Strong convergence theorem for quasi-ϕ-asymptotically nonexpansive mappings in the intermediate sense in Banach spaces

Abstract

In this paper, we modify Halpern and Mann’s iterations for finding a fixed point of an infinite family of quasi-ϕ-asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.

MSC:47H09, 47J25.

1 Introduction

Let E be a real Banach space with the dual space $E^{\ast }$ and let C be a nonempty closed convex subset of E. We denote by $R^{+}$ and R the set of all nonnegative real numbers and the set of all real numbers, respectively. Also, we denote by J the normalized duality mapping from E to $2^{E^{\ast }}$ defined by

$$Jx=\{x^{\ast }\in E^{\ast }:〈x,x^{\ast }〉={\parallel x\parallel }^2={\parallel x^{\ast }\parallel }^2\},\mathrm{\forall }x\in E,$$

(1.1)

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. Recall that if E is smooth, then J is single-valued and norm-to-weak^∗ continuous, and that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E. We shall denote by J the single-valued duality mapping.

A Banach space E is said to be strictly convex if $\frac{\parallel x+y\parallel }{2}\le 1$ for all $x,y\in U=\{z\in E:\parallel z\parallel =1\}$ with $x\ne y$. E is said to be uniformly convex if, for each $\epsilon \in (0,2]$, there exists $\delta >0$ such that $\frac{\parallel x+y\parallel }{2}\le 1-\delta$ for all $x,y\in U$ with $\parallel x-y\parallel \ge \epsilon$. E is said to be smooth if the limit

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

exists for all $x,y\in U$. E is said to be uniformly smooth if the above limit exists uniformly in $x,y\in U$.

Remark 1.1 The following basic properties of a Banach space E can be found in [1]:

• (i) If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.

• (ii) If E is a reflective and strictly convex Banach space, then $J^{-1}$ is norm-to-weak^∗ continuous.

• (iii) If E is a smooth, reflective and strictly convex Banach space, then the normalized duality mapping $J:E\to 2^{E^{\ast }}$ is single-valued, one-to-one and surjective.

• (iv) A Banach space E is uniformly smooth if and only if $E^{\ast }$ is uniformly convex. If E is uniformly smooth, then it is smooth and reflective.

• (v) Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence $\{x_n\}\subset E$, if $x_n\rightharpoonup x\in E$ and $\parallel x_n\parallel \to \parallel x\parallel$, then $x_n\to x$. See [1, 2] for more details.

• (vi) If E is a strictly convex and reflective Banach space with a strictly convex dual $E^{\ast }$ and $J^{\ast }:E^{\ast }\to E$ is the normalized duality mapping in $E^{\ast }$, then $J^{-1}=J^{\ast }$, $J{J}^{\ast }=I_{E^{\ast }}$ and $J^{\ast }J=I_E$.

Next, we assume that E is a smooth, reflective and strictly convex Banach space. Consider the functional defined as in [3, 4] by

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

(1.2)

It is clear that in a Hilbert space H, (1.2) reduces to $\varphi (x,y)={\parallel x-y\parallel }^2$, $\mathrm{\forall }x,y\in H$.

It is obvious from the definition of ϕ that

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

(1.3)

and

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

(1.4)

Following Alber [3], the generalized projection ${\mathrm{\Pi }}_C:E\to C$ is defined by

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

(1.5)

That is, ${\mathrm{\Pi }}_{C}x=\overline{x}$, where $\overline{x}$ is the unique solution to the minimization problem $\varphi (\overline{x},x)={inf}_{y\in C}\varphi (y,x)$.

The existence and uniqueness of the operator ${\mathrm{\Pi }}_C$ follows from the properties of the functional $\varphi (x,y)$ and strict monotonicity of the mapping J (see, e.g., [1–5]). In a Hilbert space H, ${\mathrm{\Pi }}_C=P_C$.

Let H be a real Hilbert space, let D be a nonempty subset of H, and let $T:D\to D$ be a nonlinear mapping. The symbol $F(T)$ stands for the fixed point set of T. Recall the following. T is said to be nonexpansive if

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

(1.6)

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

$$\parallel p-Ty\parallel \le \parallel p-y\parallel ,\mathrm{\forall }p\in F(T),\mathrm{\forall }y\in D.$$

(1.7)

T is said to be asymptotically nonexpansive if there exists a sequence $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ with ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ such that

$$\parallel T^{n}x-T^{n}y\parallel \le (1+{\mu }_n)\parallel x-y\parallel ,\mathrm{\forall }x,y\in D,\mathrm{\forall }n\ge 1.$$

(1.8)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [6]. Since 1972, a host of authors have studied the convergence of iterative algorithms for such a class of mappings.

T is said to be asymptotically quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and there exists a sequence $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ with ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ such that

$$\parallel p-T^{n}y\parallel \le (1+{\mu }_n)\parallel p-y\parallel ,\mathrm{\forall }p\in F(T),\mathrm{\forall }y\in D,\mathrm{\forall }n\ge 1.$$

(1.9)

Let C be a nonempty closed convex subset of E, and let T be a mapping from C into itself. A point $p\in C$ is called an asymptotically fixed point of T [7] if there exists a sequence $\{x_n\}\subset C$ such that $x_n\rightharpoonup p$ and $\parallel x_n-T{x}_n\parallel \to 0$. The set of asymptotical fixed points of T will be denoted by $\hat{F}(T)$. A point $p\in C$ is said to be a strong asymptotic fixed point of T, if there exists a sequence $\{x_n\}\subset C$ such that $x_n\to p$ and $\parallel x_n-T{x}_n\parallel \to 0$. The set of strong asymptotical fixed points of T will be denoted by $\tilde{F}(T)$.

A mapping $T:C\to C$ is said to be relatively nonexpansive [8–10] if $F(T)\ne \mathrm{\varnothing }$, $F(T)=\hat{F}(T)$ and $\varphi (p,Tx)\le \varphi (p,x)$, $\mathrm{\forall }x\in C$, $p\in F(T)$.

A mapping $T:C\to C$ is said to be relatively asymptotically nonexpansive if

$$\begin{array}{r}F(T)\ne \mathrm{\varnothing },F(T)=\hat{F}(T)\text{and}\\ \varphi (p,T^{n}x)\le (1+{\mu }_n)\varphi (p,x),\mathrm{\forall }x\in C,p\in F(T),\end{array}$$

(1.10)

where $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ is a sequence such that ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$.

A mapping $T:C\to C$ is said to be quasi-ϕ-nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and $\varphi (p,Tx)\le \varphi (p,x)$, $\mathrm{\forall }x\in C$, $p\in F(T)$.

A mapping $T:C\to C$ is said to be quasi-ϕ-asymptotically nonexpansive if $F(T)\ne \mathrm{\varnothing }$, and there exists a real sequence $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ with ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ such that

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

(1.11)

Remark 1.2 From the definition, it is easy to know that

• (i) Each relatively nonexpansive mapping is closed;

• (ii) The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true;

• (iii) The class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true. (See [11–15] for more details.)

Asymptotically (quasi-)nonexpansive mappings in the intermediate sense were first considered by Bruck et al. [16]. Very recently Qin and Wang [17] introduced the concept of the asymptotically (quasi-)ϕ-nonexpansive mappings in the intermediate sense as follows:

• (1) T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}(\parallel T^{n}x-T^{n}y\parallel -\parallel x-y\parallel )\le 0.$$

  (1.12)

It is worth mentioning that the class of asymptotically nonexpansive in the intermediate sense mappings may not be Lipschitzian continuous; see [16, 18, 19].

• (2) T is said to be asymptotically quasi-nonexpansive in the intermediate sense if $F(T)\ne \mathrm{\varnothing }$ and the following inequality holds:

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{p\in F(T),y\in C}{sup}(\parallel p-T^{n}y\parallel -\parallel p-y\parallel )\le 0.$$

  (1.13)

• (3) T is said to be an asymptotically ϕ-nonexpansive mapping in the intermediate sense if and only if

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}(\varphi (T^{n}x,T^{n}y)-\varphi (x,y))\le 0.$$

  (1.14)

• (4) $T:C\to C$ is said to be quasi-ϕ-asymptotically nonexpansive mapping in the intermediate sense if and only if $F(T)\ne \mathrm{\varnothing }$ and

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{p\in F(T),x\in C}{sup}(\varphi (p,T^{n}x)-\varphi (p,x))\le 0.$$

  (1.15)

Remark 1.3 The asymptotically (quasi-)ϕ-nonexpansive mapping in the intermediate sense is a generalization of the asymptotically (quasi-)nonexpansive mapping in the intermediate sense in the framework of Banach spaces.

Definition 1.4 An infinite family of mappings ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ is said to be uniformly quasi-ϕ-asymptotically nonexpansive in the intermediate sense if ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\ne \mathrm{\varnothing }$ for each $i\ge 1$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{p\in F(T_i),x\in C}{sup}(\varphi (p,T_i^{n}x)-\varphi (p,x))\le 0.$$

(1.16)

If we define

$${\xi }_n=max\{0,\underset{p\in F(T_i),x\in C}{sup}(\varphi (p,T_i^{n}x)-\varphi (p,x))\},$$

then ${\xi }_n\to 0$ as $n\to \mathrm{\infty }$. It follows that (1.16) is reduced to

$$\varphi (p,T_i^{n}x)\le \varphi (p,x)+{\xi }_n,\mathrm{\forall }p\in F(T_i),\mathrm{\forall }x\in C,\mathrm{\forall }n\ge 1.$$

(1.17)

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping. In 1953, Mann [20] introduced the iteration as follows: a sequence $\{x_n\}$ defined by

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,$$

(1.18)

where the initial guess $x_1\in C$ is arbitrary and $\{a_n\}$ is a real sequence in $[0,1]$. It is known that under appropriate settings the sequence $\{x_n\}$ converges weakly to a fixed point of T. However, for nonexpansive mappings, even in a Hilbert space, the Mann iteration may fail to converge strongly; for example, see [21].

Some attempts to construct the iteration method guaranteeing the strong convergence have been made. For example, Halpern [22] proposed the following so-called Halpern iteration:

$$x_{n+1}={\alpha }_{n}u+(1-{\alpha }_n)T{x}_n,$$

(1.19)

where $u\in C$ is fixed, $x_1\in C$ is arbitrarily chosen and $\{a_n\}$ is a real sequence in $[0,1]$.

Recently, Nilsrakoo and Saejung [23] modified Halpern and Mann’s iterations introduced the following iteration to find a fixed point of the relatively nonexpansive mappings in the Banach space:

$$x_{n+1}={\mathrm{\Pi }}_C{J}^{-1}({\alpha }_{n}Ju+(1-{\alpha }_n)({\beta }_{n}J{x}_n+(1-{\beta }_n)JT{x}_n)).$$

(1.20)

They proved that $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T)}u$, where $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences in $(0,1)$, ${\mathrm{\Pi }}_{F(T)}$ is the generalized projection from E onto $F(T)$.

Iteration methods for approximating fixed points of asymptotically nonexpansive mappings, quasi-ϕ-nonexpansive mapping, quasi-ϕ-asymptotically nonexpansive mapping have been further studied by authors (see, e.g., [6, 24–29]).

Quite recently, Qin and Wang [17] introduced the following iterative scheme to find a fixed point of the quasi-ϕ-asymptotically nonexpansive mappings in the intermediate sense in a reflective, strictly convex and smooth Banach space such that both E and $E^{\ast }$ have the Kadec-Klee property:

$$\{\begin{array}{c}{x}_0\in E\text{chosen arbitrarily},\hfill \\ C_{(1,i)}=C,\hfill \\ C_1={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(1,i)},\hfill \\ x_1={\mathrm{\Pi }}_{C_1}{x}_0,\hfill \\ C_{(n+1,i)}=\{u\in C_{(n,i)}:\varphi (x_n,T_i^n{x}_n)\le 2〈x_n-u,J{x}_n-J{T}_i^n{x}_n〉+{\xi }_{(n,i)}\},\hfill \\ C_{n+1}={\bigcap }_{i\in \mathrm{\Lambda }}{C}_{(n+1,i)},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,\mathrm{\forall }n\ge 0,\hfill \end{array}$$

where

$${\xi }_{(n,i)}=max\{0,\underset{p\in F(T_i),x\in C}{sup}(\varphi (p,T_i^{n}x)-\varphi (p,x))\}.$$

They proved that the sequence $\{x_n\}$ converges strongly to $\overline{x}={\mathrm{\Pi }}_{{\bigcap }_{i\in \mathrm{\Lambda }}F(T_i)}{x}_0$.

Inspired and motivated by the recent work of Bruck [16], Qin and Wang [17], Nilsrakoo and Saejung [23], Chang et al. [24], etc., in this paper, we modify Halpern and Mann’s iterations for finding a fixed point of an infinite family of quasi-ϕ-asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.

2 Preliminaries

Throughout this paper, let E be a real Banach space with the dual space $E^{\ast }$ and let C be a nonempty closed convex subset of E. We denote the strong convergence, weak convergence of a sequence $\{x_n\}$ to a point $x\in E$ by $x_n\to x$, $x_n\rightharpoonup x$, respectively, and $F(T)$ is the fixed point set of a mapping T.

Lemma 2.1 [30]

Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let $\{x_n\}$ and $\{y_n\}$ be two sequences in C such that $x_n\to p$ and $\varphi (x_n,y_n)\to 0$, where ϕ is the functional defined by (1.2), then $y_n\to p$.

Lemma 2.2 [3]

Let E be a smooth, strictly convex and reflective Banach space and let C be a nonempty closed convex subset of E. Then the following conclusions hold:

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

• (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 $x=y$.

Lemma 2.3 [31]

Let E be a uniformly convex Banach space, r be a positive number and $B_r(0)$ be a closed ball of E. Then, for any sequence ${\{x_i\}}_{i=1}^{\mathrm{\infty }}\subset B_r(0)$ and for any sequence ${\{{\lambda }_i\}}_{i=1}^{\mathrm{\infty }}$ of positive numbers with ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n=1$, there exists a continuous, strictly increasing and convex function $g:[0,2r]\to [0,\mathrm{\infty })$, $g(0)=0$ such that for any positive integer $i\ne 1$, the following holds:

$${\parallel \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{x}_n\parallel }^2\le \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{\parallel x_n\parallel }^2-{\lambda }_1{\lambda }_{i}g(\parallel x_1-x_i\parallel ).$$

(2.1)

3 Main results

Theorem 3.1 Let C be a nonempty, closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in the intermediate sense and for each $i\ge 1$, let $T_i$ be uniformly $L_i$-Lipschitzian continuous. $\{x_n\}$ is defined by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},C_0=C,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_0+(1-{\alpha }_n)J{z}_n),\hfill \\ z_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n),\hfill \\ C_{n+1}=\{\nu \in C_n:\varphi (\nu ,y_n)\le {\alpha }_n\varphi (\nu ,x_0)+(1-{\alpha }_n)\varphi (\nu ,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,n\ge 0,\hfill \end{array}$$

(3.1)

where ${\xi }_n=max\{0,{sup}_{p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i),x\in C}(\varphi (p,T_i^{n}x)-\varphi (p,x))\}$, ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$, $\{{\beta }_{n,0},{\beta }_{n,i}\}$ and $\{{\alpha }_n\}$ are sequences in $[0,1]$ satisfying the following conditions:

• (1) for each $n\ge 0$, ${\beta }_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$ for any $i\ge 1$;

• (3) $0\le {\alpha }_n\le \alpha <1$ for some $\alpha \in (0,1)$.

If ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$ is a nonempty and bounded subset of C, then the sequence $\{x_n\}$ converges strongly to $p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$, where $p={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$.

Proof We shall divide the proof into six steps.

Step 1. We show that ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$ and $C_n$ are closed and convex for each $n\ge 0$.

Using the similar methods given in the proof of Theorem 3.1 by Qin and Wang [17], the conclusion that $F(T_i)$ is closed and convex subset of C for each $i\ge 1$ can be easily obtained. Therefore ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$ is closed and convex in C.

Again, by the assumption, $C_0=C$ is closed and convex. Suppose that $C_n$ is closed and convex for some $n\ge 1$. Since for any $z\in C_n$, we know

$$\begin{array}{r}\varphi (z,y_n)\le {\alpha }_n\varphi (z,x_0)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\\ \iff 2{\alpha }_n〈z,J{x}_0〉+2(1-{\alpha }_n)〈z,J{x}_n〉-2〈z,J{y}_n〉\\ \phantom{\iff }\le {\alpha }_n{\parallel x_0\parallel }^2+(1-{\alpha }_n){\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2+{\xi }_n.\end{array}$$

(3.2)

Hence the set $C_{n+1}$ = {$z\in C_n$ : $2{\alpha }_n〈z,J{x}_0〉+2(1-{\alpha }_n)〈z,J{x}_n〉-2〈z,J{y}_n〉$ ≤ ${\alpha }_n{\parallel x_0\parallel }^2+(1-{\alpha }_n){\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2+{\xi }_n$} is closed and convex. Therefore ${\mathrm{\Pi }}_{C_n}{x}_0$ and ${\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$ are well defined.

Step 2. We show that ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\subset C_n$ for all $n\ge 0$.

It is obvious that ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\subset C_0=C$. Suppose that ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\subset C_n$ for some $n\ge 1$. Since E is uniformly smooth, $E^{\ast }$ is uniformly convex. For any given $q\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\subset C_n$, we observe that

$$\begin{array}{rcl}\varphi (q,y_n)& =& \varphi (q,J^{-1}({\alpha }_{n}J{x}_0+(1-{\alpha }_n)J{z}_n))\\ =& {\parallel q\parallel }^2-2〈q,{\alpha }_{n}J{x}_0+(1-{\alpha }_n)J{z}_n〉+{\parallel {\alpha }_{n}J{x}_0+(1-{\alpha }_n)J{z}_n\parallel }^2\\ \le & {\parallel q\parallel }^2-2{\alpha }_n〈q,J{x}_0〉-2(1-{\alpha }_n)〈q,J{z}_n〉+{\alpha }_n{\parallel x_0\parallel }^2+(1-{\alpha }_n){\parallel z_n\parallel }^2\\ =& {\alpha }_n\varphi (q,x_0)+(1-{\alpha }_n)\varphi (q,z_n).\end{array}$$

(3.3)

On the other hand, it follows from Lemma 2.3 that for any positive integer $l>1$ and for any $q\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$, we have

$$\begin{array}{rcl}\varphi (q,z_n)& =& \varphi (q,J^{-1}({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n))\\ =& {\parallel q\parallel }^2-2〈q,{\beta }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n〉+{\parallel {\beta }_{n,0}J{x}_n+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n\parallel }^2\\ \le & {\parallel q\parallel }^2-2{\beta }_{n,0}〈q,J{x}_n〉-2\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}〈q,J{T}_i^n{x}_n〉+{\beta }_{n,0}{\parallel x_n\parallel }^2\\ +\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}{\parallel T_i^n{x}_n\parallel }^2-{\beta }_{n,0}{\beta }_{n,l}g\left(\parallel J{x}_n-J{T}_l^n{x}_n\parallel \right)\\ =& {\beta }_{n,0}\varphi (q,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}\varphi (q,T_i^n{x}_n)-{\beta }_{n,0}{\beta }_{n,l}g\left(\parallel J{x}_n-J{T}_l^n{x}_n\parallel \right)\\ \le & {\beta }_{n,0}\varphi (q,x_n)+\sum _{i=1}^{\mathrm{\infty }}{\beta }_{n,i}\{\varphi (q,x_n)+{\xi }_n\}-{\beta }_{n,0}{\beta }_{n,l}g\left(\parallel J{x}_n-J{T}_l^n{x}_n\parallel \right)\\ \le & \varphi (q,x_n)+{\xi }_n-{\beta }_{n,0}{\beta }_{n,l}g\left(\parallel J{x}_n-J{T}_l^n{x}_n\parallel \right).\end{array}$$

(3.4)

Substituting (3.4) into (3.3), we get

$$\begin{array}{rcl}\varphi (q,y_n)& \le & {\alpha }_n\varphi (q,x_0)+(1-{\alpha }_n)\varphi (q,z_n)\\ \le & {\alpha }_n\varphi (q,x_0)+(1-{\alpha }_n)[\varphi (q,x_n)+{\xi }_n-{\beta }_{n,0}{\beta }_{n,l}g\left(\parallel J{x}_n-J{T}_l^n{x}_n\parallel \right)]\\ \le & {\alpha }_n\varphi (q,x_0)+(1-{\alpha }_n)\varphi (q,x_n)+{\xi }_n.\end{array}$$

(3.5)

This shows that $q\in C_{n+1}$. Further this implies that ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\subset C_{n+1}$ and hence ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\subset C_n$ for all $n\ge 0$. Since ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$ is nonempty, $C_n$ is a nonempty closed convex subset of E and hence ${\mathrm{\Pi }}_{C_n}$ exists for all $n\ge 0$. This implies that the sequence $\{x_n\}$ is well defined.

Step 3. We show that $\{x_n\}$ is bounded and $\{\varphi (x_n,x_0)\}$ is a convergent sequence.

It follows from (3.1) and Lemma 2.2 that

$$\begin{array}{rcl}\varphi (x_n,x_0)& =& \varphi ({\mathrm{\Pi }}_{C_n}{x}_0,x_0)\\ \le & \varphi (p,x_0)-\varphi (p,x_n)\\ \le & \varphi (p,x_0),\mathrm{\forall }p\in C_{n+1},\mathrm{\forall }n\ge 0.\end{array}$$

(3.6)

From the definition of $C_{n+1}$ that $x_n={\mathrm{\Pi }}_{C_n}{x}_0$ and $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0$, we have

$$\varphi (x_n,x_0)\le \varphi (x_{n+1},x_0),\mathrm{\forall }n\ge 0.$$

(3.7)

Therefore, $\{\varphi (x_n,x_0)\}$ is nondecreasing and bounded. So, $\{\varphi (x_n,x_0)\}$ is a convergent sequence, without loss of generality, we can assume that ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_0)=d\ge 0$. In particular, by (1.3), the sequence $\{{(\parallel x_n\parallel -\parallel x_0\parallel )}^2\}$ is bounded. This implies $\{x_n\}$ is also bounded.

Step 4. We prove that $\{x_n\}$ converges strongly to some point $p\in C$.

Since $\{x_n\}$ is bounded and E is reflective, there exists a subsequence $\{x_{n_i}\}\subset \{x_n\}$ such that $x_{n_i}\rightharpoonup p$ (some point in C). Since $C_n$ is closed and convex and $C_{n+1}\subset C_n$, this implies that $C_n$ is weakly closed and $p\in C_n$ for each $n\ge 0$. From $x_{n_i}={\mathrm{\Pi }}_{C_{n_i}}{x}_0$, we have

$$\varphi (x_{n_i},x_0)\le \varphi (p,x_0),\mathrm{\forall }{n}_i\ge 0.$$

(3.8)

Since the norm $\parallel \cdot \parallel$ is weakly lower semi-continuous, we have

$$\begin{array}{rcl}\underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (x_{n_i},x_0)& =& \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\{{\parallel x_{n_i}\parallel }^2-2〈x_{n_i},J{x}_0〉+{\parallel x_0\parallel }^2\}\\ \ge & {\parallel p\parallel }^2-2〈p,J{x}_0〉+{\parallel x_0\parallel }^2\\ =& \varphi (p,x_0),\end{array}$$

(3.9)

and so

$$\varphi (p,x_0)\le \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (x_{n_i},x_0)\le \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (x_{n_i},x_0)\le \varphi (p,x_0).$$

(3.10)

This implies that ${lim}_{n_i\to \mathrm{\infty }}\varphi (x_{n_i},x_0)\to \varphi (p,x_0)$, and so $\parallel x_n\parallel \to \parallel p\parallel$. Since $x_{n_i}\rightharpoonup p$, in view of the Kadec-Klee property of E, it follows that

$$\underset{n_i\to \mathrm{\infty }}{lim}{x}_{n_i}=p.$$

(3.11)

Since $\{\varphi (x_n,x_0)\}$ is convergent, this together with ${lim}_{n_i\to \mathrm{\infty }}\varphi (x_{n_i},x_0)\to \varphi (p,x_0)$, we have ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_0)\to \varphi (p,x_0)$. If there exists a subsequence $\{x_{n_j}\}\subset \{x_n\}$ such that $x_{n_j}\to q$, then from Lemma 2.2(a), we have that

$$\begin{array}{rcl}\varphi (p,q)& =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},x_{n_j})\\ =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}\varphi (x_{n_i},{\mathrm{\Pi }}_{C_{n_j}}{x}_0)\\ \le & \underset{n_i,n_j\to \mathrm{\infty }}{lim}(\varphi (x_{n_i},x_0)-\varphi ({\mathrm{\Pi }}_{C_{n_j}}{x}_0,x_0))\\ =& \underset{n_i,n_j\to \mathrm{\infty }}{lim}(\varphi (x_{n_i},x_0)-\varphi (x_{n_i},x_0))\\ =& \varphi (p,x_0)-\varphi (p,x_0)\\ =& 0.\end{array}$$

(3.12)

This implies that $p=q$ and

$$\underset{n\to \mathrm{\infty }}{lim}{x}_n=p.$$

(3.13)

Step 5. We show that $p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$.

Since $x_{n+1}\in C_{n+1}$, it follows from (3.1) and (3.13) that

$$\varphi (x_{n+1},y_n)\le {\alpha }_n\varphi (x_{n+1},x_0)+(1-{\alpha }_n)\varphi (x_{n+1},x_n)+{\xi }_n\to 0(\text{as }n\to \mathrm{\infty }).$$

(3.14)

Since $x_n\to p$, by Lemma 2.1

$$\underset{n\to \mathrm{\infty }}{lim}{y}_n=p.$$

(3.15)

By (3.3) and (3.4), for any $q\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$, we have

$$\varphi (q,y_n)\le {\alpha }_n\varphi (q,x_0)+(1-{\alpha }_n)\varphi (q,x_n)+{\xi }_n-(1-{\alpha }_n){\beta }_{n,0}{\beta }_{n,l}g\left(\parallel J{x}_n-J{T}_l^n{x}_n\parallel \right).$$

(3.16)

So, as $n\to \mathrm{\infty }$,

$$\begin{array}{rl}(1-{\alpha }_n){\beta }_{n,0}{\beta }_{n,l}g\parallel J{x}_n-J{T}_l^n{x}_n\parallel & \le {\alpha }_n\varphi (q,x_0)+(1-{\alpha }_n)\varphi (q,x_n)+{\xi }_n-\varphi (q,y_n)\\ \to 0.\end{array}$$

(3.17)

Therefore,

$$\underset{n\to \mathrm{\infty }}{lim}(1-{\alpha }_n){\beta }_{n,0}{\beta }_{n,l}g\parallel J{x}_n-J{T}_l^n{x}_n\parallel =0.$$

(3.18)

In view of the property of g, we have

$$\parallel J{x}_n-J{T}_l^n{x}_n\parallel \to 0(\text{as }n\to \mathrm{\infty }).$$

(3.19)

Since $J{x}_n\to Jp$, this implies that ${lim}_{n\to \mathrm{\infty }}J{T}_l^n{x}_n=Jp$. Remark 1.1(ii) yields

$$T_l^n{x}_n\rightharpoonup p(\text{as }n\to \mathrm{\infty }).$$

(3.20)

Again, since

$$\parallel T_l^n{x}_n\parallel -\parallel p\parallel =\parallel J\left(T_l^n{x}_n\right)\parallel -\parallel Jp\parallel \le \parallel J\left(T_l^n{x}_n\right)-Jp\parallel \to 0(\text{as }n\to \mathrm{\infty }),$$

this together with (3.20) and the Kadec-Klee property of E shows that

$$\underset{n\to \mathrm{\infty }}{lim}{T}_l^n{x}_n=p.$$

(3.21)

By the assumption that $T_l$ is uniformly $L_l$-Lipschitz continuous, we have

$$\begin{array}{rcl}\parallel T_l^{n+1}{x}_n-T_l^n{x}_n\parallel & \le & \parallel T_l^{n+1}{x}_n-T_l^{n+1}{x}_{n+1}\parallel +\parallel T_l^{n+1}{x}_{n+1}-x_{n+1}\parallel \\ +\parallel x_{n+1}-x_n\parallel +\parallel x_n-T_l^n{x}_n\parallel \\ \le & (L_l+1)\parallel x_{n+1}-x_n\parallel +\parallel T_l^{n+1}{x}_{n+1}-x_{n+1}\parallel +\parallel x_n-T_l^n{x}_n\parallel .\end{array}$$

This together with (3.21) and $x_n\to p$ shows that ${lim}_{n\to \mathrm{\infty }}\parallel T_l^{n+1}{x}_n-T_l^n{x}_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}{T}_l^{n+1}{x}_n=p$, that is, ${lim}_{n\to \mathrm{\infty }}{T}_l{T}_l^n{x}_n=p$. In view of the closeness of $T_l$, it follows that $T_{l}p=p$, that is, $p\in F(T_l)$. By the arbitrariness of $l\ge 1$, we have $p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$.

Step 6. We prove that $x_n\to p={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$.

Let $q={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$. From $x_n={\mathrm{\Pi }}_{C_n}{x}_0$ and $q\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\subset C_n$, we have

$$\varphi (x_n,x_0)\le \varphi (q,x_0),\mathrm{\forall }n\ge 0.$$

(3.22)

This implies that

$$\varphi (p,x_0)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,x_0)\le \varphi (q,x_0).$$

(3.23)

By the definition of $p={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$, we have $p=q$. Therefore, $x_n\to p={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$. This completes the proof. □

In Theorem 3.1, as $T_i=T$ for each $i\in N$, we can obtain the following corollary.

Corollary 3.2 Let C be a nonempty, closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let $T:C\to C$ be a closed uniformly L-Lipschitzian continuous and uniformly quasi-ϕ-asymptotically nonexpansive mapping in the intermediate sense such that $F(T)$ is a nonempty and bounded subset of C. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},C_0=C,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_0+(1-{\alpha }_n)J{z}_n),\hfill \\ z_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{T}^n{x}_n),\hfill \\ C_{n+1}=\{\nu \in C_n:\varphi (\nu ,y_n)\le {\alpha }_n\varphi (\nu ,x_0)+(1-{\alpha }_n)\varphi (\nu ,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,n\ge 0,\hfill \end{array}$$

(3.24)

where ${\xi }_n=max\{0,{sup}_{p\in F(T),x\in C}(\varphi (p,T^{n}x)-\varphi (p,x))\}$, ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$, $\{{\alpha }_n\}$ is a sequence in $[0,\alpha ]$, $\{{\beta }_n\}\subset (0,1)$ satisfies that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n(1-{\beta }_n)>0$, then the sequence $\{x_n\}$ converges strongly to $p\in F(T)$, where $p={\mathrm{\Pi }}_{F(T)}{x}_0$.

Corollary 3.3 Let C be a nonempty, closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in the intermediate sense and for each $i\ge 1$, $T_i$ is uniformly $L_i$-Lipschitzian continuous. $\{x_n\}$ is defined by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},C_0=C,\hfill \\ z_n=J^{-1}({\beta }_{n,0}J{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}J{T}_i^n{x}_n),\hfill \\ C_{n+1}=\{\nu \in C_n:\varphi (\nu ,y_n)\le \varphi (\nu ,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,n\ge 0,\hfill \end{array}$$

(3.25)

where ${\xi }_n=max\{0,{sup}_{p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i),x\in C}(\varphi (p,T_i^{n}x)-\varphi (p,x))\}$, ${\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto $C_{n+1}$, $\{{\beta }_{n,0},{\beta }_{n,i}\}$ and $\{{\alpha }_n\}$ are sequences in $[0,1]$ satisfying the following conditions:

• (1) for each $n\ge 0$ ${\beta }_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$ for any $i\ge 1$;

• (3) $0\le {\alpha }_n\le \alpha <1$ for some $\alpha \in (0,1)$.

If ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$ is a nonempty and bounded subset of C, then the sequence $\{x_n\}$ converges strongly to $p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$, where $p={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$.

Proof Setting ${\alpha }_n\equiv 0$ in Theorem 3.1, then we get that $y_n=z_n$. Thus, from the method of the proof of Theorem 3.1, we obtain Corollary 3.3 immediately. □

In the Hilbert space, the following corollary can be directly obtained from Theorem 3.1.

Corollary 3.4 Let C be a nonempty, closed and convex subset of a Hilbert space E. Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be an infinite family of closed and uniformly $L_i$-Lipschitzian continuous and uniformly asymptotically quasi-nonexpansive mappings in the intermediate sense such that ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$ is a nonempty and bounded subset of C. Let $\{x_n\}$ be the sequence generated by

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},C_0=C,\hfill \\ y_n={\alpha }_n{x}_0+(1-{\alpha }_n)z_n,\hfill \\ z_n={\beta }_{n,0}{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}{T}_i^n{x}_n,\hfill \\ C_{n+1}=\{\nu \in C_n:{\parallel \nu -y_n\parallel }^2\le {\alpha }_n{\parallel \nu -x_0\parallel }^2+(1-{\alpha }_n){\parallel \nu -x_n\parallel }^2+{\xi }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0,n\ge 0,\hfill \end{array}$$

(3.26)

where ${\xi }_n=max\{0,{sup}_{p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i),x\in C}({\parallel p-T_i^{n}x\parallel }^2-{\parallel p-x\parallel }^2)\}$, $P_{C_{n+1}}$ is the metric projection of E onto $C_{n+1}$, $\{{\beta }_{n,0},{\beta }_{n,i}\}$ and $\{{\alpha }_n\}$ are sequences in $[0,1]$ satisfying the following conditions:

• (1) for each $n\ge 0$ ${\beta }_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{\beta }_{n,i}=1$;

• (2) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_{n,0}{\beta }_{n,i}>0$ for any $i\ge 1$;

• (3) $0\le {\alpha }_n\le \alpha <1$ for some $\alpha \in (0,1)$.

Then the sequence $\{x_n\}$ converges strongly to $p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$, where $p={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)}{x}_0$.