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# Fixed point theory for nonlinear mappings in Banach spaces and applications

Fixed Point Theory and Applications20142014:108

https://doi.org/10.1186/1687-1812-2014-108

• Accepted: 21 April 2014
• Published:

## Abstract

The purpose of this research is to study a finite family of the set of solutions of variational inequality problems and to prove a convergence theorem for the set of such problems and the sets of fixed points of nonexpansive and strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. We also prove a fixed point theorem for finite families of nonexpansive and strictly pseudo-contractive mappings in the last section.

## Keywords

• nonexpansive mapping
• strictly pseudo-contractive mapping
• inverse-strongly monotone
• variational inequality problems
• uniformly convex
• 2-uniformly smooth

## 1 Introduction

Let E and ${E}^{\ast }$ be a Banach space and the dual space of E, respectively, and let C be a nonempty closed convex subset of E. Throughout this paper, we use ‘→’ and ‘’ to denote strong and weak convergence, respectively. The duality mapping $J:E\to {2}^{{E}^{\ast }}$ is defined by $J\left(x\right)=\left\{{x}^{\ast }\in {E}^{\ast }:〈x,{x}^{\ast }〉={\parallel x\parallel }^{2},\parallel x\parallel =\parallel {x}^{\ast }\parallel \right\}$ for all $x\in E$.

Definition 1.1 Let E be a Banach space. Then a function ${\delta }_{X}:\left[0,2\right]\to \left[0,1\right]$ is said to be the modulus of convexity of E if
${\delta }_{E}\left(ϵ\right)=inf\left\{1-\parallel \frac{x+y}{2}\parallel :\parallel x\parallel \le 1,\parallel y\parallel \le 1,\parallel x-y\parallel \ge ϵ\right\}.$

If ${\delta }_{E}\left(ϵ\right)>0$ for all $ϵ\in \left(0,2\right]$, then E is uniformly convex.

The function ${\rho }_{E}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is said to be the modulus of smoothness of E if
${\rho }_{E}\left(t\right)=sup\left\{\frac{\parallel x+y\parallel +\parallel x-y\parallel }{2}-1:\parallel x\parallel =1,\parallel y\parallel =t\right\},\phantom{\rule{1em}{0ex}}t\ge 0.$

If ${lim}_{t\to 0}\frac{{\rho }_{E}\left(t\right)}{t}=0$, then E is uniformly smooth. It is well known that every uniformly smooth Banach space is smooth and if E is smooth, then J is single-valued which is denoted by j. A Banach space E is said to be q-uniformly smooth if there exists a fixed constant $c>0$ such that ${\rho }_{E}\left(t\right)\le c{t}^{q}$. If E is q-uniformly smooth, then $q\le 2$ and E is uniformly smooth.

A mapping $T:C\to C$ is called nonexpansive if
$\parallel Tx-Ty\parallel \le \parallel x-y\parallel$

for all $x,y\in C$.

T is called η-strictly pseudo-contractive if there exists a constant $\eta \in \left(0,1\right)$ such that
$〈Tx-Ty,j\left(x-y\right)〉\le {\parallel x-y\parallel }^{2}-\eta {\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2}$
(1.1)
for every $x,y\in C$ and for some $j\left(x-y\right)\in J\left(x-y\right)$. It is clear that (1.1) is equivalent to the following:
$〈\left(I-T\right)x-\left(I-T\right)y,j\left(x-y\right)〉\ge \eta {\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2}$
(1.2)

for every $x,y\in C$ and for some $j\left(x-y\right)\in J\left(x-y\right)$. Let C and D be nonempty subsets of a Banach space E such that C is nonempty closed convex and $D\subset C$, then a mapping $P:C\to D$ is sunny (see ) provided $P\left(x+t\left(x-P\left(x\right)\right)\right)=P\left(x\right)$ for all $x\in C$ and $t\ge 0$, whenever $x+t\left(x-P\left(x\right)\right)\in C$. A mapping $P:C\to D$ is called a retraction if $Px=x$ for all $x\in D$. Furthermore, P is a sunny nonexpansive retraction from C onto D if P is a retraction from C onto D which is also sunny and nonexpansive.

A subset D of C is called a sunny nonexpansive retract of C (see ) if there exists a sunny nonexpansive retraction from C onto D.

An operator A of C into E is said to be if there exists $j\left(x-y\right)\in J\left(x-y\right)$ such that
$〈Ax-Ay,j\left(x-y\right)〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$
A mapping $A:C\to E$ is said to be α-inverse strongly accretive if there exist $j\left(x-y\right)\in J\left(x-y\right)$ and $\alpha >0$ such that
$〈Ax-Ay,j\left(x-y\right)〉\ge \alpha {\parallel Ax-Ay\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$
A mapping $A:C\to E$ is called γ-strongly accretive if there exist $j\left(x-y\right)\in J\left(x-y\right)$ and a constant $\gamma >0$ such that
$〈Ax-Ay,j\left(x-y\right)〉\ge \gamma {\parallel x-y\parallel }^{2}$

for all $x,y\in C$.

In 2006, Aoyama et al.  studied the variational inequality problem in Banach spaces. Such a problem is to find a point ${x}^{\ast }\in C$ such that for some $j\left(x-{x}^{\ast }\right)\in J\left(x-{x}^{\ast }\right)$,
$〈A{x}^{\ast },j\left({x}^{\ast }-x\right)〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C.$
(1.3)
The set of solutions of (1.3) in Banach spaces is denoted by $S\left(C,A\right)$, that is,
$S\left(C,A\right)=\left\{u\in C:〈Au,J\left(v-u\right)〉\ge 0,\mathrm{\forall }v\in C\right\}.$
(1.4)

They introduced the strong convergence theorem involving the variational inequality problem in Banach spaces as follows.

Theorem 1.1 Let E be a uniformly convex and 2-uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let ${Q}_{C}$ be a sunny nonexpansive retraction from E onto C, let $\alpha >0$, and let A be an α-inverse strongly accretive operator of C into E with $S\left(C,A\right)\ne \mathrm{\varnothing }$. Suppose ${x}_{1}=x\in C$ and $\left\{{x}_{n}\right\}$ is given by
${x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right){Q}_{C}\left({x}_{n}-{\lambda }_{n}A{x}_{n}\right)$

for every $n=1,2,\dots$ , where $\left\{{\lambda }_{n}\right\}$ is a sequence of positive real numbers and $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left[0,1\right]$. If $\left\{{\lambda }_{n}\right\}$ and $\left\{{\alpha }_{n}\right\}$ are chosen so that ${\lambda }_{n}\in \left[a,\frac{\alpha }{{K}^{2}}\right]$ for some $a>0$ and ${\alpha }_{n}\in \left[b,c\right]$ for some b, c with $0, then $\left\{{x}_{n}\right\}$ converges weakly to some element z of $S\left(C,A\right)$, where K is the 2-uniformly smoothness constant of E.

Many authors have studied the variational inequality problem; see, for example, . The variational inequality problem is an important tool for studying fixed point theory, equilibrium problems, optimization problems and partial differential equations with applications principally drawn from mechanics; see, e.g., [9, 10].

Recently, Kangtunyakarn  introduced a new mapping in uniformly convex and 2-smooth Banach spaces to prove a strong convergence theorem for finding a common element of the set of fixed points of finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of solutions of variational inequality problems as follows.

Theorem 1.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ${Q}_{C}$ be a sunny nonexpansive retraction from E onto C, and let A, B be α and β-inverse strongly accretive mappings of C into E, respectively. Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strict pseudo-contractions of C into itself, and let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of nonexpansive mappings of C into itself with $\mathcal{F}={\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap S\left(C,A\right)\cap S\left(C,B\right)\ne \mathrm{\varnothing }$ and $\kappa =min\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$ with ${K}^{2}\le \kappa$, where K is the 2-uniformly smooth constant of E. Let ${\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, where $I=\left[0,1\right]$, ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$, ${\alpha }_{1}^{j}\in \left(0,1\right]$, ${\alpha }_{2}^{j}\in \left[0,1\right]$ and ${\alpha }_{3}^{j}\in \left(0,1\right)$ for all $j=1,2,\dots ,N$. Let ${S}^{A}$ be the ${S}^{A}$-mapping generated by ${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$. Let $\left\{{x}_{n}\right\}$ be the sequence generated by ${x}_{1}$, $u\in C$ and
${x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-aA\right){x}_{n}+{\delta }_{n}{Q}_{C}\left(I-bB\right){x}_{n}+{\eta }_{n}{S}^{A}{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,$
(1.5)
where $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\},\left\{{\gamma }_{n}\right\},\left\{{\delta }_{n}\right\},\left\{{\eta }_{n}\right\}\in \left[0,1\right]$ and ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}+{\delta }_{n}+{\eta }_{n}=1$ and satisfy the following conditions:
1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

2. (ii)

$\left\{{\gamma }_{n}\right\},\left\{{\delta }_{n}\right\},\left\{{\eta }_{n}\right\}\subseteq \left[c,d\right]\subset \left(0,1\right)$ for some $c,d>0$;

3. (iii)

${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{\eta }_{n+1}-{\eta }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|<\mathrm{\infty }$;

4. (iv)

$0<{lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}\le {lim sup}_{n\to \mathrm{\infty }}{\beta }_{n}<1$;

5. (v)

$a\in \left(0,\frac{\alpha }{{K}^{2}}\right)$ and $b\in \left(0,\frac{\beta }{{K}^{2}}\right)$.

Then $\left\{{x}_{n}\right\}$ converges strongly to ${z}_{0}={Q}_{\mathcal{F}}u$, where ${Q}_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto .

For every $i=1,2,\dots ,N$, let ${A}_{i}:C\to H$ be a mapping. From (1.3), we introduce the combination of variational inequality problems in Banach spaces as follows: to find a point ${x}^{\ast }\in C$ such that for some $j\left(x-{x}^{\ast }\right)\in J\left(x-{x}^{\ast }\right)$,
$〈\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}^{\ast },j\left({x}^{\ast }-x\right)〉\ge 0$
(1.6)
for all $x\in C$ and ${a}_{i}$ is a positive real number for all $i=1,2,\dots ,N$ with ${\sum }_{i=1}^{N}{a}_{i}=1$. The set of solutions of (1.6) in Banach spaces is denoted by $S\left(C,{\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right)$, that is,
$S\left(C,\sum _{i=1}^{N}{a}_{i}{A}_{i}\right)=\left\{u\in C:〈\sum _{i=1}^{N}{a}_{i}{A}_{i}u,J\left(v-u\right)〉\ge 0,\mathrm{\forall }v\in C\right\}.$
(1.7)

By using (1.6) we prove the convergence theorem for a finite family of the set of solutions of variational inequality problems and two sets of fixed points of nonlinear mappings in a Banach space.

## 2 Preliminaries

The following lemmas are important tools to prove our main results in the next section.

Lemma 2.1 (See )

Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
${\parallel x+y\parallel }^{2}\le {\parallel x\parallel }^{2}+2〈y,J\left(x\right)〉+2{\parallel Ky\parallel }^{2}$

for any $x,y\in E$.

Lemma 2.2 (See )

Let X be a uniformly convex Banach space and ${B}_{r}=\left\{x\in X:\parallel x\parallel \le r\right\}$, $r>0$. Then there exists a continuous, strictly increasing and convex function $g:\left[0,\mathrm{\infty }\right]\to \left[0,\mathrm{\infty }\right]$, $g\left(0\right)=0$ such that
${\parallel \alpha x+\beta y+\gamma z\parallel }^{2}\le \alpha {\parallel x\parallel }^{2}+\beta {\parallel y\parallel }^{2}+\gamma {\parallel z\parallel }^{2}-\alpha \beta g\left(\parallel x-y\parallel \right)$

for all $x,y,z\in {B}_{r}$ and all $\alpha ,\beta ,\gamma \in \left[0,1\right]$ with $\alpha +\beta +\gamma =1$.

Remark 2.3 For every $i=1,2,\dots ,N$, if ${x}_{i}\in {B}_{r}\left(0\right)$, from Lemma 2.2, we have ${\parallel {\sum }_{i=1}^{N}{a}_{i}{x}_{i}\parallel }^{2}\le {\sum }_{i=1}^{N}{a}_{i}{\parallel {x}_{i}\parallel }^{2}$, where ${a}_{i}\in \left[0,1\right]$ and ${\sum }_{i=1}^{N}{a}_{i}=1$.

Lemma 2.4 (See )

Let C be a nonempty closed convex subset of a smooth Banach space E. Let ${Q}_{C}$ be a sunny nonexpansive retraction from E onto C, and let A be an accretive operator of C into E. Then, for all $\lambda >0$,
$S\left(C,A\right)=F\left({Q}_{C}\left(I-\lambda A\right)\right).$

Lemma 2.5 (See )

Let $r>0$. If E is uniformly convex, then there exists a continuous, strictly increasing and convex function $g:\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$, $g\left(0\right)=0$ such that for all $x,y\in {B}_{r}\left(0\right)=\left\{x\in E:\parallel x\parallel \le r\right\}$ and for any $\alpha \in \left[0,1\right]$, we have ${\parallel \alpha x+\left(1-\alpha \right)y\parallel }^{2}\le \alpha {\parallel x\parallel }^{2}+\left(1-\alpha \right){\parallel y\parallel }^{2}-\alpha \left(1-\alpha \right)g\left(\parallel x-y\parallel \right)$.

Lemma 2.6 (See )

Let C be a closed and convex subset of a real uniformly smooth Banach space E, and let $T:C\to C$ be a nonexpansive mapping with a nonempty fixed point $F\left(T\right)$. If $\left\{{x}_{n}\right\}\subset C$ is a bounded sequence such that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-T{x}_{n}\parallel =0$, then there exists a unique sunny nonexpansive retraction ${Q}_{F\left(T\right)}:C\to F\left(T\right)$ such that
$\underset{n\to \mathrm{\infty }}{lim sup}〈u-{Q}_{F\left(T\right)}u,J\left({x}_{n}-{Q}_{F\left(T\right)}u\right)〉\le 0$

for any given $u\in C$.

Lemma 2.7 (See )

Let $\left\{{s}_{n}\right\}$ be a sequence of nonnegative real numbers satisfying ${s}_{n+1}\le \left(1-{\alpha }_{n}\right){s}_{n}+{\delta }_{n}$, $\mathrm{\forall }n\ge 0$, where $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left(0,1\right)$ and $\left\{{\delta }_{n}\right\}$ is a sequence such that
1. (1)

${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$,

2. (2)

${lim sup}_{n\to \mathrm{\infty }}\frac{{\delta }_{n}}{{\alpha }_{n}}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n}|<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{s}_{n}=0$.

Lemma 2.8 

Let C be a closed convex subset of a strictly convex Banach space E. Let ${T}_{1}$, ${T}_{2}$ and ${T}_{3}$ be three nonexpansive mappings from C into itself with $F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\cap F\left({T}_{3}\right)\ne \mathrm{\varnothing }$. Define a mapping S by
$Sx=\alpha {T}_{1}x+\beta {T}_{2}x+\gamma {T}_{3}x,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C,$

where α, β, γ is a constant in $\left(0,1\right)$ and $\alpha +\beta +\gamma =1$. Then S is a nonexpansive mapping and $F\left(S\right)=F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\cap F\left({T}_{3}\right)$.

Lemma 2.9 Let C be a nonempty closed convex subset of a real smooth Banach space E. For every $i=1,2,\dots ,N$, let ${A}_{i}:C\to E$ be an ${\alpha }_{i}$-strongly accretive mapping with $\overline{\alpha }={min}_{i=1,2,\dots ,N}\left\{{\alpha }_{i}\right\}$ and ${\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)\ne \mathrm{\varnothing }$. Then $S\left(C,{\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right)={\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)$, where ${a}_{i}\in \left[0,1\right]$ and ${\sum }_{i=1}^{N}{a}_{i}=1$.

Proof It is easy to see that ${\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)\subseteq S\left(C,{\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right)$. Let ${x}_{0}\in S\left(C,{\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right)$ and ${x}^{\ast }\in {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)$. Then there exist $j\left(y-{x}^{\ast }\right)\in J\left(y-{x}^{\ast }\right)$ and $j\left(y-{x}_{0}\right)\in J\left(y-{x}_{0}\right)$ such that
$〈\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}_{0},j\left(y-{x}_{0}\right)〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C$
(2.1)
and
$〈\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}^{\ast },j\left(y-{x}^{\ast }\right)〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
(2.2)
From (2.1), (2.2) and ${x}_{0},{x}^{\ast }\in C$, we have
$〈\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}_{0},j\left({x}^{\ast }-{x}_{0}\right)〉\ge 0$
(2.3)
and
$〈\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}^{\ast },j\left({x}_{0}-{x}^{\ast }\right)〉\ge 0.$
(2.4)
From (2.3) and (2.4), we have
$\begin{array}{rcl}0& \le & 〈\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}_{0}-\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}^{\ast },j\left({x}^{\ast }-{x}_{0}\right)〉\\ =& -〈\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}^{\ast }-\sum _{i=1}^{N}{a}_{i}{A}_{i}{x}_{0},j\left({x}^{\ast }-{x}_{0}\right)〉\\ =& -\sum _{i=1}^{N}{a}_{i}〈{A}_{i}{x}^{\ast }-{A}_{i}{x}_{0},j\left({x}^{\ast }-{x}_{0}\right)〉\le -\sum _{i=1}^{N}{a}_{i}{\alpha }_{i}{\parallel {x}^{\ast }-{x}_{0}\parallel }^{2}\\ \le & -\sum _{i=1}^{N}{a}_{i}\overline{\alpha }{\parallel {x}^{\ast }-{x}_{0}\parallel }^{2}=-\overline{\alpha }{\parallel {x}^{\ast }-{x}_{0}\parallel }^{2}.\end{array}$

It implies that ${x}^{\ast }={x}_{0}$, that is, ${x}_{0}\in {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)$. Therefore $S\left(C,{\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right)\subseteq {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)$. □

## 3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ${Q}_{C}$ be a sunny nonexpansive retraction from E onto C. For every $i=1,2,\dots ,N$, let ${A}_{i}:C\to E$ be ${\alpha }_{i}$-strongly accretive and ${L}_{i}$-Lipschitz continuous with $\overline{\alpha }={min}_{i=1,2,\dots ,N}{\alpha }_{i}$ and $\overline{L}={max}_{i=1,2,\dots ,N}{L}_{i}$. Let $T:C\to C$ be a nonexpansive mapping and $S:C\to C$ be an η-strictly pseudo-contractive mapping with ${K}^{2}\le \eta$, where K is the 2-uniformly smooth constant of E. Assume that $\mathcal{F}=F\left(T\right)\cap F\left(S\right)\cap {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be a sequence generated by u, ${x}_{1}\in C$ and
$\left\{\begin{array}{c}{z}_{n}={c}_{n}{x}_{n}+\left(1-{c}_{n}\right)S{x}_{n},\hfill \\ {y}_{n}={b}_{n}{x}_{n}+\left(1-{b}_{n}\right)T{z}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-\lambda {\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
(3.1)
where ${a}_{i}\in \left[0,1\right]$ for all $i=1,2,\dots ,N$ and $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\},\left\{{\gamma }_{n}\right\}\subseteq \left[0,1\right]$ with ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}=1$ for all $n\in \mathbb{N}$ satisfy the following conditions:
1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

2. (ii)

$0 for some $a,b>0$, $\mathrm{\forall }n\in \mathbb{N}$ and ${\sum }_{i=1}^{N}{a}_{i}=1$;

3. (iii)

$0\le \lambda {K}^{2}\le \frac{\overline{\alpha }}{{\overline{L}}^{2}}$;

4. (iv)

${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{b}_{n+1}-{b}_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{c}_{n+1}-{c}_{n}|<\mathrm{\infty }$.

Then $\left\{{x}_{n}\right\}$ converges strongly to ${z}_{0}={Q}_{\mathcal{F}}u$, where ${Q}_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto .

Proof First, we show that ${\sum }_{i=1}^{N}{a}_{i}{A}_{i}$ is an $\frac{\overline{\alpha }}{{\overline{L}}^{2}}$-inverse strongly monotone mapping.

Let $x,y\in C$, there exists $j\left(x-y\right)\in J\left(x-y\right)$ and
$\begin{array}{rcl}〈\sum _{i=1}^{N}{a}_{i}{A}_{i}x-\sum _{i=1}^{N}{a}_{i}{A}_{i}y,j\left(x-y\right)〉& =& \sum _{i=1}^{N}{a}_{i}〈{A}_{i}x-{A}_{i}y,j\left(x-y\right)〉\\ \ge & \sum _{i=1}^{N}{a}_{i}{\alpha }_{i}{\parallel x-y\parallel }^{2}\\ \ge & \sum _{i=1}^{N}{a}_{i}\frac{{\alpha }_{i}}{{L}_{i}^{2}}{\parallel {A}_{i}x-{A}_{i}y\parallel }^{2}\\ \ge & \frac{\overline{\alpha }}{{\overline{L}}^{2}}\sum _{i=1}^{N}{a}_{i}{\parallel {A}_{i}x-{A}_{i}y\parallel }^{2}\\ \ge & \frac{\overline{\alpha }}{{\overline{L}}^{2}}{\parallel \sum _{i=1}^{N}{a}_{i}{A}_{i}x-\sum _{i=1}^{N}{a}_{i}{A}_{i}y\parallel }^{2}.\end{array}$
(3.2)
Next, we show that ${Q}_{C}\left(I-\lambda {\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right)$ is a nonexpansive mapping. From (3.2), we have
$\begin{array}{r}{\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right)x-{Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right)y\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y-\lambda \left(\sum _{i=1}^{N}{a}_{i}{A}_{i}x-\sum _{i=1}^{N}{a}_{i}{A}_{i}y\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}-2\lambda 〈\sum _{i=1}^{N}{a}_{i}{A}_{i}x-\sum _{i=1}^{N}{a}_{i}{A}_{i}y,j\left(x-y\right)〉\\ \phantom{\rule{2em}{0ex}}+2{K}^{2}{\lambda }^{2}{\parallel \sum _{i=1}^{N}{a}_{i}\left({A}_{i}x-{A}_{i}y\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}-2\lambda \frac{\overline{\alpha }}{{\overline{L}}^{2}}{\parallel \sum _{i=1}^{N}{a}_{i}{A}_{i}x-\sum _{i=1}^{N}{a}_{i}{A}_{i}y\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}+2{K}^{2}{\lambda }^{2}{\parallel \sum _{i=1}^{N}{a}_{i}\left({A}_{i}x-{A}_{i}y\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}={\parallel x-y\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}-2\lambda \left(\frac{\overline{\alpha }}{{\overline{L}}^{2}}-{K}^{2}\lambda \right){\parallel \sum _{i=1}^{N}{a}_{i}{A}_{i}x-\sum _{i=1}^{N}{a}_{i}{A}_{i}y\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}\end{array}$
for all $x,y\in C$. Let ${x}^{\ast }\in \mathcal{F}$, we have
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}^{\ast }\parallel & =& \parallel {\alpha }_{n}\left(u-{x}^{\ast }\right)+{\beta }_{n}\left({x}_{n}-{x}^{\ast }\right)+{\gamma }_{n}\left({Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}^{\ast }\right)\parallel \\ \le & {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +{\beta }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +{\gamma }_{n}\parallel {y}_{n}-{x}^{\ast }\parallel \\ =& {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +{\beta }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +{\gamma }_{n}\parallel {b}_{n}\left({x}_{n}-p\right)+\left(1-{b}_{n}\right)\left(T{z}_{n}-{x}^{\ast }\right)\parallel \\ \le & {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +{\beta }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel \\ +{\gamma }_{n}\left({b}_{n}\parallel {x}_{n}-p\parallel +\left(1-{b}_{n}\right)\parallel {z}_{n}-{x}^{\ast }\parallel \right)\\ =& {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +{\beta }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +{\gamma }_{n}\left({b}_{n}\parallel {x}_{n}-p\parallel \\ +\left(1-{b}_{n}\right)\parallel {c}_{n}\left({x}_{n}-{x}^{\ast }\right)+\left(1-{c}_{n}\right)\left(S{x}_{n}-{x}^{\ast }\right)\parallel \right).\end{array}$
(3.3)
Since S is a strictly pseudo-contractive mapping, we have
$\begin{array}{rcl}{\parallel {c}_{n}\left({x}_{n}-{x}^{\ast }\right)+\left(1-{c}_{n}\right)\left(S{x}_{n}-{x}^{\ast }\right)\parallel }^{2}& =& {\parallel {x}_{n}-{x}^{\ast }+\left(1-{c}_{n}\right)\left(S{x}_{n}-{x}_{n}\right)\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+2\left(1-{c}_{n}\right)〈S{x}_{n}-{x}_{n},j\left({x}_{n}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left(1-{c}_{n}\right)}^{2}{\parallel S{x}_{n}-{x}_{n}\parallel }^{2}\\ =& {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-2\left(1-{c}_{n}\right)〈\left(I-S\right){x}_{n},j\left({x}_{n}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left(1-{c}_{n}\right)}^{2}{\parallel \left(I-S\right){x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-2\left(1-{c}_{n}\right)\eta {\parallel \left(I-S\right){x}_{n}\parallel }^{2}\\ +2{K}^{2}{\left(1-{c}_{n}\right)}^{2}{\parallel \left(I-S\right){x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-2{\left(1-{c}_{n}\right)}^{2}\eta {\parallel \left(I-S\right){x}_{n}\parallel }^{2}\\ +2{K}^{2}{\left(1-{c}_{n}\right)}^{2}{\parallel \left(I-S\right){x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-2{\left(1-{c}_{n}\right)}^{2}\left(\eta -{K}^{2}\right){\parallel \left(I-S\right){x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}.\end{array}$
(3.4)
From (3.3) and (3.4), we have
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}^{\ast }\parallel & \le & {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +{\beta }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +{\gamma }_{n}\left({b}_{n}\parallel {x}_{n}-p\parallel \\ +\left(1-{b}_{n}\right)\parallel {c}_{n}\left({x}_{n}-{x}^{\ast }\right)+\left(1-{c}_{n}\right)\left(S{x}_{n}-{x}^{\ast }\right)\parallel \right)\\ \le & {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +\left(1-{\alpha }_{n}\right)\parallel {x}_{n}-{x}^{\ast }\parallel \\ \le & max\left\{\parallel u-{x}^{\ast }\parallel ,\parallel {x}_{1}-{x}^{\ast }\parallel \right\}.\end{array}$

From induction we can conclude that $\left\{{x}_{n}\right\}$ is bounded and so are $\left\{{y}_{n}\right\}$, $\left\{{z}_{n}\right\}$.

Next, we show that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n+1}-{x}_{n}\parallel =0$.

For every $n\in \mathbb{N}$, we have
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}_{n}\parallel & =& \parallel {\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}\\ -{\alpha }_{n-1}u-{\beta }_{n-1}{x}_{n-1}-{\gamma }_{n-1}{Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n-1}\parallel \\ \le & |{\alpha }_{n}-{\alpha }_{n-1}|\parallel u\parallel +{\beta }_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel {x}_{n-1}\parallel \\ +{\gamma }_{n}\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n-1}\parallel \\ +|{\gamma }_{n}-{\gamma }_{n-1}|\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n-1}\parallel \\ \le & |{\alpha }_{n}-{\alpha }_{n-1}|\parallel u\parallel +{\beta }_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel {x}_{n-1}\parallel \\ +{\gamma }_{n}\parallel {y}_{n}-{y}_{n-1}\parallel +|{\gamma }_{n}-{\gamma }_{n-1}|\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n-1}\parallel .\end{array}$
(3.5)
From the definition of ${y}_{n}$, we have
$\begin{array}{rcl}\parallel {y}_{n}-{y}_{n-1}\parallel & =& \parallel {b}_{n}{x}_{n}+\left(1-{b}_{n}\right)T{z}_{n}-{b}_{n-1}{x}_{n-1}-\left(1-{b}_{n-1}\right)T{z}_{n-1}\parallel \\ \le & {b}_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel {x}_{n-1}\parallel +\left(1-{b}_{n}\right)\parallel T{z}_{n}-T{z}_{n-1}\parallel \\ +|{b}_{n}-{b}_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & {b}_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel {x}_{n-1}\parallel +\left(1-{b}_{n}\right)\parallel {z}_{n}-{z}_{n-1}\parallel \\ +|{b}_{n}-{b}_{n-1}|\parallel T{z}_{n-1}\parallel .\end{array}$
(3.6)
From the definition of ${z}_{n}$, we have
$\begin{array}{rcl}\parallel {z}_{n}-{z}_{n-1}\parallel & =& \parallel {c}_{n}{x}_{n}+\left(1-{c}_{n}\right)S{x}_{n}-{c}_{n-1}{x}_{n-1}-\left(1-{c}_{n-1}\right)S{x}_{n-1}\parallel \\ =& \parallel {c}_{n}\left({x}_{n}-{x}_{n-1}\right)+\left({c}_{n}-{c}_{n-1}\right){x}_{n-1}+\left(1-{c}_{n}\right)\left(S{x}_{n}-S{x}_{n-1}\right)\\ +\left({c}_{n-1}-{c}_{n}\right)S{x}_{n-1}\parallel \\ \le & \parallel {c}_{n}\left({x}_{n}-{x}_{n-1}\right)+\left(1-{c}_{n}\right)\left(S{x}_{n}-S{x}_{n-1}\right)\parallel +|{c}_{n}-{c}_{n-1}|\parallel {x}_{n-1}\parallel \\ +|{c}_{n}-{c}_{n-1}|\parallel S{x}_{n-1}\parallel .\end{array}$
(3.7)
Since S is an η-strictly pseudo-contractive mapping, we have
$\begin{array}{r}{\parallel {c}_{n}\left({x}_{n}-{x}_{n-1}\right)+\left(1-{c}_{n}\right)\left(S{x}_{n}-S{x}_{n-1}\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}={\parallel {x}_{n}-{x}_{n-1}-\left(1-{c}_{n}\right)\left(\left(I-S\right){x}_{n}-\left(I-S\right){x}_{n-1}\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-{x}_{n-1}\parallel }^{2}-2\left(1-{c}_{n}\right)〈\left(I-S\right){x}_{n}-\left(I-S\right){x}_{n-1},j\left({x}_{n}-{x}_{n-1}\right)〉\\ \phantom{\rule{2em}{0ex}}+2{K}^{2}{\left(1-{c}_{n}\right)}^{2}{\parallel \left(I-S\right){x}_{n}-\left(I-S\right){x}_{n-1}\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-{x}_{n-1}\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}-2\left(1-{c}_{n}\right)\eta {\parallel \left(I-S\right){x}_{n}-\left(I-S\right){x}_{n-1}\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}+2{K}^{2}{\left(1-{c}_{n}\right)}^{2}{\parallel \left(I-S\right){x}_{n}-\left(I-S\right){x}_{n-1}\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-{x}_{n-1}\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}-2{\left(1-{c}_{n}\right)}^{2}\left(\eta -{K}^{2}\right){\parallel \left(I-S\right){x}_{n}-\left(I-S\right){x}_{n-1}\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-{x}_{n-1}\parallel }^{2}.\end{array}$
(3.8)
From (3.6), (3.7) and (3.8), we have
$\begin{array}{rcl}\parallel {y}_{n}-{y}_{n-1}\parallel & \le & {b}_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel {x}_{n-1}\parallel +\left(1-{b}_{n}\right)\parallel {z}_{n}-{z}_{n-1}\parallel \\ +|{b}_{n}-{b}_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & {b}_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel {x}_{n-1}\parallel +\left(1-{b}_{n}\right)\left(\parallel {c}_{n}\left({x}_{n}-{x}_{n-1}\right)\\ +\left(1-{c}_{n}\right)\left(S{x}_{n}-S{x}_{n-1}\right)\parallel +|{c}_{n}-{c}_{n-1}|\parallel {x}_{n-1}\parallel \\ +|{c}_{n}-{c}_{n-1}|\parallel S{x}_{n-1}\parallel \right)+|{b}_{n}-{b}_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & {b}_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel {x}_{n-1}\parallel \\ +\left(1-{b}_{n}\right)\left(\parallel {x}_{n}-{x}_{n-1}\parallel +|{c}_{n}-{c}_{n-1}|\parallel {x}_{n-1}\parallel \\ +|{c}_{n}-{c}_{n-1}|\parallel S{x}_{n-1}\parallel \right)+|{b}_{n}-{b}_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & \parallel {x}_{n}-{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel {x}_{n-1}\parallel +|{c}_{n}-{c}_{n-1}|\parallel {x}_{n-1}\parallel \\ +|{c}_{n}-{c}_{n-1}|\parallel S{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel T{z}_{n-1}\parallel .\end{array}$
(3.9)
From (3.9) and (3.5), we have
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}_{n}\parallel & \le & |{\alpha }_{n}-{\alpha }_{n-1}|\parallel u\parallel +{\beta }_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel {x}_{n-1}\parallel \\ +{\gamma }_{n}\parallel {y}_{n}-{y}_{n-1}\parallel +|{\gamma }_{n}-{\gamma }_{n-1}|\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n-1}\parallel \\ \le & \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-{x}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel {x}_{n-1}\parallel +|{\alpha }_{n}-{\alpha }_{n-1}|\parallel u\parallel \\ +|{\gamma }_{n}-{\gamma }_{n-1}|\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n-1}\parallel \\ +|{b}_{n}-{b}_{n-1}|\parallel {x}_{n-1}\parallel +|{c}_{n}-{c}_{n-1}|\parallel {x}_{n-1}\parallel \\ +|{c}_{n}-{c}_{n-1}|\parallel S{x}_{n-1}\parallel +|{b}_{n}-{b}_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-{x}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|M+|{\alpha }_{n}-{\alpha }_{n-1}|M\\ +|{\gamma }_{n}-{\gamma }_{n-1}|M+2|{b}_{n}-{b}_{n-1}|M+2|{c}_{n}-{c}_{n-1}|M,\end{array}$
where $M={max}_{n\in \mathbb{N}}\left\{\parallel {x}_{n}\parallel ,\parallel u\parallel ,\parallel {Q}_{C}\left(I-\lambda {\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}\parallel ,\parallel S{x}_{n}\parallel ,\parallel T{z}_{n}\parallel \right\}$. Applying Lemma 2.7, conditions (i) and (iv), we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-{x}_{n}\parallel =0.$
(3.10)
Next, we show that
$\underset{n\to \mathrm{\infty }}{lim}\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){x}_{n}-{x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-T{x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-S{x}_{n}\parallel =0.$
From the definition of ${x}_{n}$, we have
$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& =& {\parallel {\alpha }_{n}\left(u-{x}^{\ast }\right)+{\beta }_{n}\left({x}_{n}-{x}^{\ast }\right)+{\gamma }_{n}\left({Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}^{\ast }\right)\parallel }^{2}\\ \le & {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\beta }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\gamma }_{n}{\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}^{\ast }\parallel }^{2}\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ \le & {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\beta }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\gamma }_{n}{\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ =& {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\beta }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\gamma }_{n}{\parallel {b}_{n}\left({x}_{n}-{x}^{\ast }\right)+\left(1-{b}_{n}\right)\left(T{z}_{n}-{x}^{\ast }\right)\parallel }^{2}\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ \le & {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\beta }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\\ +{\gamma }_{n}\left({b}_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left(1-{b}_{n}\right){\parallel {z}_{n}-{x}^{\ast }\parallel }^{2}-{b}_{n}\left(1-{b}_{n}\right){g}_{2}\left(\parallel {x}_{n}-T{z}_{n}\parallel \right)\right)\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ \le & {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\beta }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\\ +{\gamma }_{n}\left({b}_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left(1-{b}_{n}\right)\left({\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\\ -2{\left(1-{c}_{n}\right)}^{2}\left(\eta -{K}^{2}\right){\parallel S{x}_{n}-{x}_{n}\parallel }^{2}\right)-{b}_{n}\left(1-{b}_{n}\right){g}_{2}\left(\parallel {x}_{n}-T{z}_{n}\parallel \right)\right)\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ \le & {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\beta }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\\ +{\gamma }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-2{\gamma }_{n}\left(1-{b}_{n}\right){\left(1-{c}_{n}\right)}^{2}\left(\eta -{K}^{2}\right){\parallel S{x}_{n}-{x}_{n}\parallel }^{2}\\ -{\gamma }_{n}{b}_{n}\left(1-{b}_{n}\right){g}_{2}\left(\parallel {x}_{n}-T{z}_{n}\parallel \right)\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ \le & {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\\ -2{\gamma }_{n}\left(1-{b}_{n}\right){\left(1-{c}_{n}\right)}^{2}\left(\eta -{K}^{2}\right){\parallel S{x}_{n}-{x}_{n}\parallel }^{2}\\ -{\gamma }_{n}{b}_{n}\left(1-{b}_{n}\right){g}_{2}\left(\parallel {x}_{n}-T{z}_{n}\parallel \right)\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right).\end{array}$
It implies that
$\begin{array}{r}2{\gamma }_{n}\left(1-{b}_{n}\right){\left(1-{c}_{n}\right)}^{2}\left(\eta -{K}^{2}\right){\parallel S{x}_{n}-{x}_{n}\parallel }^{2}+{\gamma }_{n}{b}_{n}\left(1-{b}_{n}\right){g}_{2}\left(\parallel {x}_{n}-T{z}_{n}\parallel \right)\\ \phantom{\rule{2em}{0ex}}+{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+\left(\parallel {x}_{n}-{x}^{\ast }\parallel +\parallel {x}_{n+1}-{x}^{\ast }\parallel \right)\parallel {x}_{n+1}-{x}_{n}\parallel .\end{array}$
From (3.10), conditions (i) and (ii), we have
$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}\parallel S{x}_{n}-{x}_{n}\parallel & =\underset{n\to \mathrm{\infty }}{lim}{g}_{1}\left(\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \right)\\ =\underset{n\to \mathrm{\infty }}{lim}{g}_{2}\left(\parallel {x}_{n}-T{z}_{n}\parallel \right)=0.\end{array}$
(3.11)
From the properties of ${g}_{1}$ and ${g}_{2}$, we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-T{z}_{n}\parallel =0.$
(3.12)
From (3.11) and the definition of ${z}_{n}$, we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {z}_{n}-{x}_{n}\parallel =0.$
(3.13)
From (3.12) and the definition of ${y}_{n}$, we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {y}_{n}-{x}_{n}\parallel =0.$
(3.14)
From (3.13) and (3.14), we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {y}_{n}-{z}_{n}\parallel =0.$
(3.15)
From (3.12) and (3.14), we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{y}_{n}\parallel =0.$
(3.16)
Then
$\begin{array}{rcl}\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){x}_{n}-{x}_{n}\parallel & \le & \parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){x}_{n}-{Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}\parallel \\ +\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{y}_{n}\parallel +\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{x}_{n}\parallel .\end{array}$
From (3.12) and (3.14), we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){x}_{n}-{x}_{n}\parallel =0.$
(3.17)
Since
$\begin{array}{rcl}\parallel T{x}_{n}-{x}_{n}\parallel & \le & \parallel T{x}_{n}-T{z}_{n}\parallel +\parallel T{z}_{n}-{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{z}_{n}\parallel +\parallel T{z}_{n}-{x}_{n}\parallel ,\end{array}$
from (3.12) and (3.13), we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel T{x}_{n}-{x}_{n}\parallel =0.$
(3.18)
Define the mapping $G:C\to C$ by $Gx=\alpha {Q}_{C}\left(I-\lambda {\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right)x+\beta Tx+\gamma Wx$, where $Wx=cx+\left(1-c\right)Sx$ for all $x\in C$ and $\alpha ,\beta ,\gamma ,c\in \left[0,1\right]$ with $\alpha +\beta +\gamma =1$. We show that W is a nonexpansive mapping. Let $x,y\in C$, we have
$\begin{array}{rcl}{\parallel Wx-Wy\parallel }^{2}& =& {\parallel c\left(x-y\right)+\left(1-c\right)\left(Sx-Sy\right)\parallel }^{2}\\ =& {\parallel x-y-\left(1-c\right)\left(\left(I-S\right)x-\left(I-S\right)y\right)\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}-2\left(1-c\right)〈\left(I-S\right)x-\left(I-S\right)y,j\left(x-y\right)〉\\ +2{K}^{2}{\left(1-c\right)}^{2}{\parallel \left(I-S\right)x-\left(I-S\right)y\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}-2\left(1-c\right)\eta {\parallel \left(I-S\right)x-\left(I-S\right)y\parallel }^{2}\\ +2{K}^{2}{\left(1-c\right)}^{2}{\parallel \left(I-S\right)x-\left(I-S\right)y\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}-2{\left(1-c\right)}^{2}\left(\eta -{K}^{2}\right){\parallel \left(I-S\right)x-\left(I-S\right)y\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}.\end{array}$
Then W is a nonexpansive mapping. It is easy to see that the mapping G is nonexpansive. From the definition of W, we have
$F\left(S\right)=F\left(W\right).$
(3.19)
From (3.19) Lemmas 2.8, 2.9 and the definition of G, we have $F\left(G\right)=F\left(T\right)\cap F\left(S\right)\cap {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)=\mathcal{F}$. Since
$\begin{array}{rcl}\parallel G{x}_{n}-{x}_{n}\parallel & \le & \alpha \parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){x}_{n}-{x}_{n}\parallel +\beta \parallel T{x}_{n}-{x}_{n}\parallel +\gamma \parallel W{x}_{n}-{x}_{n}\parallel \\ =& \alpha \parallel {Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){x}_{n}-{x}_{n}\parallel +\beta \parallel T{x}_{n}-{x}_{n}\parallel +\gamma \left(1-c\right)\parallel S{x}_{n}-{x}_{n}\parallel ,\end{array}$
and (3.11), (3.17) and (3.18), we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel G{x}_{n}-{x}_{n}\parallel =0.$
(3.20)
From Lemma 2.6, we have
$\underset{n\to \mathrm{\infty }}{lim sup}〈u-{z}_{0},j\left({x}_{n}-{z}_{0}\right)〉\le 0,$
(3.21)

where ${z}_{0}={Q}_{\mathcal{F}}u$.

Finally, we show that the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${z}_{0}={Q}_{\mathcal{F}}u$. From the definition of ${x}_{n}$, we have
$\begin{array}{rcl}{\parallel {x}_{n+1}-{z}_{0}\parallel }^{2}& =& {\parallel {\alpha }_{n}\left(u-{z}_{0}\right)+{\beta }_{n}\left({x}_{n}-{z}_{0}\right)+{\gamma }_{n}\left({Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{z}_{0}\right)\parallel }^{2}\\ \le & {\parallel {\beta }_{n}\left({x}_{n}-{z}_{0}\right)+{\gamma }_{n}\left({Q}_{C}\left(I-\lambda \sum _{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n}-{z}_{0}\right)\parallel }^{2}\\ +2{\alpha }_{n}〈u-{z}_{0},j\left({x}_{n+1}-{z}_{0}\right)〉\\ \le & \left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{z}_{0}\parallel }^{2}+2{\alpha }_{n}〈u-{z}_{0},j\left({x}_{n+1}-{z}_{0}\right)〉.\end{array}$

From Lemma 2.7 and condition (i), we can conclude that the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${z}_{0}={Q}_{\mathcal{F}}u$. This completes the proof. □

The following corollary is a direct sequel of Theorem 3.1. Therefore, we omit the proof.

Corollary 3.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ${Q}_{C}$ be a sunny nonexpansive retraction from E onto C. Let $A:C\to E$ be α-strongly accretive and L-Lipschitz continuous. Let $T:C\to C$ be a nonexpansive mapping and $S:C\to C$ be an η-strictly pseudo-contractive mapping with ${K}^{2}\le \eta$, where K is the 2-uniformly smooth constant of E. Assume that $\mathcal{F}=F\left(T\right)\cap F\left(S\right)\cap S\left(C,A\right)\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be the sequence generated by $u,{x}_{1}\in C$ and
$\left\{\begin{array}{c}{z}_{n}={c}_{n}{x}_{n}+\left(1-{c}_{n}\right)S{x}_{n},\hfill \\ {y}_{n}={b}_{n}{x}_{n}+\left(1-{b}_{n}\right)T{z}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-\lambda A\right){y}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
where $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\},\left\{{\gamma }_{n}\right\}\subseteq \left[0,1\right]$ with ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}=1$ for all $n\in \mathbb{N}$ satisfy the following conditions:
1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

2. (ii)

$0, for some $a,b>0$, $\mathrm{\forall }n\in \mathbb{N}$;

3. (iii)

$0\le \lambda {K}^{2}\le \frac{\alpha }{{L}^{2}}$;

4. (iv)

${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{b}_{n+1}-{b}_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{c}_{n+1}-{c}_{n}|<\mathrm{\infty }$.

Then $\left\{{x}_{n}\right\}$ converges strongly to ${z}_{0}={Q}_{\mathcal{F}}u$, where ${Q}_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto .

## 4 Applications

Using the concepts of the ${S}^{A}$-mapping and Theorem 3.1, we prove the strong convergence theorem for the set of fixed points of two finite families of nonlinear mappings. We need the following definition and lemma to prove our result.

Definition 4.1 

Let C be a nonempty convex subset of a real Banach space. Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ and ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be two finite families of the mappings of C into itself. For each $j=1,2,\dots ,N$, let ${\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, where $I\in \left[0,1\right]$ and ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$. Define the mapping ${S}^{A}:C\to C$ as follows:
$\begin{array}{c}{U}_{0}={T}_{1}=I,\hfill \\ {U}_{1}={T}_{1}\left({\alpha }_{1}^{1}{S}_{1}{U}_{0}+{\alpha }_{2}^{1}{U}_{0}+{\alpha }_{3}^{1}I\right),\hfill \\ {U}_{2}={T}_{2}\left({\alpha }_{1}^{2}{S}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I\right),\hfill \\ {U}_{3}={T}_{3}\left({\alpha }_{1}^{3}{S}_{3}{U}_{2}+{\alpha }_{2}^{3}{U}_{2}+{\alpha }_{3}^{3}I\right),\hfill \\ ⋮\hfill \\ {U}_{N-1}={T}_{N-1}\left({\alpha }_{1}^{N-1}{S}_{N-1}{U}_{N-2}+{\alpha }_{2}^{N-1}{U}_{N-2}+{\alpha }_{3}^{N-1}I\right),\hfill \\ {S}^{A}={U}_{N}={T}_{N}\left({\alpha }_{1}^{N}{S}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I\right).\hfill \end{array}$

This mapping is called the ${S}^{A}$-mapping generated by ${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$.

Lemma 4.1 

Let C be a nonempty closed convex subset of a 2-uniformly smooth and uniformly convex Banach space. Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strict pseudo-contractions of C into itself, and let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of nonexpansive mappings of C into itself with ${\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\ne \mathrm{\varnothing }$ and $\kappa =min\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$ with ${K}^{2}\le \kappa$, where K is the 2-uniformly smooth constant of E. Let ${\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, where $I=\left[0,1\right]$, ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$, ${\alpha }_{1}^{j}\in \left(0,1\right]$, ${\alpha }_{2}^{j}\in \left[0,1\right]$ and ${\alpha }_{3}^{j}\in \left(0,1\right)$ for all $j=1,2,\dots ,N$. Let ${S}^{A}$ be the ${S}^{A}$-mapping generated by ${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$. Then $F\left({S}^{A}\right)={\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$ and ${S}^{A}$ is a nonexpansive mapping.

Theorem 4.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ${Q}_{C}$ be a sunny nonexpansive retraction from E onto C. For every $i=1,2,\dots ,N$, let ${A}_{i}:C\to E$ be ${\alpha }_{i}$-strongly accretive and ${L}_{i}$-Lipschitz continuous with $\overline{\alpha }={min}_{i=1,2,\dots ,N}{\alpha }_{i}$ and $\overline{L}={max}_{i=1,2,\dots ,N}{L}_{i}$. Let $S:C\to C$ be an η-strictly pseudo-contractive mapping with ${K}^{2}\le \eta$, where K is the 2-uniformly smooth constant of E. Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strict pseudo-contractions of C into itself, and let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of nonexpansive mappings of C into itself with $\kappa =min\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$ with ${K}^{2}\le \kappa$. Let ${\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, where $I=\left[0,1\right]$, ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$, ${\alpha }_{1}^{j}\in \left(0,1\right]$, ${\alpha }_{2}^{j}\in \left[0,1\right]$ and ${\alpha }_{3}^{j}\in \left(0,1\right)$ for all $j=1,2,\dots ,N$. Let ${S}^{A}$ be the ${S}^{A}$-mapping generated by ${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$. Assume that $\mathcal{F}=F\left(S\right)\cap {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be the sequence generated by $u,{x}_{1}\in C$ and
$\left\{\begin{array}{c}{z}_{n}={c}_{n}{x}_{n}+\left(1-{c}_{n}\right)S{x}_{n},\hfill \\ {y}_{n}={b}_{n}{x}_{n}+\left(1-{b}_{n}\right){S}^{A}{z}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-\lambda {\sum }_{i=1}^{N}{a}_{i}{A}_{i}\right){y}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
(4.1)
where ${a}_{i}\in \left[0,1\right]$ for all $i=1,2,\dots ,N$ and $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\},\left\{{\gamma }_{n}\right\}\subseteq \left[0,1\right]$ with ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}=1$ for all $n\in \mathbb{N}$ and satisfy the following conditions:
1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

2. (ii)

$0 for some $a,b>0$, $\mathrm{\forall }n\in \mathbb{N}$ and ${\sum }_{i=1}^{N}{a}_{i}=1$;

3. (iii)

$0\le \lambda {K}^{2}\le \frac{\overline{\alpha }}{{\overline{L}}^{2}}$;

4. (iv)

${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{b}_{n+1}-{b}_{n}|,{\sum }_{n=1}^{\mathrm{\infty }}|{c}_{n+1}-{c}_{n}|<\mathrm{\infty }$.

Then $\left\{{x}_{n}\right\}$ converges strongly to ${z}_{0}={Q}_{\mathcal{F}}u$, where ${Q}_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto .

Proof From Lemma 4.1 and Theorem 3.1, we can reach the desired conclusion. □

## Declarations

### Acknowledgements

This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, 10520, Thailand

## References 