n\}$ be a sequence generated by

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

(3.21)

Then the sequence $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_1$.

Proof Since $\mathcal{T}:=\{T(t):t\ge 0\}$ is a closed, quasi-ϕ-nonexpansive semigroup, by Remark 2.5, it is a closed, uniformly Lipschitzian and quasi-ϕ-asymptotically nonexpansive semigroup with the sequence $\{k_n=1\}$. Hence, ${\xi }_n=(k_n-1){sup}_{u\in \mathcal{F}}\varphi (u,x_n)=0$. Therefore, the conditions appearing in Theorem 3.1: ‘$\mathcal{F}$ is a bounded subset in C’ and ‘$\mathcal{T}:=\{T(t):t\ge 0\}$ is uniformly Lipschitzian’ are of no use here. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.3 can be obtained from Theorem 3.2 immediately. □

Remark 3.4 Theorems 3.1, 3.2 and 3.3 improve and extend the corresponding results of Suzuki [4], Xu [5], Chang et al. [6–8, 22, 23, 30], Cho et al. [10], Thong [11], Buong [12], Mann [13], Halpern [14], Qin et al. [15], Nakajo et al. [18] and others.