Convergence of a hybrid iterative method for finite families of generalized quasi-ϕ-asymptotically nonexpansive mappings

Abstract

Strong convergence theorem for finite families of generalized quasi-ϕ-asymptotically nonexpansive mappings is proved in a real uniformly convex and uniformly smooth Banach space using a new modified hybrid iterative algorithm.

MSC:47H09, 47J25.

1 Introduction

Let E be a real Banach space and $E^{\ast }$ be the dual space of E. The normalized duality mapping $J:E\to 2^{E^{\ast }}$ is defined by

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

for all $x\in E$, where $〈\cdot ,\cdot 〉$ denotes the duality pairing. A Banach space E is said to be uniformly convex if given $ϵ\in (0,2]$, there exists $\delta >0$ such that for all $x,y\in E$ with $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $\parallel x-y\parallel \ge ϵ$, we have $\parallel \frac{x+y}{2}\parallel \le 1-\delta$. E is strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. The space 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 $x,y\in U$, where $U:=\{z\in E:\parallel z\parallel =1\}$. It is also uniformly smooth if the limit exists uniformly for $x,y\in U$. It is well known that if E is strictly convex, smooth and reflexive, then the duality map J is one-to-one, single-valued and onto. Also if E is uniformly smooth, then J is norm-to-norm uniformly continuous on bounded subsets of E.

Let C be a nonempty, closed, convex subset of E. Let $T:C\to C$ be a map, a point $x\in C$ is called a fixed point of T if $Tx=x$ and the set of all fixed points of T is denoted by $F(T)$. We recall that a point $p\in C$ is called an asymptotic fixed point of T if there exists a sequence $\{x_n\}\subset C$ which converges weakly to p and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$. The mapping T is called Lipschitz if there exists $L>0$ such that $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel$ for all $x,y\in C$, and if $L=1$, then T is called nonexpansive. T is asymptotically nonexpansive if there exists a sequence $\{t_n\}\subset [1,\mathrm{\infty })$ such that $t_n\to 1$ as $n\to \mathrm{\infty }$ and $\parallel T^{n}x-T^{n}y\parallel \le t_n\parallel x-y\parallel$ for all $n\in \mathbb{N}$ and for all $x,y\in C$. The map T is quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and for all $x\in C$, $q\in F(T)$, $\parallel Tx-q\parallel \le \parallel x-q\parallel$ and is called asymptotically quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and $\parallel T^{n}x-q\parallel \le t_n\parallel x-q\parallel$ for all $x\in C$, $q\in F(T)$ and the sequence $\{t_n\}\subset [1,\mathrm{\infty })$ satisfies $t_n\to 1$ as $n\to \mathrm{\infty }$. The mapping T is called generalized asymptotically quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$, there exist sequences $\{s_n\}\subset [0,1]$, $\{t_n\}\subset [1,\mathrm{\infty })$ with $s_n\to 0$, $t_n\to 1$ as $n\to \mathrm{\infty }$ and $\parallel T^{n}x-q\parallel \le t_n\parallel x-q\parallel +s_n$ for all $x\in C$, $q\in F(T)$ and $n\in \mathbb{N}$.

The map T is said to be

• (i) asymptotically regular on C if ${lim}_{n\to \mathrm{\infty }}\parallel T^{n+1}x-T^{n}x\parallel =0$ for all $x\in C$,

• (ii) uniformly asymptotically regular on C if ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }x\in K}\parallel T^{n+1}x-T^{n}x\parallel =0$ holds for any bounded subset K of C.

For a positive real number L, the map T is called uniformly L-Lipschitzian if $\parallel T^{n}x-T^{n}y\parallel \le L\parallel x-y\parallel$ for all $x,y\in C$ and $n\in \mathbb{N}$.

It is clear from these definitions that every nonexpansive mapping with a fixed point is quasi-nonexpansive and all asymptotically nonexpansive maps with fixed points are asymptotically quasi-nonexpansive. Recently, the class of generalized asymptotically quasi-nonexpansive mappings was introduced and studied by Shahzad and Zegeye [21]. They proved that every asymptotically quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping and the inclusion is proper. The class of quasi-nonexpansive mappings was introduced and studied first in 1967 by Diaz and Metcalf [7]. Goebel and Kirk [8] introduced the class of asymptotically nonexpansive mappings and proved that if C is a nonempty, closed, convex and bounded subset of a uniformly convex Banach space E, then an asymptotically nonexpansive mapping $T:C\to C$ has a fixed point.

Kirk [16], proved that if E is a reflexive Banach space with normal structure and C is a nonempty, closed, convex and bounded subset of E, a nonexpansive map $T:C\to C$ has a fixed point in C. This result was extended to a finite family of nonexpansive maps by Bellus and Kirk [3] and then to an infinite family of nonexpansive maps by Lim [17].

Let H be a real Hilbert space, C be a nonempty closed convex subset of H. Recall that for each $x\in H$ there exists a unique nearest point in C to x denoted by $P_{C}x$. That is, $\parallel x-P_{C}x\parallel \le \parallel x-y\parallel$ for all $y\in C$. $P_C$ is called a metric projection of H onto C.

It is well known that the metric projection is nonexpansive only in a Hilbert space. This fact actually characterizes Hilbert spaces. Alber [1], introduced a generalized projection map ${\prod }_C:E\to C$ in a Banach space which is an analogue of the metric projection in a Hilbert space.

Let E be a real normed linear space with single-valued normalized duality map. Consider the functional defined by $\varphi (x,y)={\parallel x\parallel }^2+{\parallel y\parallel }^2-2〈x,Jy〉$. We observe that in a Hilbert space, $\varphi (x,y)$ reduces to ${\parallel x-y\parallel }^2$. It is clear that for $x,y\in E$, the following inequality holds ${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2$. The generalized projection map ${\prod }_C:E\to C$ is a map that assigns to an arbitrary point $x\in E$, the minimum point of the functional $\varphi (x,\cdot )$ over C, that is, ${\prod }_{C}x=x^{\ast }$ where $\varphi (x,x^{\ast })={min}_{y\in C}\varphi (x,y)$. Existence and uniqueness of the map ${\prod }_C$ follow from the properties of the functional ϕ and the strict monotonicity of J (see, for example, [2]).

Let C be a nonempty, closed, and convex subset of E, a mapping $T:C\to C$ is said to be

• (i) relatively nonexpansive if $F(T)=\widetilde{F(T)}$ and $\varphi (q,Tx)\le \varphi (q,x)$ for all $x\in C$, $q\in F(T)$ where $\widetilde{F(T)}$ denotes the set of asymptotic fixed points of T;

• (ii) ϕ-nonexpansive if $\varphi (Tx,Ty)\le \varphi (x,y)$ for all $x,y\in C$;

• (iii) ϕ-asymptotically nonexpansive if there exists a sequence $\{t_n\}\subset [1,\mathrm{\infty })$ satisfying $t_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$ and $\varphi (T^{n}x,T^{n}y)\le t_n\varphi (x,y)$ for all $x,y\in C$, $n\in \mathbb{N}$;

• (iv) quasi-ϕ-asymptotically nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and $\varphi (q,T^{n}x)\le t_n\varphi (q,x)$ for all $x\in C$, $q\in F(T)$, $n\in \mathbb{N}$, where $\{t_n\}$ is as in (iii) above.

We shall call the map T generalized quasi-ϕ-asymptotically nonexpansive in the light of [21], if $F(T)\ne \mathrm{\varnothing }$ and there exist sequences $\{s_n\}\subset [0,1]$, $\{t_n\}\subset [1,\mathrm{\infty })$ with $s_n\to 0$, $t_n\to 1$ as $n\to \mathrm{\infty }$ and $\varphi (q,T^{n}x)\le t_n\varphi (q,x)+s_n$ for all $x\in C$, $q\in F(T)$ and $n\in \mathbb{N}$.

Existence and approximations of fixed points of mappings of nonexpansive type and their generalizations were studied by numerous authors, see, for example, [3, 5, 7, 8, 10, 11, 14–17, 19, 21, 27] and the references therein.

In 1986, Das and Debata [6] studied the Ishikawa-like scheme defined by $x_1\in C$,

$$x_{n+1}={\alpha }_{n}S[{\beta }_{n}T{x}_n+(1-{\beta }_n)x_n]+(1-{\alpha }_n)x_n,$$

(1.1)

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $[a,b]$ such that $0<a<b<1$. They studied the scheme for two quasi-nonexpansive maps S and T and proved strong convergence of the sequence $\{x_n\}$ to a common fixed point of S and T in a real strictly convex Banach space. Takahashi and Tamura [25] proved strong and weak convergence of the sequence defined by (1.1) to a common fixed point of a pair of nonexpansive mappings T and S using a weaker condition on the maps.

Using a similar scheme, Wang [26] proved strong and weak convergence theorems for a pair of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space.

Shahzad and Udomene [22] proved the necessary and sufficient conditions for the strong convergence of the scheme of type (1.1) to a common fixed point of two uniformly continuous asymptotically quasi-nonexpansive mappings in a real Banach space.

Chidume and Ali [4] introduced and proved strong convergence of the scheme defined by

$$\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}=P[(1-{\alpha }_{1n})x_n+{\alpha }_{1n}{T}_1{(P{T}_1)}^{n-1}{y}_{n+m-2}],\hfill \\ y_{n+m-2}=P[(1-{\alpha }_{2n})x_n+{\alpha }_{2n}{T}_2{(P{T}_2)}^{n-1}{y}_{n+m-3}],\hfill \\ ⋮\hfill \\ y_n=P[(1-{\alpha }_{mn})x_n+{\alpha }_{mn}{T}_m{(P{T}_m)}^{n-1}{x}_n],n\ge 1\hfill \end{array}$$

to a common fixed point of a finite family of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space.

Khan et al. [13] introduced and studied the following scheme:

$$\{\begin{array}{c}{x}_1\in K,y_{0n}=x_n,\hfill \\ x_{n+1}=(1-{\alpha }_{kn})x_n+{\alpha }_{kn}{T}_k^n{y}_{k-1n},\hfill \\ y_{k-1n}=(1-{\alpha }_{k-1n})x_n+{\alpha }_{k-1n}{T}_{k-1}^n{y}_{k-2n},\hfill \\ ⋮\hfill \\ y_{1n}=(1-{\alpha }_{1n})x_n+{\alpha }_{1n}{T}_1^n{y}_{0n},n\ge 1\hfill \end{array}$$

for a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in a Banach space.

It is known that only weak convergence theorems were proved for nonexpansive maps even in Hilbert spaces using Mann and Ishikawa type schemes.

In 2000 Solodov and Svaiter [23] introduced a hybrid proximal point type iterative scheme and proved the strong convergence of the scheme to a zero of a maximal monotone operator.

In 2003 Nakajo and Takahashi [19] proposed a hybrid Mann scheme for nonexpansive mappings and nonexpansive semigroups and proved strong convergence theorems.

Kim and Xu [14] generalized the result of Nakajo and Takahashi by proving strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. Plubtieng and Ughchittrakool [20] introduced an Ishikawa type hybrid scheme for two asymptotically nonexpansive mappings and two asymptotically nonexpansive semigroups.

Takahashi et al. [24] studied a simpler hybrid scheme for nonexpansive mappings in Hilbert spaces. Inchan and Plubtieng [10], adopted this simpler scheme of Takahashi et al. with little modification for two nonexpansive maps and two nonexpansive semigroups. They proved the following theorem:

Theorem 1.1 ([10])

Let H be a real Hilbert space and let C be a nonempty, closed, convex, and bounded subset of H. Let $S,T:C\to C$ be two asymptotically nonexpansive mappings with sequences $\{s_n\}$ and $\{t_n\}$ respectively and $F=F(S)\cap F(T)\ne \mathrm{\varnothing }$. Let $x_0\in C$. Then the sequence $\{x_n\}$ generated by

$$\{\begin{array}{c}{y}_n={\alpha }_n{x}_n+(1-{\alpha }_n)T_1^n{z}_n,\hfill \\ z_n={\beta }_n{x}_n+(1-{\beta }_n)S^n{x}_n,\hfill \\ C_{n+1}=\{z\in C_n:{\parallel y_n-z\parallel }^2\le {\parallel x_n-z\parallel }^2+{\theta }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0,n\in \mathbb{N},\hfill \end{array}$$

(1.2)

converges strongly to $z_0=P_F{x}_0$, where ${\theta }_n=(1-{\alpha }_n)[(t_n^2-1)+(1-{\beta }_n)t_n^2(s_n^2-1)]{(diamC)}^2\to 0$ as $n\to \mathrm{\infty }$ and $0\le {\alpha }_n\le a<1$, $0<b\le {\beta }_n\le c<1$ for all $n\in \mathbb{N}$.

Kimura and Takahashi [15] proved strong convergence theorem for the family of relatively nonexpansive mappings in strictly convex Banach spaces having Kadec-Klee property and Frechet differentiable norm.

Recently, Zhou et al. [28] have proved strong convergence theorem for the family $T_i:C\to C$, $i\in I$ of quasi-ϕ-asymptotically nonexpansive mappings, where C is a nonempty, closed, convex and bounded subset of a uniformly smooth and uniformly convex Banach space E.

More recently, Xu et al. [27] have studied a modified hybrid scheme for fixed point of families of quasi-ϕ-asymptotically nonexpansive mappings. They proved the following theorem:

Theorem 1.2 ([27])

Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let $T_i:C\to C$, $i\in I$ be a family of closed and quasi-ϕ-asymptotically nonexpansive mappings such that $F:={\bigcap }_{i\in I}F(T_i)\ne \mathrm{\varnothing }$. Assume that every $T_i$, $i\in I$ is asymptotically regular on C. Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\gamma }_n\}$ be real sequences in $[0,1]$ such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\gamma }_n>0$. Define a sequence $\{x_n\}$ in C by:

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},\hfill \\ y_{ni}=j^{-1}({\alpha }_{n}j{x}_0+{\beta }_{n}j{x}_n+{\gamma }_{n}j{T}_i^n{x}_n),\hfill \\ C_0=C,\hfill \\ C_{n,i}=\{z\in C_{n-1}:\varphi (z,y_{n,i})\le {\alpha }_n\varphi (z,x_0)+({\beta }_n+{\gamma }_n{k}_{n,i})\varphi (z,x_n)\},\hfill \\ C_n={\bigcap }_{i\in I}{C}_{n,i},\hfill \\ x_{n+1}={\prod }_{C_n}{x}_0.\hfill \end{array}$$

(1.3)

Then, $\{x_n\}$ converges strongly to ${\prod }_F{x}_0$, where ${\prod }_F$ is the generalized projection from E onto F.

Motivated by these results, we have the purpose in this paper to study a new modified hybrid iterative scheme and prove a strong convergence theorem for a finite family of generalized quasi-ϕ-asymptotically nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space. Our theorems improve and unify several recent important results.

2 Preliminaries

Consider a sequence $\{C_n\}$ of nonempty closed and convex subsets of a reflexive Banach space E. Let $s-lim{C}_n$ denotes the set of all strong limits of sequences $\{x_n\}$ satisfying $x_n\in C_n$ for all $n\in \mathbb{N}$ and $w-lim{C}_n$ be the set of all weak limits of sequences $\{y_i\}$ satisfying $y_i\in C_{n_i}$ for all $i\in \mathbb{N}$ where $\{C_{n_i}\}$ is some subsequence of $\{C_n\}$. The sequence $\{C_n\}$ is said to converge to $C^{\ast }$ in the sense of Mosco [18] if $s-lim{C}_n=w-lim{C}_n=C^{\ast }$. The Mosco limit of $\{C_n\}$ is denoted by $M-lim{C}_n$.

We shall make use of the following important results in the sequel.

Lemma 2.1 (Kamimura and Takahashi [12])

Let E be a real smooth and uniformly convex Banach space and $\{x_n\}$, $\{y_n\}$ be two sequences of E. If ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,y_n)=0$ and either $\{x_n\}$ or $\{y_n\}$ is bounded, then ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$.

Lemma 2.2 (Ibaraki, Kimura and Takahashi [9])

Let C be a nonempty closed convex subset of a real uniformly smooth and uniformly convex Banach space E. Let $\{C_n\}$ be a sequence of nonempty closed convex subsets of C. If $M-lim{C}_n=C^{\ast }$ exists and is nonempty, then $\{{\prod }_{C_n}x\}$ converges strongly to $\{{\prod }_{C^{\ast }}x\}$ for each $x\in E$.

The result in [9] is more general than the one presented here, but this is sufficient for our purpose.

Lemma 2.3 Let C be a nonempty closed convex subset of a real smooth Banach space and $T:C\to C$ be a closed generalized quasi-ϕ-asymptotically nonexpansive mapping. Then $F(T)$ is closed and convex.

Proof By the closedness assumption on T and the definition of ϕ, the result follows immediately. □

3 Main results

Theorem 3.1 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty, bounded, closed and convex subset of E. Let $T_k:C\to C$, $k=1,2,3,\dots ,m$ be a finite family of closed generalized quasi-ϕ-asymptotically nonexpansive maps with corresponding sequences $\{t_{kn}\}$ and $\{s_{kn}\}$, $k=1,2,3,\dots ,m$ such that $t_{kn}\to 1$ and $s_{kn}\to 0$ as $n\to \mathrm{\infty }$. Let $F={\bigcap }_{k=1}^{m}F(T_k)\ne \mathrm{\varnothing }$ and let $t_n={max}_{1\le k\le m}{t}_{kn}$, $n\in \mathbb{N}$. Assume also that the maps $T_k$, $k=1,2,\dots ,m$ are uniformly asymptotically regular. Let $x_0\in C$ be arbitrary and $C_0=C$ and let $M={sup}_{x,y\in C}\varphi (x,y)$. For $k=1,2,\dots ,m$, let $\{{\beta }_{kn}\}$ be sequences in $(a,b)$ for some $a,b\in (0,1)$, $a<b$. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_1=x_0,\hfill \\ z_{1n}=j^{-1}({\beta }_{1n}j{x}_n+(1-{\beta }_{1n})j{T}_1^n{x}_n),\hfill \\ z_{kn}=j^{-1}({\beta }_{kn}j{x}_n+(1-{\beta }_{kn})j{T}_k^n{z}_{(k-1)n}),k=2,3,4,\dots ,m,\hfill \\ C_{kn}=\{v\in C_{n-1}:\varphi (v,z_{kn})\le \varphi (v,x_n)+{\gamma }_{kn}\},k=1,2,3,\dots ,m,\hfill \\ C_n={\bigcap }_{k=1}^m{C}_{kn},\hfill \\ x_{n+1}={\prod }_{C_n}{x}_0,n\ge 1,\hfill \end{array}$$

(3.1)

where ${\gamma }_{kn}=(t_n-1)(1-{\beta }_{kn})[1+t_{kn}(1-{\beta }_{(k-1)n})[1+t_{(k-1)n}(1-{\beta }_{(k-2)n})\times [1+t_{(k-2)n}(1-{\beta }_{(k-3)n})[\cdots [1+t_{2n}(1-{\beta }_{1n})]\cdots ]]]]M+{\sum }_{i=1}^k{s}_{in}{\prod }_{j=i}^k(1-{\beta }_{jn}){\prod }_{l=i+1}^k{t}_{ln}$. Then the sequence $\{x_n\}$ converges strongly to $x^{\ast }={\prod }_F{x}_0$.

Proof We start by showing that $F\subset C_n$ $\mathrm{\forall }n\in \mathbb{N}\cup \{0\}$. We do this by induction. $F\subset C_0$ by definition. We suppose that $F\subset C_N$ for some $N\in \mathbb{N}\cup \{0\}$. We observe that for $v\in F$, using convexity of ${\parallel \cdot \parallel }^2$ and (3.1), we have

$$\begin{array}{rcl}\varphi (v,z_{1n})& =& {\parallel v\parallel }^2+{\parallel j^{-1}({\beta }_{1n}j{x}_n+(1-{\beta }_{1n})j{T}_1^n{x}_n)\parallel }^2\\ -2〈v,j\left(j^{-1}({\beta }_{1n}j{x}_n+(1-{\beta }_{1n})j{T}_1^n{x}_n)\right)〉\\ =& {\beta }_{1n}{\parallel v\parallel }^2+(1-{\beta }_{1n}){\parallel v\parallel }^2+{\parallel {\beta }_{1n}j{x}_n+(1-{\beta }_{1n})j{T}_1^n{x}_n\parallel }^2\\ -2〈v,{\beta }_{1n}j{x}_n+(1-{\beta }_{1n})j{T}_1^n{x}_n〉\\ \le & {\beta }_{1n}({\parallel v\parallel }^2+{\parallel x_n\parallel }^2-2〈v,j{x}_n〉)\\ +(1-{\beta }_{1n})({\parallel v\parallel }^2+{\parallel j{T}_1^n{x}_n\parallel }^2-2〈v,j{T}_1^n{x}_n〉)\\ =& {\beta }_{1n}\varphi (v,x_n)+(1-{\beta }_{1n})\varphi (v,T_1^n{x}_n)\\ \le & \varphi (v,x_n)-(1-{\beta }_{1n})\varphi (v,x_n)+(1-{\beta }_{1n})t_{1n}\varphi (v,x_n)+(1-{\beta }_{1n})s_{1n}\\ =& \varphi (v,x_n)+(t_{1n}-1)(1-{\beta }_{1n})\varphi (v,x_n)+s_{1n}(1-{\beta }_{1n})\\ \le & \varphi (v,x_n)+(t_n-1)(1-{\beta }_{1n})\varphi (v,x_n)+s_{1n}(1-{\beta }_{1n})\end{array}$$

and

$$\begin{array}{rcl}\varphi (v,z_{2n})& =& {\parallel v\parallel }^2+{\parallel j^{-1}({\beta }_{2n}j{x}_n+(1-{\beta }_{2n})j{T}_2^n{z}_{1n})\parallel }^2\\ -2〈v,j\left(j^{-1}({\beta }_{2n}j{x}_n+(1-{\beta }_{2n})j{T}_2^n{z}_{1n})\right)〉\\ \le & \varphi (v,x_n)-(1-{\beta }_{2n})\varphi (v,x_n)+(1-{\beta }_{2n})t_{2n}\varphi (v,z_{1n})+(1-{\beta }_{2n})s_{2n}\\ \le & \varphi (v,x_n)-(1-{\beta }_{2n})\varphi (v,x_n)\\ +(1-{\beta }_{2n})t_{2n}[\varphi (v,x_n)+(t_n-1)(1-{\beta }_{1n})\varphi (v,x_n)+(1-{\beta }_{1n})s_{1n}]\\ +(1-{\beta }_{2n})s_{2n}\\ =& \varphi (v,x_n)+(t_{2n}-1)(1-{\beta }_{2n})\varphi (v,x_n)+(1-{\beta }_{2n})t_{2n}[(t_n-1)(1-{\beta }_{1n})\varphi (v,x_n)]\\ +(1-{\beta }_{1n})(1-{\beta }_{2n})s_{1n}{t}_{2n}+(1-{\beta }_{2n})s_{2n}\\ \le & \varphi (v,x_n)+(t_n-1)(1-{\beta }_{2n})[1+t_{2n}(1-{\beta }_{1n})]\varphi (v,x_n)\\ +s_{1n}(1-{\beta }_{1n})(1-{\beta }_{2n})t_{2n}+s_{2n}(1-{\beta }_{2n}).\end{array}$$

Similarly,

$$\begin{array}{rcl}\varphi (v,z_{3n})& =& {\parallel v\parallel }^2+{\parallel j^{-1}({\beta }_{3n}j{x}_n+(1-{\beta }_{3n})j{T}_3^n{z}_{2n})\parallel }^2\\ -2〈v,j\left(j^{-1}({\beta }_{3n}j{x}_n+(1-{\beta }_{3n})j{T}_3^n{z}_{2n})\right)〉\\ \le & \varphi (v,x_n)-(1-{\beta }_{3n})\varphi (v,x_n)+(1-{\beta }_{3n})t_{3n}\varphi (v,z_{2n})+(1-{\beta }_{3n})s_{3n}\\ \le & \varphi (v,x_n)-(1-{\beta }_{3n})\varphi (v,x_n)\\ +(1-{\beta }_{3n})t_{3n}[\varphi (v,x_n)+(t_n-1)(1-{\beta }_{2n})[1+t_{2n}(1-{\beta }_{1n})]\varphi (v,x_n)\\ +(1-{\beta }_{1n})(1-{\beta }_{2n})s_{1n}{t}_{2n}+(1-{\beta }_{2n})s_{2n}]+(1-{\beta }_{3n})s_{3n}\\ \le & \varphi (v,x_n)+(t_n-1)(1-{\beta }_{3n})\varphi (v,x_n)\\ +(1-{\beta }_{3n})t_{3n}[(t_n-1)(1-{\beta }_{2n})[1+t_{2n}(1-{\beta }_{1n})]]\varphi (v,x_n)\\ +s_{1n}(1-{\beta }_{1n})(1-{\beta }_{2n})(1-{\beta }_{3n})t_{2n}{t}_{3n}+s_{2n}(1-{\beta }_{2n})(1-{\beta }_{3n})t_{3n}+s_{3n}(1-{\beta }_{3n})\\ =& \varphi (v,x_n)+(t_n-1)(1-{\beta }_{3n})[1+t_{3n}(1-{\beta }_{2n})[1+t_{2n}(1-{\beta }_{1n})]]\varphi (v,x_n)\\ +\sum _{i=1}^3{s}_{in}\prod _{j=i}^3(1-{\beta }_{jn})\prod _{l=i+1}^3{t}_{ln}.\end{array}$$

Continuing in this way, we get for $k=4,5,\dots ,m$,

$$\begin{array}{rcl}\varphi (v,z_{kn})& \le & \varphi (v,x_n)+(t_n-1)(1-{\beta }_{kn})[1+t_{kn}(1-{\beta }_{(k-1)n})\\ \times [1+t_{(k-1)n}(1-{\beta }_{(k-2)n})[1+t_{(k-2)n}(1-{\beta }_{(k-3)n})\\ \times [\cdots [1+t_{2n}(1-{\beta }_{1n})]\cdots ]\left]\right]]M\\ +\sum _{i=1}^k{s}_{in}\prod _{j=i}^k(1-{\beta }_{jn})\prod _{l=i+1}^k{t}_{ln}\\ =& \varphi (v,x_n)+{\gamma }_{kn}.\end{array}$$

So $\varphi (v,z_{k(N+1)})\le \varphi (v,x_{N+1})+{\gamma }_{k(N+1)}$ for any $v\in F$ and $k\in \{1,2,\dots ,m\}$. This and the induction hypothesis give that $F\subset C_{k(N+1)}$ for all $k\in \{1,2,\dots ,m\}$. Therefore, $F\subset C_{N+1}$ and hence $F\subset C_n$ for all $n\in \mathbb{N}$.

Also by induction and using the fact that $\varphi (\cdot ,x)$ is continuous on E for any $x\in E$, it follows that $C_{kn}$ is closed for each $n\in \mathbb{N}$ and $k\in \{1,2,\dots ,m\}$, and consequently, $C_n$ is closed for each $n\in \mathbb{N}$.

We now prove that $C_n$ is convex for all $n\in \mathbb{N}$. We observe that $s\in C_{kn}$ is equivalent to $s\in C_{n-1}$ and ${\parallel z_{nk}\parallel }^2-{\parallel x_n\parallel }^2\le 2〈s,j{x}_n-j{z}_{nk}〉+{\gamma }_{kn}$. So the convexity of $C_{kn}$ for each $k\in \{1,2,\dots ,m\}$ and for each $n\in \mathbb{N}$ follows immediately by induction. Thus $C_n$ is convex for each $n\in \mathbb{N}$.

We now show that the sequence $\{x_n\}$ converges. Since $\{C_n\}$ is a decreasing sequence of closed, convex subsets of E, such that ${\bigcap }_{n=1}^{\mathrm{\infty }}{C}_n\ne \mathrm{\varnothing }$, then the Mosco limit $M-{lim}_{n\to \mathrm{\infty }}{C}_n$ exists and $M-{lim}_{n\to \mathrm{\infty }}{C}_n={\bigcap }_{n=1}^{\mathrm{\infty }}{C}_n$. By Lemma 2.2, the sequence $\{x_n\}$ converges to $x^{\ast }:={\prod }_{C^{\ast }}{x}_0$, where $C^{\ast }={\bigcap }_{n=1}^{\mathrm{\infty }}{C}_n$.

We observe that

$$\underset{n\to \mathrm{\infty }}{lim}{\gamma }_{kn}=0\text{for each }k\in \{1,2,\dots ,m\}$$

(3.2)

and from the fact that $\{x_n\}$ is convergent, we easily deduce that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},x_n)=0.$$

(3.3)

Since $x_{n+1}\in C_n$, we get that for each $k\in \{1,2,\dots ,m\}$, $\varphi (x_{n+1},z_{kn})\le \varphi (x_{n+1},x_n)+{\gamma }_{kn}$, and so from (3.2) and (3.3), we obtain ${lim}_{n\to \mathrm{\infty }}\varphi (x_{n+1},z_{kn})=0$ for each $k\in \{1,2,\dots ,m\}$. By Lemma 2.1, we get that ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-z_{kn}\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$. So, for each $k\in \{1,2,\dots ,m\}$,

$$\parallel x_n-z_{kn}\parallel \le \parallel x_{n+1}-z_{kn}\parallel +\parallel x_n-x_{n+1}\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

(3.4)

Since j is norm-to-norm uniformly continuous on bounded subsets of E, we get that, for each $k\in \{1,2,\dots ,m\}$, ${lim}_{n\to \mathrm{\infty }}\parallel j{x}_n-j{z}_{kn}\parallel =0$. Using (3.1) we obtain that

$$\parallel j{x}_n-j{T}_1^n{x}_n\parallel =\frac{1}{(1-{\beta }_{1n})}\parallel j{x}_n-j{z}_{1n}\parallel \to 0\text{as }n\to \mathrm{\infty },$$

and for $k\in \{2,3,\dots ,m\}$,

$$\parallel j{x}_n-j{T}_k^n{z}_{(k-1)n}\parallel =\frac{1}{(1-{\beta }_{kn})}\parallel j{x}_n-j{z}_{kn}\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

Using these and the fact that $j^{-1}$ is norm-to-norm uniformly continuous on bounded subsets of $E^{\ast }$, we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T_1^n{x}_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T_2^n{z}_{1n}\parallel =\cdots =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T_m^n{z}_{(m-1)n}\parallel =0.$$

(3.5)

Since $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$, we obtain that $T_1^n{x}_n\to x^{\ast },T_2^n{z}_{1n}\to x^{\ast },\dots ,T_m^n{z}_{(m-1)n}\to x^{\ast }$, as $n\to \mathrm{\infty }$. By the uniform asymptotic regularity of each of the maps $T_k$, $k=1,2,\dots ,m$, we get

$$\parallel T_1^{n+1}{x}_n-x^{\ast }\parallel \le \parallel T_1^{n+1}{x}_n-T_1^n{x}_n\parallel +\parallel T_1^n{x}_n-x^{\ast }\parallel \to 0\text{as }n\to \mathrm{\infty },$$

and

$$\begin{array}{rcl}\parallel T_k^{n+1}{z}_{(k-1)n}-x^{\ast }\parallel & \le & \parallel T_k^{n+1}{z}_{(k-1)n}-T_k^n{z}_{(k-1)n}\parallel \\ +\parallel T_k^n{z}_{(k-1)n}-x^{\ast }\parallel \to 0\text{as }n\to \mathrm{\infty },\end{array}$$

for $k=2,3,\dots ,m$. These imply $T_1(T_1^n{x}_n)\to x^{\ast }$ and $T_k(T_k^n{z}_{(k-1)n})\to x^{\ast }$ as $n\to \mathrm{\infty }$, and for $k=2,3,\dots ,m$. By the closedness of each of the maps $T_k$, $k=1,2,\dots ,m$, we have that $x^{\ast }\in F$.

As F is a nonempty closed convex subset of $C^{\ast }:={\bigcap }_{n=1}^{\mathrm{\infty }}{C}_n$, we obtain that $x^{\ast }={\prod }_F{x}_0$. This completes the proof. □

The conditions of closedness and uniform asymptotic regularity on the maps ${\{T_k\}}_{k=1}^m$ can be replaced by the condition that each of the maps ${\{T_k\}}_{k=1}^m$ is uniformly Lipschitz. So we have the following theorem:

Theorem 3.2 Let E, C, ${\{T_k\}}_{k=1}^m$, F, $\{t_{kn}\}$, $\{s_{kn}\}$, and $\{x_n\}$ be as in Theorem  3.1 with the exception that ${\{T_k\}}_{k=1}^m$ are uniformly $L_k$, $k=1,2,\dots ,m$, Lipschitzian instead of uniformly asymptotically regular and closed. Then the sequence $\{x_n\}$ converges strongly to $x^{\ast }={\prod }_F{x}_0$.

Proof The proof that $F\in C_n$, $C_n$ is closed, convex for each $n\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty }}{x}_n=x^{\ast }$ follows as in Theorem 3.1. Also relations (3.2), (3.3), (3.4) and (3.5) are obtainable as in Theorem 3.1. We only need to show that $x^{\ast }\in F$. Let $L:={max}_{1\le k\le m}{L}_k$, then using (3.4) and (3.5) we get

$$\begin{array}{rcl}\parallel T_1^n{x}_n-x_n\parallel & \le & \parallel T_1^n{x}_n-T_1^n{z}_{1n}\parallel +\parallel T_1^n{z}_{1n}-x_n\parallel \\ \le & L\parallel x_n-z_{1n}\parallel +\parallel T_1^n{z}_{1n}-x_n\parallel \to 0\text{as }n\to \mathrm{\infty },\end{array}$$

(3.6)

and for $k=2,3,\dots ,m$,

$$\begin{array}{rcl}\parallel T_k^n{x}_n-x_n\parallel & \le & \parallel T_k^n{x}_n-T_k^n{z}_{(k-1)n}\parallel +\parallel T_1^n{z}_{(k-1)n}-x_n\parallel \\ \le & L\parallel x_n-z_{(k-1)n}\parallel +\parallel T_k^n{z}_{(k-1)n}-x_n\parallel \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(3.7)

So we obtain

$$\begin{array}{rcl}\parallel T_1{x}_n-x_n\parallel & \le & \parallel T_1{x}_n-T_1^{n+1}{x}_n\parallel +\parallel T_1^{n+1}{x}_n-T_1^{n+1}{x}_{n+1}\parallel \\ +\parallel T_1^{n+1}{x}_{n+1}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel \\ \le & L\parallel x_n-T_1^n{x}_n\parallel +L\parallel x_n-x_{n+1}\parallel \\ +\parallel x_{n+1}-T_1^{n+1}{x}_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel \to 0\text{as }n\to \mathrm{\infty },\end{array}$$

(3.8)

and also for $k=2,3,\dots ,m$,

$$\begin{array}{rcl}\parallel T_k{x}_n-x_n\parallel & \le & \parallel T_k{x}_n-T_k^{n+1}{z}_{(k-1)n}\parallel +\parallel T_k^{n+1}{z}_{(k-1)n}-T_k^{n+1}{x}_n\parallel \\ +\parallel T_k^{n+1}{x}_n-T_k^{n+1}{x}_{n+1}\parallel +\parallel T_k^{n+1}{x}_{n+1}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel \\ \le & L\parallel x_n-T_k^n{z}_{(k-1)n}\parallel +L\parallel z_{(k-1)n}-x_n\parallel \\ +(1+L)\parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T_k^{n+1}{x}_{n+1}\parallel \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(3.9)

Finally, using these, the fact that $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$, and the continuity of $T_k$ for each k, we obtain that $x^{\ast }\in F$ and this completes the proof. □

The following corollaries follow from Theorems 3.1 and 3.2.

Corollary 3.3 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty, bounded, closed and convex subset of E. Let $T_k:C\to C$, $k=1,2,3,\dots ,m$ be a finite family of quasi-ϕ-asymptptically nonexpansive maps with corresponding sequences $\{t_{kn}\}$, $k=1,2,3,\dots ,m$, such that $t_{kn}\to 1$, as $n\to \mathrm{\infty }$. Let $F={\bigcap }_{k=1}^{m}F(T_k)\ne \mathrm{\varnothing }$ and let $t_n={max}_{1\le k\le m}{t}_{kn},n\in \mathbb{N}$. Assume also that the maps $T_k$, $k=1,2,\dots ,m$ are either closed and uniformly asymptotically regular on C or uniformly Lipschitzian on C. Let $x_0\in C$ be arbitrary and $C_0=C$. For $k=1,2,\dots ,m$, let $\{{\beta }_{kn}\}$ be sequences in $(a,b)$ for some $a,b\in (0,1)$, $a<b$. Let $\{x_n\}$ be a sequence generated by (3.1). Then the sequence $\{x_n\}$ converges strongly to $x^{\ast }={\prod }_F{x}_0$.

Corollary 3.4 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty, bounded, closed and convex subset of E. Let $T_k:C\to C$, $k=1,2,3,\dots ,m$ be a finite family of ϕ-asymptotically nonexpansive maps with corresponding sequences $\{t_{kn}\}$, $k=1,2,3,\dots ,m$, such that $t_{kn}\to 1$, as $n\to \mathrm{\infty }$. Let $F={\bigcap }_{k=1}^{m}F(T_k)\ne \mathrm{\varnothing }$ and let $t_n={max}_{1\le k\le m}{t}_{kn}$. Assume also that the maps $T_k$, $k=1,2,\dots ,m$ are either closed and uniformly asymptotically regular on C or uniformly Lipschitzian on C. Let $x_0\in C$ be arbitrary and $C_0=C$. For $k=1,2,\dots ,m$, let $\{{\beta }_{kn}\}$ be sequences in $(a,b)$ for some $a,b\in (0,1)$, $a<b$. Let $\{x_n\}$ be a sequence generated by (3.1). Then the sequence $\{x_n\}$ converges strongly to $x^{\ast }={\prod }_F{x}_0$.

Corollary 3.5 Let H be a real Hilbert space, C be a nonempty, bounded, closed and convex subset of H. Let $T_k:C\to C$, $k=1,2,3,\dots ,m$ be a finite family of generalized asymptotically quasi-nonexpansive maps with corresponding sequences $\{t_{kn}\}$ and $\{s_{kn}\}$, $k=1,2,3,\dots ,m$ such that $t_{kn}\to 1$ and $s_{kn}\to 0$ as $n\to \mathrm{\infty }$. Let $F={\bigcap }_{k=1}^{m}F(T_k)\ne \mathrm{\varnothing }$ and let $t_n={max}_{1\le k\le m}{t}_{kn}$. Assume also that the maps $T_k$, $k=1,2,\dots ,m$ are either closed and uniformly asymptotically regular on C or uniformly $L_k$, $k=1,2,\dots ,m$ Lipschitzian on C. Let $x_0\in C$ be arbitrary and $C_0=C$. For $k=1,2,\dots ,m$, let $\{{\beta }_{kn}\}$ be sequences in $(a,b)$ for some $a,b\in (0,1)$, $a<b$. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_1=x_0\hfill \\ z_{1n}={\beta }_{1n}{x}_n+(1-{\beta }_{1n})T_1^n{x}_n,\hfill \\ z_{kn}={\beta }_{kn}{x}_n+(1-{\beta }_{kn})T_k^n{z}_{(k-1)n},k=2,3,4,\dots ,m,\hfill \\ C_{kn}=\{v\in C_{n-1}:{\parallel z_{kn}-v\parallel }^2\le {\parallel x_n-v\parallel }^2+{\gamma }_{kn}\},\hfill \\ C_n={\bigcap }_{k=1}^k{C}_{kn},\hfill \\ x_{n+1}=P_{C_n}{x}_0,n\in \mathbb{N},\hfill \end{array}$$

(3.10)

where ${\gamma }_{kn}=(t_n^2-1)(1-{\beta }_{kn})[1+t_{kn}^2(1-{\beta }_{k-1n})[1+t_{(k-1)n}^2(1-{\beta }_{(k-2)n})\times [1+t_{(k-2)n}^2(1-{\beta }_{(k-3)n})[\cdots [1+t_{2n}^2(1-{\beta }_{1n})]\cdots ]]]]{(diamC)}^2+{\sum }_{i=1}^k{s}_{in}{\prod }_{j=i}^k(1-{\beta }_{jn}){\prod }_{l=i+1}^k{t}_{ln}$. Then, the sequence $\{x_n\}$ converges strongly to $x^{\ast }=P_F{x}_0$.

Corollary 3.6 Let H be a real Hilbert space, C be a nonempty,closed and convex subset of H. Let $T_k:C\to C$, $k=1,2,3,\dots ,m$ be a finite family of asymptotically nonexpansive maps with corresponding sequences $\{t_{kn}\}$, $k=1,2,3,\dots ,m$, such that $t_{kn}\to 1$ as $n\to \mathrm{\infty }$. Let $F={\bigcap }_{k=1}^{m}F(T_k)\ne \mathrm{\varnothing }$ and let $t_n={max}_{1\le k\le m}{t}_{kn}$. Let $x_0\in C$ be arbitrary and $C_1=C$. For $k=1,2,\dots ,m$, let $\{{\beta }_{kn}\}$ be sequences in $(a,b)$ for some $a,b\in (0,1)$, $a<b$. Let $\{x_n\}$ be a sequence generated by (3.10). Then the sequence $\{x_n\}$ converges to $P_F{x}_0$.

Remark 3.7 Theorem 3.1 and Corollary 3.5 extend and improve several important recent results. For instance, Corollary 3.5 is an improvement and generalization of Theorem 1.1 and Theorem 3.1 of [20].

Remark 3.8 It is not clear whether Theorem 3.1 and Corollary 3.5 hold without the boundedness assumption on C.