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# A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces

Fixed Point Theory and Applications20132013:157

https://doi.org/10.1186/1687-1812-2013-157

• Accepted: 28 May 2013
• Published:

## Abstract

In this paper we introduce a new mapping in a uniformly convex and 2-smooth Banach space to prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and two sets of solutions of variational inequality problems. Moreover, we also obtain a strong convergence theorem for a finite family of the set of solutions of variational inequality problems and the set of fixed points of a finite family of strictly pseudo-contractive mappings by using our main result.

## Keywords

• nonexpansive mapping
• strictly pseudo-contractive mapping
• variational inequality problem

## 1 Introduction

Throughout this paper, we use E and ${E}^{\ast }$ to denote a real Banach space and a dual space of E, respectively. For any pair $x\in E$ and $f\in {E}^{\ast }$, $〈x,f〉$ instead of $f\left(x\right)$. 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 $x\in E$. It is well known that if E is a Hilbert space, then $J=I$, where I is the identity mapping. Recall the following definitions.

Definition 1.1 A Banach space E is said to be uniformly convex iff for any ϵ, $0<ϵ\le 2$, the inequalities $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $\parallel x-y\parallel \ge ϵ$ imply there exists a $\delta >0$ such that $\parallel \frac{x+y}{2}\parallel \le 1-\delta$.

Definition 1.2 A Banach space E is said to be smooth if for each $x\in {S}_{E}=\left\{x\in E:\parallel x\parallel =1\right\}$, there exists a unique functional ${j}_{x}\in {E}^{\ast }$ such that $〈x,{j}_{x}〉=\parallel x\parallel$ and $\parallel {j}_{x}\parallel =1$.

It is obvious that if E is smooth, then J is single-valued which is denoted by j.

Definition 1.3 Let E be a Banach space. Then a function ${\rho }_{E}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is said to be the modulus of smoothness of E if
${\rho }_{E}\left(t\right)=sup\left\{\frac{\parallel x+y\parallel +\parallel x-y\parallel }{2}-1:\parallel x\parallel =1,\parallel y\parallel =t\right\}.$
A Banach space E is said to be uniformly smooth if
$\underset{t\to 0}{lim}\frac{{\rho }_{E}\left(t\right)}{t}=0.$

It is well known that every uniformly smooth Banach space is smooth.

Let $q>1$. A Banach space E is said to be q-uniformly smooth if there exists a fixed constant $c>0$ such that ${\rho }_{E}\left(t\right)\le c{t}^{q}$. It is easy to see that if E is q-uniformly smooth, then $q\le 2$ and E is uniformly smooth.

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

for all $x,y\in C$.

T is called an η-strictly pseudo-contractive mapping 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 $j\left(x-y\right)\in J\left(x-y\right)$. We give some examples for a strictly pseudo-contractive mapping as follows.

Example 1.1 Let be a real line endowed with the Euclidean norm and let $C=\left(0,\mathrm{\infty }\right)$. Define the mapping $T:C\to C$ by
$Tx=\frac{2{x}^{2}}{3+2x},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C.$

Then T is a $\frac{1}{9}$-strictly pseudo-contractive mapping.

Example 1.2 (See [1])

Let be a real line endowed with the Euclidean norm. Let $C=\left[-1,1\right]$ and let $T:C\to C$ be defined by

Then T is a λ-strictly pseudo-contractive mapping where $\lambda \le min\left\{{\lambda }_{1},{\lambda }_{2}\right\}$ and ${\lambda }_{1}\le \frac{1}{2}$, ${\lambda }_{2}<1$.

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 [2] 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.

Subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D.

An operator A of C into E is said to be accretive 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
$〈Ax-Ay,j\left(x-y\right)〉\ge \alpha {\parallel Ax-Ay\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

Remark 1.3 From (1.1) and (1.2), if T is an η-strictly pseudo-contractive mapping, then $I-T$ is η-inverse strongly accretive.

The variational inequality problem in a Banach space is to find a point ${x}^{\ast }\in C$ such that for some $j\left(x-{x}^{\ast }\right)\in J\left(x-{x}^{\ast }\right)$,
$〈A{x}^{\ast },j\left(x-{x}^{\ast }\right)〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C.$
(1.3)
This problem was considered by Aoyama et al. [3]. The set of solutions of the variational inequality in a Banach space is denoted by $S\left(C,A\right)$, that is,
$S\left(C,A\right)=\left\{u\in C:〈Au,J\left(v-u\right)〉\ge 0,\phantom{\rule{0.25em}{0ex}}\mathrm{\forall }v\in C\right\}.$
(1.4)

Several problems in pure and applied science, numerous problems in physics and economics reduce to finding an element in (1.4); see, for instance, [46].

Recall that normal Mann’s iterative process was introduced by Mann [7] in 1953. The normal Mann’s iterative process generates a sequence $\left\{{x}_{n}\right\}$ in the following manner:
$\left\{\begin{array}{c}{x}_{1}\in C,\hfill \\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
(1.5)

where the sequence $\left\{{\alpha }_{n}\right\}\subset \left(0,1\right)$. If T is a nonexpansive mapping with a fixed point and the control sequence $\left\{{\alpha }_{n}\right\}$ is chosen so that ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}\left(1-{\alpha }_{n}\right)=\mathrm{\infty }$, then the sequence $\left\{{x}_{n}\right\}$ generated by normal Mann’s iterative process (1.5) converges weakly to a fixed point of T.

In 1967, Halpern has introduced the iteration method guaranteeing the strong convergence as follows:
$\left\{\begin{array}{c}{x}_{1}\in C,\hfill \\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{1}+{\alpha }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
(1.6)

where $\left\{{\alpha }_{n}\right\}\subset \left(0,1\right)$. Such an iteration is called Halpern iteration if T is a nonexpansive mapping with a fixed point. He also pointed out that the conditions ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$ are necessary for the strong convergence of $\left\{{x}_{n}\right\}$ to a fixed point of T.

Many authors have modified the iteration (1.6) for a strong convergence theorem; see, for instance, [810].

In 2008, Zhou [11] proved a strong convergence theorem for the modification of normal Mann’s iteration algorithm generated by a strict pseudo-contraction in a real 2-uniformly smooth Banach space as follows.

Theorem 1.4 Let C be a closed convex subset of a real 2-uniformly smooth Banach space E and let $T:C\to C$ be a λ-strict pseudo-contraction such that $F\left(T\right)\ne \mathrm{\varnothing }$. Given $u,{x}_{0}\in C$ and sequences $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$, $\left\{{\gamma }_{n}\right\}$ and $\left\{{\delta }_{n}\right\}$ in $\left(0,1\right)$, the following control conditions are satisfied:
Let a sequence $\left\{{x}_{n}\right\}$ be generated by
$\left\{\begin{array}{c}{y}_{n}={\alpha }_{n}T{x}_{n}+\left(1-{\alpha }_{n}\right){x}_{n},\hfill \\ {x}_{n+1}={\beta }_{n}u+{\gamma }_{n}{x}_{n}+{\delta }_{n}{y}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.\hfill \end{array}$

Then $\left\{{x}_{n}\right\}$ converges strongly to ${x}^{\ast }\in F\left(T\right)$, where ${x}^{\ast }={Q}_{F\left(T\right)}\left(u\right)$ and ${Q}_{F\left(T\right)}:C\to F\left(T\right)$ is the unique sunny nonexpansive retraction from C onto $F\left(T\right)$.

In 2006, Aoyama et al. introduced a Halpern-type iterative sequence and proved that such a sequence converges strongly to a common fixed point of nonexpansive mappings as follows.

Theorem 1.5 Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let C be a nonempty closed convex subset of E. Let $\left\{{T}_{n}\right\}$ be a sequence of nonexpansive mappings of C into itself such that ${\bigcap }_{n=1}^{N}F\left({T}_{i}\right)$ is nonempty and let $\left\{{\alpha }_{n}\right\}$ be a sequence of $\left[0,1\right]$ such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$. Let $\left\{{x}_{n}\right\}$ be a sequence of C defined as follows: ${x}_{1}=x\in C$ and
${x}_{n+1}={\alpha }_{n}x+\left(1-{\alpha }_{n}\right){T}_{n}{x}_{n}$
for every $n\in \mathbb{N}$. Suppose that ${\sum }_{n=1}^{\mathrm{\infty }}sup\left\{\parallel {T}_{n+1}z-{T}_{n}z\parallel :z\in B\right\}<\mathrm{\infty }$ for any bounded subset B of C. Let T be a mapping of C into itself defined by $Tz={lim}_{n\to \mathrm{\infty }}{T}_{n}z$ for all $z\in C$ and suppose that $F\left(T\right)={\bigcap }_{n=1}^{\mathrm{\infty }}F\left({T}_{n}\right)$. If either
$\begin{array}{r}\text{(i)}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|<\mathrm{\infty }\mathit{\text{or}}\\ \text{(ii)}\phantom{\rule{1em}{0ex}}{\alpha }_{n}\in \left(0,1\right]\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}n\in \mathbb{N}\mathit{\text{and}}\underset{n\to \mathrm{\infty }}{lim}\frac{{\alpha }_{n}}{{\alpha }_{n+1}},\end{array}$

then $\left\{{x}_{n}\right\}$ converges strongly to Qx, where Q is the sunny nonexpansive retraction of E onto $F\left(T\right)={\bigcap }_{i=1}^{\mathrm{\infty }}F\left({T}_{n}\right)$.

In 2005, Aoyama et al. [3] proved a weak convergence theorem for finding a solution of problem (1.3) as follows.

Theorem 1.6 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 that ${x}_{1}=x\in C$ 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.

In 2009, Kangtunykarn and Suantai [12] introduced the S-mapping generated by a finite family of mappings and real numbers as follows.

Definition 1.4 Let C be a nonempty convex subset of a real Banach space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of 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 $S:C\to C$ as follows:
$\begin{array}{r}{U}_{0}=I,\\ {U}_{1}={\alpha }_{1}^{1}{T}_{1}{U}_{0}+{\alpha }_{2}^{1}{U}_{0}+{\alpha }_{3}^{1}I,\\ {U}_{2}={\alpha }_{1}^{2}{T}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I,\\ {U}_{3}={\alpha }_{1}^{3}{T}_{3}{U}_{2}+{\alpha }_{2}^{3}{U}_{2}+{\alpha }_{3}^{3}I,\\ ⋮\\ {U}_{N-1}={\alpha }_{1}^{N-1}{T}_{N-1}{U}_{N-2}+{\alpha }_{2}^{N-1}{U}_{N-2}+{\alpha }_{3}^{N-1}I,\\ S={U}_{N}={\alpha }_{1}^{N}{T}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I.\end{array}$
(1.7)

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

For every $i=1,2,\dots ,N$, put ${\alpha }_{3}^{j}=0$ in (1.7), then the S-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$ reduces to the K-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1}^{1},{\alpha }_{1}^{2},\dots ,{\alpha }_{1}^{N}$, which is defined by Kangtunyakarn and Suantai [13].

Recently, Kangtunyakarn [14] introduced an iterative scheme by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of an η-strictly pseudo-contractive mapping and a nonexpansive mapping as follows.

Theorem 1.7 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let ${Q}_{C}$ be the sunny nonexpansive retraction from E onto C. For every $i=1,2,\dots ,N$, let ${A}_{i}:C\to E$ be an ${\alpha }_{i}$-inverse strongly accretive mapping. Define a mapping ${G}_{i}:C\to C$ by ${Q}_{C}\left(I-{\lambda }_{i}{A}_{i}\right)x={G}_{i}x$ for all $x\in C$ and $i=1,2,\dots ,N$, where ${\lambda }_{i}\in \left(0,\frac{{\alpha }_{i}}{{K}^{2}}\right)$, K is the 2-uniformly smooth constant of E. Let $B:C\to C$ be the K-mapping generated by ${G}_{1},{G}_{2},\dots ,{G}_{N}$ and ${\rho }_{1},{\rho }_{2},\dots ,{\rho }_{N}$, where ${\rho }_{i}\in \left(0,1\right)$, $\mathrm{\forall }i=1,2,\dots ,N-1$ and ${\rho }_{N}\in \left(0,1\right]$. Let $T:C\to C$ be a nonexpansive mapping and $S:C\to C$ be an η-strictly pseudo-contractive mapping with $\mathcal{F}=F\left(S\right)\cap F\left(T\right)\cap {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)\ne \mathrm{\varnothing }$. Define a mapping ${B}_{A}:C\to C$ by $T\left(\left(1-\alpha \right)I+\alpha S\right)x={B}_{A}x$, $\mathrm{\forall }x\in C$ and $\alpha \in \left(0,\frac{\eta }{{K}^{2}}\right)$. Let $\left\{{x}_{n}\right\}$ be a sequence generated by ${x}_{1}\in C$ and
${x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+{\beta }_{n}{x}_{n}+{\gamma }_{n}B{x}_{n}+{\delta }_{n}{B}_{A}{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,$
(1.8)
where $f:C\to C$ is a contractive mapping and $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\},\left\{{\gamma }_{n}\right\},\left\{{\delta }_{n}\right\}\subseteq \left[0,1\right]$, ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}+{\delta }_{n}=1$ and satisfy the following conditions:
$\begin{array}{r}\text{(i)}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}=0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty },\\ \text{(ii)}\phantom{\rule{1em}{0ex}}\left\{{\gamma }_{n}\right\},\left\{{\delta }_{n}\right\}\subseteq \left[c,d\right]\subset \left(0,1\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for some}}c,d>0\mathit{\text{and}}\mathrm{\forall }n\ge 1,\\ \text{(iii)}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_{n}|<\mathrm{\infty },\\ \text{(iv)}\phantom{\rule{1em}{0ex}}0<\underset{n\to \mathrm{\infty }}{lim inf}{\beta }_{n}\le \underset{n\to \mathrm{\infty }}{lim sup}{\beta }_{n}<1.\end{array}$
Then the sequence $\left\{{x}_{n}\right\}$ converses strongly to $q\in \mathcal{F}$, which solves the following variational inequality:
$〈q-f\left(q\right),j\left(q-p\right)〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in \mathcal{F}.$

Question How can we prove a strong convergence theorem for the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and the set of solutions of variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space?

Motivated by the S-mapping, we define a new mapping in the next section to answer the above question, and from Theorems 1.4, 1.5, 1.6 and 1.7 we modify the Halpern iteration for finding a common element of two sets of solutions of (1.3) and the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. Moreover, by using our main result, we also obtain a strong convergence theorem for a finite family of the set of solutions of (1.3) and the set of fixed points of a finite family of strictly pseudo-contractive mappings.

## 2 Preliminaries

In this section we collect and prove the following lemmas to use in our main result.

Lemma 2.1 (See [15])

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 [16])

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 $\alpha +\beta +\gamma =1$.

Lemma 2.3 (See [3])

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 2.4 (See [15])

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 2.5 (See [17])

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 2.6 (See [18])

Let $\left\{{s}_{n}\right\}$ be a sequence of nonnegative real numbers satisfying
${s}_{n+1}=\left(1-{\alpha }_{n}\right){s}_{n}+{\delta }_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,$
where $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left(0,1\right)$ and $\left\{{\delta }_{n}\right\}$ is a sequence such that
$\begin{array}{r}\left(1\right)\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty },\\ \left(2\right)\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty }}{lim sup}\frac{{\delta }_{n}}{{\alpha }_{n}}\le 0\phantom{\rule{1em}{0ex}}\mathit{\text{or}}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\delta }_{n}|<\mathrm{\infty }.\end{array}$

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

From the S-mapping, we define the mapping generated by two sets of finite families of the mappings and real numbers as follows.

Definition 2.1 Let C be a nonempty convex subset of a Banach space. Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ and ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be two finite families of 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 ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$. We define the mapping ${S}^{A}:C\to C$ as follows:
$\begin{array}{r}{U}_{0}={T}_{1}=I,\\ {U}_{1}={T}_{1}\left({\alpha }_{1}^{1}{S}_{1}{U}_{0}+{\alpha }_{2}^{1}{U}_{0}+{\alpha }_{3}^{1}I\right),\\ {U}_{2}={T}_{2}\left({\alpha }_{1}^{2}{S}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I\right),\\ {U}_{3}={T}_{3}\left({\alpha }_{1}^{3}{S}_{3}{U}_{2}+{\alpha }_{2}^{3}{U}_{2}+{\alpha }_{3}^{3}I\right),\\ ⋮\\ {U}_{N-1}={T}_{N-1}\left({\alpha }_{1}^{N-1}{S}_{N-1}{U}_{N-2}+{\alpha }_{2}^{N-1}{U}_{N-2}+{\alpha }_{3}^{N-1}I\right),\\ {S}^{A}={U}_{N}={T}_{N}\left({\alpha }_{1}^{N}{S}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I\right).\end{array}$
(2.1)

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 2.7 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.

Proof Let ${x}_{0}\in F\left({S}^{A}\right)$ and ${x}^{\ast }\in {\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$, we have
$\begin{array}{rcl}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}& =& {\parallel {T}_{N}\left({\alpha }_{1}^{N}{S}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I\right){x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {\alpha }_{1}^{N}\left({S}_{N}{U}_{N-1}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{2}^{N}\left({U}_{N-1}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{3}^{N}\left({x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ =& \parallel \left(1-{\alpha }_{3}^{N}\right)\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left({S}_{N}{U}_{N-1}{x}_{0}-{x}^{\ast }\right)+\frac{{\alpha }_{2}^{N}}{1-{\alpha }_{3}^{N}}\left({U}_{N-1}{x}_{0}-{x}^{\ast }\right)\right)\\ +{\alpha }_{3}^{N}\left({x}_{0}-{x}^{\ast }\right){\parallel }^{2}\\ \le & \left(1-{\alpha }_{3}^{N}\right){\parallel \frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left({S}_{N}{U}_{N-1}{x}_{0}-{x}^{\ast }\right)+\frac{{\alpha }_{2}^{N}}{1-{\alpha }_{3}^{N}}\left({U}_{N-1}{x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{3}^{N}\right){\parallel \frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left({S}_{N}{U}_{N-1}{x}_{0}-{x}^{\ast }\right)+\left(1-\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)\left({U}_{N-1}{x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{3}^{N}\right){\parallel \frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left({S}_{N}{U}_{N-1}{x}_{0}-{U}_{N-1}{x}_{0}\right)+{U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{3}^{N}\right)\left({\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}〈{S}_{N}{U}_{N-1}{x}_{0}-{U}_{N-1}{x}_{0},j\left({U}_{N-1}{x}_{0}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)}^{2}{\parallel {S}_{N}{U}_{N-1}{x}_{0}-{U}_{N-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{3}^{N}\right)\left({\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}+2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}〈{S}_{N}{U}_{N-1}{x}_{0}-{x}^{\ast },j\left({U}_{N-1}{x}_{0}-{x}^{\ast }\right)〉\\ +2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}〈{x}^{\ast }-{U}_{N-1}{x}_{0},j\left({U}_{N-1}{x}_{0}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)}^{2}{\parallel {S}_{N}{U}_{N-1}{x}_{0}-{U}_{N-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{3}^{N}\right)\left({\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left({\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}-\kappa {\parallel \left(I-{S}_{N}\right){U}_{N-1}{x}_{0}\parallel }^{2}\right)\\ -2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}{\parallel {x}^{\ast }-{U}_{N-1}{x}_{0}\parallel }^{2}+2{K}^{2}{\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)}^{2}{\parallel {S}_{N}{U}_{N-1}{x}_{0}-{U}_{N-1}{x}_{0}\parallel }^{2}\right)\\ +{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{3}^{N}\right)\left({\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}-2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\kappa {\parallel \left(I-{S}_{N}\right){U}_{N-1}{x}_{0}\parallel }^{2}\\ +2{K}^{2}{\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)}^{2}{\parallel {S}_{N}{U}_{N-1}{x}_{0}-{U}_{N-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{3}^{N}\right)\left({\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)\right){\parallel \left(I-{S}_{N}\right){U}_{N-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{3}^{N}\right){\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{3}^{N}\right)\left(\left(1-{\alpha }_{3}^{N-1}\right){\parallel {U}_{N-2}{x}_{0}-{x}^{\ast }\parallel }^{2}+{\alpha }_{3}^{N-1}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=N-1}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {U}_{N-2}{x}_{0}-{x}^{\ast }\parallel }^{2}+\left(1-\prod _{j=N-1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ ⋮\\ \le & \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {U}_{2}{x}_{0}-{x}^{\ast }\parallel }^{2}+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {T}_{2}\left({\alpha }_{1}^{2}{S}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I\right){x}_{0}-{x}^{\ast }\parallel }^{2}\\ +\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {\alpha }_{1}^{2}\left({S}_{2}{U}_{1}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{2}^{2}\left({U}_{1}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{3}^{2}\left({x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\parallel \left(1-{\alpha }_{3}^{2}\right)\left(\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left({S}_{2}{U}_{1}{x}_{0}-{x}^{\ast }\right)+\frac{{\alpha }_{2}^{2}}{1-{\alpha }_{3}^{2}}\left({U}_{1}{x}_{0}-{x}^{\ast }\right)\right)\\ +{\alpha }_{3}^{2}\left({x}_{0}-{x}^{\ast }\right){\parallel }^{2}+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{2}\right)\parallel \frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left({S}_{2}{U}_{1}{x}_{0}-{x}^{\ast }\right)\\ +\frac{{\alpha }_{2}^{2}}{1-{\alpha }_{3}^{2}}\left({U}_{1}{x}_{0}-{x}^{\ast }\right){\parallel }^{2}+{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{2}\right){\parallel \frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left({S}_{2}{U}_{1}{x}_{0}-{x}^{\ast }\right)+\left(1-\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\right)\left({U}_{1}{x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{2}\right){\parallel \frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left({S}_{2}{U}_{1}{x}_{0}-{U}_{1}{x}_{0}\right)+{U}_{1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{2}\right)\left({\parallel {U}_{1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +2\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}〈{S}_{2}{U}_{1}{x}_{0}-{U}_{1}{x}_{0},j\left({U}_{1}{x}_{0}-{x}^{\ast }\right)〉\\ +2{K}^{2}\left(\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\right){\parallel {S}_{2}{U}_{1}{x}_{0}-{U}_{1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)\\ +\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{2}\right)\left({\parallel {U}_{1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\right)\right){\parallel \left(I-{S}_{2}\right){U}_{1}{x}_{0}\parallel }^{2}\right)\\ +{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(1-{\alpha }_{3}^{2}\right)\left({\parallel {U}_{1}{x}_{0}-{x}^{\ast }\parallel }^{2}+{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)\\ +\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {U}_{1}{x}_{0}-{x}^{\ast }\parallel }^{2}+\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {\alpha }_{1}^{1}\left({S}_{1}{U}_{0}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{2}^{1}\left({U}_{0}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{3}^{1}\left({x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {\alpha }_{1}^{1}\left({S}_{1}{x}_{0}-{x}^{\ast }\right)+\left(1-{\alpha }_{1}^{1}\right)\left({x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {\alpha }_{1}^{1}\left({S}_{1}{x}_{0}-{x}_{0}\right)+{x}_{0}-{x}^{\ast }\parallel }^{2}+\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}+2{\alpha }_{1}^{1}〈{S}_{1}{x}_{0}-{x}_{0},j\left({x}_{0}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left({\alpha }_{1}^{1}\right)}^{2}{\parallel {S}_{1}{x}_{0}-{x}_{0}\parallel }^{2}\right)+\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}+2{\alpha }_{1}^{1}〈{S}_{1}{x}_{0}-{x}^{\ast },j\left({x}_{0}-{x}^{\ast }\right)〉\\ +2{\alpha }_{1}^{1}〈{x}^{\ast }-{x}_{0},j\left({x}_{0}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left({\alpha }_{1}^{1}\right)}^{2}{\parallel {S}_{1}{x}_{0}-{x}_{0}\parallel }^{2}\right)+\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}+2{\alpha }_{1}^{1}\left(\parallel {x}_{0}-{x}^{\ast }\parallel -\kappa {\parallel {S}_{1}{x}_{0}-{x}_{0}\parallel }^{2}\right)\\ -2{\alpha }_{1}^{1}{\parallel {x}^{\ast }-{x}_{0}\parallel }^{2}\\ +2{K}^{2}{\left({\alpha }_{1}^{1}\right)}^{2}{\parallel {S}_{1}{x}_{0}-{x}_{0}\parallel }^{2}\right)+\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}-2{\alpha }_{1}^{1}\left(\kappa -{K}^{2}{\alpha }_{1}^{1}\right){\parallel {S}_{1}{x}_{0}-{x}_{0}\parallel }^{2}\right)\\ +\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& {\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)2{\alpha }_{1}^{1}\left(\kappa -{K}^{2}{\alpha }_{1}^{1}\right){\parallel {S}_{1}{x}_{0}-{x}_{0}\parallel }^{2}\\ \le & {\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}.\end{array}$
(2.2)
For every $j=1,2,\dots ,N$ and (2.2), we have
${\parallel {U}_{j}{x}_{0}-{x}^{\ast }\parallel }^{2}\le {\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}.$
(2.3)
For every $k=1,2,\dots ,N-1$ and (2.2) we have
$\begin{array}{rcl}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}& \le & \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {U}_{k}{x}_{0}-{x}^{\ast }\parallel }^{2}+\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {T}_{k}\left({\alpha }_{1}^{k}{S}_{k}{U}_{k-1}+{\alpha }_{2}^{k}{U}_{k-1}+{\alpha }_{3}^{k}I\right){x}_{0}-{x}^{\ast }\parallel }^{2}\\ +\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {\alpha }_{1}^{k}\left({S}_{k}{U}_{k-1}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{2}^{k}\left({U}_{k-1}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{3}^{k}\left({x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\parallel \left(1-{\alpha }_{3}^{k}\right)\left(\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\left({S}_{k}{U}_{k-1}{x}_{0}-{x}^{\ast }\right)+\frac{{\alpha }_{2}^{k}}{1-{\alpha }_{3}^{k}}\left({U}_{k-1}{x}_{0}-{x}^{\ast }\right)\right)\\ +{\alpha }_{3}^{k}\left({x}_{0}-{x}^{\ast }\right){\parallel }^{2}+\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right){\parallel \frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\left({S}_{k}{U}_{k-1}{x}_{0}-{x}^{\ast }\right)+\frac{{\alpha }_{2}^{k}}{1-{\alpha }_{3}^{k}}\left({U}_{k-1}{x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right)\parallel \frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\left({S}_{k}{U}_{k-1}{x}_{0}-{x}^{\ast }\right)\\ +\left(1-\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\right)\left({U}_{k-1}{x}_{0}-{x}^{\ast }\right){\parallel }^{2}\\ +{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right){\parallel \frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\left({S}_{k}{U}_{k-1}{x}_{0}-{U}_{k-1}{x}_{0}\right)+{U}_{k-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right)\left({\parallel {U}_{k-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +2\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}〈{S}_{k}{U}_{k-1}{x}_{0}-{U}_{k-1}{x}_{0},j\left({U}_{k-1}{x}_{0}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left(\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\right)}^{2}{\parallel {S}_{k}{U}_{k-1}{x}_{0}-{U}_{k-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)\\ +\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right)\left({\parallel {U}_{k-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +2\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}〈{S}_{k}{U}_{k-1}{x}_{0}-{x}^{\ast },j\left({U}_{k-1}{x}_{0}-{x}^{\ast }\right)〉\\ +2\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}〈{x}^{\ast }-{U}_{k-1}{x}_{0},j\left({U}_{k-1}{x}_{0}-{x}^{\ast }\right)〉\\ +2{K}^{2}{\left(\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\right)}^{2}{\parallel {S}_{k}{U}_{k-1}{x}_{0}-{U}_{k-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)\\ +\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right)\left({\parallel {U}_{k-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ +2\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\left({\parallel {U}_{k-1}{x}_{0}-{x}^{\ast }\parallel }^{2}-\kappa \parallel \left(I-{S}_{k}\right){U}_{k-1}{x}_{0}\parallel \right)\\ -2\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}{\parallel {x}^{\ast }-{U}_{k-1}{x}_{0}\parallel }^{2}\\ +2{K}^{2}{\left(\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\right)}^{2}{\parallel {S}_{k}{U}_{k-1}{x}_{0}-{U}_{k-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)\\ +\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right)\left({\parallel {U}_{k-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\right)\right){\parallel \left(I-{S}_{k}\right){U}_{k-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)\\ +\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{k}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{k}}{1-{\alpha }_{3}^{k}}\right)\right){\parallel \left(I-{S}_{k}\right){U}_{k-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{k}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)\\ +\left(1-\prod _{j=k+1}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2},\end{array}$
which implies that
${U}_{k-1}{x}_{0}={S}_{k}{U}_{k-1}{x}_{0}$
(2.4)

for every $k=1,2,\dots ,N-1$.

From (2.2), it implies that ${x}_{0}={S}_{1}{x}_{0}$, that is, ${x}_{0}\in F\left(S\right)$. From the definition of ${S}^{A}$, we have
${U}_{1}{x}_{0}={T}_{1}\left({\alpha }_{1}^{1}{S}_{1}{U}_{0}{x}_{0}+{\alpha }_{2}^{1}{U}_{0}{x}_{0}+{\alpha }_{3}^{1}{x}_{0}\right)={T}_{1}{x}_{0}={x}_{0}.$
(2.5)
From (2.2) and ${U}_{1}{x}_{0}={x}_{0}$, we have
$\begin{array}{rcl}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}& \le & \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{2}\right)\left({\parallel {U}_{1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\right)\right){\parallel \left(I-{S}_{2}\right){U}_{1}{x}_{0}\parallel }^{2}\right)\\ +{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{2}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\right)\right){\parallel \left(I-{S}_{2}\right){x}_{0}\parallel }^{2}\right)\\ +{\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\right)+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(1-{\alpha }_{3}^{2}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\right)\right){\parallel \left(I-{S}_{2}\right){x}_{0}\parallel }^{2}\right)\\ +\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right){\alpha }_{3}^{2}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}+\left(1-\prod _{j=3}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}-2\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{2}}{1-{\alpha }_{3}^{2}}\right)\right){\parallel \left(I-{S}_{2}\right){x}_{0}\parallel }^{2}\right)\\ +\left(1-\prod _{j=2}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}.\end{array}$

It implies that ${x}_{0}={S}_{2}{x}_{0}$.

From the definition of ${S}^{A}$ and ${x}_{0}={S}_{2}{x}_{0}$, we have
${U}_{2}{x}_{0}={T}_{2}\left({\alpha }_{1}^{2}{S}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I\right){x}_{0}={T}_{2}{x}_{0}.$
(2.6)
From the definition of ${U}_{3}$ and (2.4), we have
${U}_{3}{x}_{0}={T}_{3}\left({\alpha }_{1}^{3}{S}_{3}{U}_{2}+{\alpha }_{2}^{3}{U}_{2}+{\alpha }_{3}^{3}I\right){x}_{0}={T}_{3}\left(\left(1-{\alpha }_{3}^{3}\right){U}_{2}{x}_{0}+{\alpha }_{3}^{3}{x}_{0}\right).$
(2.7)
From (2.2), (2.6), (2.7) and E is uniformly convex, we have
$\begin{array}{rcl}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}& \le & \prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {U}_{3}{x}_{0}-{x}^{\ast }\parallel }^{2}+\left(1-\prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel {T}_{3}\left(\left(1-{\alpha }_{3}^{3}\right){U}_{2}{x}_{0}+{\alpha }_{3}^{3}{x}_{0}\right)-{x}^{\ast }\parallel }^{2}\\ +\left(1-\prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel \left(1-{\alpha }_{3}^{3}\right)\left({U}_{2}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{3}^{3}\left({x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +\left(1-\prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right){\parallel \left(1-{\alpha }_{3}^{3}\right)\left({T}_{2}{x}_{0}-{x}^{\ast }\right)+{\alpha }_{3}^{3}\left({x}_{0}-{x}^{\ast }\right)\parallel }^{2}\\ +\left(1-\prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\left(\left(1-{\alpha }_{3}^{3}\right){\parallel {T}_{2}{x}_{0}-{x}^{\ast }\parallel }^{2}+{\alpha }_{3}^{3}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ -{\alpha }_{3}^{3}\left(1-{\alpha }_{3}^{3}\right){g}_{2}\left(\parallel {T}_{2}{x}_{0}-{x}_{0}\parallel \right)\right)\\ +\left(1-\prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ \le & \prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}-{\alpha }_{3}^{3}\left(1-{\alpha }_{3}^{3}\right){g}_{2}\left(\parallel {T}_{2}{x}_{0}-{x}_{0}\parallel \right)\right)\\ +\left(1-\prod _{j=4}^{N}\left(1-{\alpha }_{3}^{j}\right)\right){\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}.\end{array}$
It implies that
${g}_{2}\left(\parallel {T}_{2}{x}_{0}-{x}_{0}\parallel \right)=0.$
(2.8)
Assume that ${T}_{2}{x}_{0}\ne {x}_{0}$, then we have $\parallel {T}_{2}{x}_{0}-{x}_{0}\parallel >0$. From the properties of ${g}_{2}$, we have
$0=g\left(0\right)
(2.9)

This is a contradiction. Then we have ${T}_{2}{x}_{0}={x}_{0}$. From (2.6), we have ${x}_{0}={T}_{2}{x}_{0}={U}_{2}{x}_{0}$.

From the definition of ${U}_{3}$, we have
${U}_{3}{x}_{0}={T}_{3}\left(\left(1-{\alpha }_{3}^{3}\right){U}_{2}{x}_{0}+{\alpha }_{3}^{3}{x}_{0}\right)={T}_{3}{x}_{0}.$
By using the same method as above, we have
${x}_{0}={U}_{3}{x}_{0}={T}_{3}{x}_{0}.$
Continuing on this way, we can conclude that
${x}_{0}={U}_{i}{x}_{0}={T}_{i}{x}_{0}$
(2.10)
for every $i=1,2,\dots ,N-1$. From (2.2) and (2.10), we have
$\begin{array}{rcl}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}& \le & \left(1-{\alpha }_{3}^{N}\right)\left({\parallel {U}_{N-1}{x}_{0}-{x}^{\ast }\parallel }^{2}\\ -2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)\right){\parallel \left(I-{S}_{N}\right){U}_{N-1}{x}_{0}\parallel }^{2}\right)+{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{3}^{N}\right)\left({\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}-2\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\left(\kappa -{K}^{2}\left(\frac{{\alpha }_{1}^{N}}{1-{\alpha }_{3}^{N}}\right)\right){\parallel \left(I-{S}_{N}\right){x}_{0}\parallel }^{2}\right)\\ +{\alpha }_{3}^{N}{\parallel {x}_{0}-{x}^{\ast }\parallel }^{2}.\end{array}$
It implies that
${x}_{0}={S}_{N}{x}_{0}.$
(2.11)
From the definition of ${S}^{A}$ and (2.10), we have
${x}_{0}={S}^{A}{x}_{0}={U}_{N}{x}_{0}={T}_{N}\left({\alpha }_{1}^{N}{S}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I\right){x}_{0}={T}_{N}{x}_{0}.$
Then we have
${x}_{0}\in \bigcap _{i=1}^{N}F\left({T}_{i}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{x}_{0}\in \bigcap _{i=1}^{N}F\left({U}_{i}\right).$
(2.12)
Since ${S}_{k}{U}_{k-1}{x}_{0}={U}_{k-1}{x}_{0}$ for every $k=1,2,\dots ,N-1$ and ${x}_{0}\in {\bigcap }_{i=1}^{N}F\left({U}_{i}\right)$, then we have
${S}_{k}{x}_{0}={x}_{0}$
for every $k=1,2,\dots ,N-1$. From (2.11), it implies that
${x}_{0}\in \bigcap _{i=1}^{N}F\left({S}_{i}\right).$
(2.13)
From (2.12) and (2.13), we have
${x}_{0}\in \bigcap _{i=1}^{N}F\left({T}_{i}\right)\cap \bigcap _{i=1}^{N}F\left({S}_{i}\right).$
(2.14)

Hence, $F\left({S}^{A}\right)\subseteq {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({S}_{i}\right)$. It is easy to see that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\subseteq F\left({S}^{A}\right)$.

Applying (2.2), we have that the mapping ${S}^{A}$ is nonexpansive. □

Lemma 2.8 [19]

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

where λ is a constant in $\left(0,1\right)$. Then S is nonexpansive and $F\left(S\right)=F\left({T}_{1}\right)\cap F\left({T}_{2}\right)$.

Applying Lemma 2.8, we have the following lemma.

Lemma 2.9 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 nonexpansive and $F\left(S\right)=F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\cap F\left({T}_{3}\right)$.

Proof For every $x\in C$ and the definition of the mapping S, we have
$\begin{array}{rcl}Sx& =& \alpha {T}_{1}x+\beta {T}_{2}x+\gamma {T}_{3}x\\ =& \alpha {T}_{1}x+\left(1-\alpha \right)\left(\frac{\beta }{1-\alpha }{T}_{2}x+\frac{\gamma }{1-\alpha }{T}_{3}x\right)\\ =& \alpha {T}_{1}x+\left(1-\alpha \right)\left(\frac{\beta }{1-\alpha }{T}_{2}x+\left(1-\frac{\beta }{1-\alpha }\right){T}_{3}x\right)\\ =& \alpha {T}_{1}x+\left(1-\alpha \right){S}_{1}x,\end{array}$
(2.15)

where ${S}_{1}=\frac{\beta }{1-\alpha }{T}_{2}+\left(1-\frac{\beta }{1-\alpha }\right){T}_{3}$. From Lemma 2.8, we have $F\left({S}_{1}\right)=F\left({T}_{2}\right)\cap F\left({T}_{3}\right)$ and ${S}_{1}$ is a nonexpansive mapping. From Lemma 2.8 and (2.15), we have $F\left(S\right)=F\left({T}_{1}\right)\cap F\left({S}_{1}\right)$ and S is a nonexpansive mapping. Hence we have $F\left(S\right)=F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\cap F\left({T}_{3}\right)$. □

## 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 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
${x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-aA\right){x}_{n}+{\delta }_{n}{Q}_{C}\left(I-bB\right){x}_{n}+{\eta }_{n}{S}^{A}{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,$
(3.1)
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:
$\begin{array}{r}\text{(i)}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}=0,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty },\\ \text{(ii)}\phantom{\rule{1em}{0ex}}\left\{{\gamma }_{n}\right\},\left\{{\delta }_{n}\right\},\left\{{\eta }_{n}\right\}\subseteq \left[c,d\right]\subset \left(0,1\right),\phantom{\rule{1em}{0ex}}\mathit{\text{for some}}c,d>0,\phantom{\rule{0.25em}{0ex}}\mathrm{\forall }n\ge 1,\\ \text{(iii)}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_{n}|,\\ \phantom{\text{(iii)}\phantom{\rule{1em}{0ex}}}\sum _{n=1}^{\mathrm{\infty }}|{\eta }_{n+1}-{\eta }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|<\mathrm{\infty },\\ \text{(iv)}\phantom{\rule{1em}{0ex}}0<\underset{n\to \mathrm{\infty }}{lim inf}{\beta }_{n}\le \underset{n\to \mathrm{\infty }}{lim sup}{\beta }_{n}<1,\\ \text{(v)}\phantom{\rule{1em}{0ex}}a\in \left(0,\frac{\alpha }{{K}^{2}}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}b\in \left(0,\frac{\beta }{{K}^{2}}\right).\end{array}$

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 ${Q}_{C}\left(I-aA\right)$ and ${Q}_{C}\left(I-bB\right)$ are nonexpansive mappings. Let $x,y\in C$, we have
$\begin{array}{rcl}{\parallel {Q}_{C}\left(I-aA\right)x-{Q}_{C}\left(I-aA\right)y\parallel }^{2}& \le & {\parallel x-y-a\left(Ax-Ay\right)\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}-2a〈Ax-Ay,j\left(x-y\right)〉+2{K}^{2}{a}^{2}{\parallel Ax-Ay\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}-2a\alpha {\parallel Ax-Ay\parallel }^{2}+2{K}^{2}{a}^{2}{\parallel Ax-Ay\parallel }^{2}\\ =& {\parallel x-y\parallel }^{2}-2a\left(\alpha -{K}^{2}a\right){\parallel Ax-Ay\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}.\end{array}$
(3.2)

Then we have ${Q}_{C}\left(I-aA\right)$ is a nonexpansive mapping. By using the same methods as (3.2), we have ${Q}_{C}\left(I-bB\right)$ is a nonexpansive mapping.

Let ${x}^{\ast }\in \mathcal{F}$. From Lemma 2.3, we have ${x}^{\ast }\in F\left({Q}_{C}\left(I-aA\right)\right)$ and ${x}^{\ast }\in F\left({Q}_{C}\left(I-bB\right)\right)$. By the definition of ${x}_{n}$, we have
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}^{\ast }\parallel & \le & {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +{\beta }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +{\gamma }_{n}\parallel {Q}_{C}\left(I-aA\right){x}_{n}-{x}^{\ast }\parallel \\ +{\delta }_{n}\parallel {Q}_{C}\left(I-bB\right){x}_{n}-{x}^{\ast }\parallel +{\eta }_{n}\parallel {S}^{A}{x}_{n}-{x}^{\ast }\parallel \\ \le & {\alpha }_{n}\parallel u-{x}^{\ast }\parallel +\left(1-{\alpha }_{n}\right)\parallel {x}_{n}-{x}^{\ast }\parallel \\ \le & max\left\{\parallel u-{x}^{\ast }\parallel ,\parallel {x}_{1}-{x}^{\ast }\parallel \right\}.\end{array}$

By induction, we have $\parallel {x}_{n}-{x}^{\ast }\parallel \le max\left\{\parallel u-{x}^{\ast }\parallel ,\parallel {x}_{1}-{x}^{\ast }\parallel \right\}$. We can imply that the sequence $\left\{{x}_{n}\right\}$ is bounded and so are $\left\{{S}^{A}{x}_{n}\right\}$, $\left\{{Q}_{C}\left(I-aA\right){x}_{n}\right\}$ and $\left\{{Q}_{C}\left(I-bB\right){x}_{n}\right\}$.

Next, we show that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n+1}-{x}_{n}\parallel =0$. From the definition of ${x}_{n}$, we have
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}_{n}\parallel & =& \parallel {\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-aA\right){x}_{n}+{\delta }_{n}{Q}_{C}\left(I-bB\right){x}_{n}+{\eta }_{n}{S}^{A}{x}_{n}\\ -{\alpha }_{n-1}u-{\beta }_{n-1}{x}_{n-1}-{\gamma }_{n-1}{Q}_{C}\left(I-aA\right){x}_{n-1}-{\delta }_{n-1}{Q}_{C}\left(I-bB\right){x}_{n-1}\\ -{\eta }_{n-1}{S}^{A}{x}_{n-1}\parallel \\ \le & |{\alpha }_{n}-{\alpha }_{n-1}|\parallel u\parallel +{\beta }_{n}\parallel {x}_{n}-{x}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel {x}_{n-1}\parallel \\ +{\gamma }_{n}\parallel {Q}_{C}\left(I-aA\right){x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n-1}\parallel +|{\gamma }_{n}-{\gamma }_{n-1}|\parallel {Q}_{C}\left(I-aA\right){x}_{n-1}\parallel \\ +{\delta }_{n}\parallel {Q}_{C}\left(I-bB\right){x}_{n}-{Q}_{C}\left(I-bB\right){x}_{n-1}\parallel +|{\delta }_{n}-{\delta }_{n-1}|\parallel {Q}_{C}\left(I-bB\right){x}_{n-1}\parallel \\ +{\eta }_{n}\parallel {S}^{A}{x}_{n}-{S}^{A}{x}_{n-1}\parallel +|{\eta }_{n-1}-{\eta }_{n}|\parallel {S}^{A}{x}_{n}\parallel \\ \le & \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-{x}_{n-1}\parallel +|{\alpha }_{n}-{\alpha }_{n-1}|\parallel u\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel {x}_{n-1}\parallel \\ +|{\gamma }_{n}-{\gamma }_{n-1}|\parallel {Q}_{C}\left(I-aA\right){x}_{n-1}\parallel +|{\delta }_{n}-{\delta }_{n-1}|\parallel {Q}_{C}\left(I-bB\right){x}_{n-1}\parallel \\ +|{\eta }_{n-1}-{\eta }_{n}|\parallel {S}^{A}{x}_{n}\parallel .\end{array}$
Applying Lemma 2.6, we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-{x}_{n}\parallel =0.$
(3.3)
Next, we show that
$\underset{n\to \mathrm{\infty }}{lim}\parallel {Q}_{C}\left(I-aA\right){x}_{n}-{x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {Q}_{C}\left(I-bB\right){x}_{n}-{x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {S}^{A}{x}_{n}-{x}_{n}\parallel =0.$
(3.4)
From the definition of ${x}_{n}$, we have
$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& =& \parallel {\alpha }_{n}\left(u-{x}^{\ast }\right)+{\beta }_{n}\left({x}_{n}-{x}^{\ast }\right)+{\gamma }_{n}\left({Q}_{C}\left(I-aA\right){x}_{n}-{x}^{\ast }\right)\\ +{\delta }_{n}\left({Q}_{C}\left(I-bB\right){x}_{n}-{x}^{\ast }\right)+{\eta }_{n}\left({S}^{A}{x}_{n}-{x}^{\ast }\right){\parallel }^{2}\\ =& \parallel {\beta }_{n}\left({x}_{n}-{x}^{\ast }\right)+{\gamma }_{n}\left({Q}_{C}\left(I-aA\right){x}_{n}-{x}^{\ast }\right)+\left({\alpha }_{n}+{\delta }_{n}+{\eta }_{n}\right)\left(\frac{{\alpha }_{n}\left(u-{x}^{\ast }\right)}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}\\ +\frac{{\delta }_{n}\left({Q}_{C}\left(I-bB\right){x}_{n}-{x}^{\ast }\right)}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}+\frac{{\eta }_{n}\left({S}^{A}{x}_{n}-{x}^{\ast }\right)}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}\right){\parallel }^{2}\\ =& {\parallel {\beta }_{n}\left({x}_{n}-{x}^{\ast }\right)+{\gamma }_{n}\left({Q}_{C}\left(I-aA\right){x}_{n}-{x}^{\ast }\right)+{c}_{n}{z}_{n}\parallel }^{2},\end{array}$

where ${c}_{n}={\alpha }_{n}+{\delta }_{n}+{\eta }_{n}$ and ${z}_{n}=\frac{{\alpha }_{n}\left(u-{x}^{\ast }\right)}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}+\frac{{\delta }_{n}\left({Q}_{C}\left(I-bB\right){x}_{n}-{x}^{\ast }\right)}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}+\frac{{\eta }_{n}\left({S}^{A}{x}_{n}-{x}^{\ast }\right)}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}$.

From Lemma 2.2, we have
$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& \le & {\beta }_{n}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\gamma }_{n}\parallel {Q}_{C}\left(I-aA\right){x}_{n}-{x}^{\ast }\parallel +{c}_{n}{\parallel {z}_{n}\parallel }^{2}\\ -{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n}\parallel \right)\\ \le & \left({\beta }_{n}+{\gamma }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n}\parallel \right)\\ +{c}_{n}\left(\frac{{\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}+\frac{{\delta }_{n}{\parallel {Q}_{C}\left(I-bB\right){x}_{n}-{x}^{\ast }\parallel }^{2}}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}+\frac{{\eta }_{n}{\parallel {S}^{A}{x}_{n}-{x}^{\ast }\parallel }^{2}}{{\alpha }_{n}+{\delta }_{n}+{\eta }_{n}}\right)\\ \le & \left({\beta }_{n}+{\gamma }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n}\parallel \right)\\ +{\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}+\left({\delta }_{n}+{\eta }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n}\parallel \right)+{\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2},\end{array}$
which implies that
$\begin{array}{rcl}{\beta }_{n}{\gamma }_{n}{g}_{1}\left(\parallel {x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n}\parallel \right)& \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}\\ \le & \left(\parallel {x}_{n}-{x}^{\ast }\parallel +\parallel {x}_{n+1}-{x}^{\ast }\parallel \right)\parallel {x}_{n+1}-{x}_{n}\parallel \\ +{\alpha }_{n}{\parallel u-{x}^{\ast }\parallel }^{2}.\end{array}$
(3.5)
From (3.3) and condition (i), we obtain
$\underset{n\to \mathrm{\infty }}{lim}{g}_{1}\left(\parallel {x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n}\parallel \right)=0.$
(3.6)
From the property of ${g}_{1}$, we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{Q}_{C}\left(I-aA\right){x}_{n}\parallel =0.$
(3.7)
By using the same method as (3.7), we can imply that
$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{Q}_{C}\left(I-bB\right){x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{S}^{A}{x}_{n}\parallel =0.$
Define $Gx=\alpha {S}^{A}x+\beta {Q}_{C}\left(I-aA\right)x+\gamma {Q}_{C}\left(I-bB\right)x$ for all $x\in C$ and $\alpha +\beta +\gamma =1$. From Lemma 2.9, we have $F\left(G\right)=F\left({Q}_{C}\left(I-aA\right)\right)\cap F\left({Q}_{C}\left(I-bB\right)\right)\cap F\left({S}^{A}\right)$. From Lemmas 2.3 and 2.7, we have $\mathcal{F}=F\left(G\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap S\left(C,A\right)\cap S\left(C,B\right)$. By the definition of G, we obtain
$\parallel G{x}_{n}-{x}_{n}\parallel \le \alpha \parallel {S}^{A}{x}_{n}-{x}_{n}\parallel +\beta \parallel {Q}_{C}\left(I-aA\right){x}_{n}-{x}_{n}\parallel +\gamma \parallel {Q}_{C}\left(I-bB\right){x}_{n}-{x}_{n}\parallel .$
From (3.4), we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel G{x}_{n}-{x}_{n}\parallel =0.$
(3.8)
From Lemma 2.5 and (3.8), we have
$\underset{n\to \mathrm{\infty }}{lim sup}〈u-{z}_{0},j\left({x}_{n}-{z}_{0}\right)〉\le 0,$
(3.9)
where ${z}_{0}={Q}_{\mathcal{F}}u$. Finally, we prove strong convergence of the sequence $\left\{{x}_{n}\right\}$ to ${z}_{0}={Q}_{\mathcal{F}}u$. From the definition of ${x}_{n}$, we have
$\begin{array}{rcl}{\parallel {x}_{n+1}-{z}_{0}\parallel }^{2}& =& \parallel {\alpha }_{n}\left(u-{z}_{0}\right)+{\beta }_{n}\left({x}_{n}-{z}_{0}\right)+{\gamma }_{n}\left({Q}_{C}\left(I-aA\right){x}_{n}-{z}_{0}\right)\\ +{\delta }_{n}\left({Q}_{C}\left(I-bB\right){x}_{n}-{z}_{0}\right)+{\eta }_{n}\left({S}^{A}{x}_{n}-{z}_{0}\right){\parallel }^{2}\\ =& \parallel {\alpha }_{n}\left(u-{z}_{0}\right)+\left(1-{\alpha }_{n}\right)\left(\frac{{\beta }_{n}\left({x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}+\frac{{\gamma }_{n}\left({Q}_{C}\left(I-aA\right){x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}\\ +\frac{{\delta }_{n}\left({Q}_{C}\left(I-bB\right){x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}+\frac{{\eta }_{n}\left({S}^{A}{x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}\right){\parallel }^{2}\\ \le & \parallel \left(1-{\alpha }_{n}\right)\left(\frac{{\beta }_{n}\left({x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}+\frac{{\gamma }_{n}\left({Q}_{C}\left(I-aA\right){x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}\\ +\frac{{\delta }_{n}\left({Q}_{C}\left(I-bB\right){x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}+\frac{{\eta }_{n}\left({S}^{A}{x}_{n}-{z}_{0}\right)}{1-{\alpha }_{n}}\right){\parallel }^{2}+2{\alpha }_{n}〈u-{x}_{0},j\left({x}_{n+1}-{z}_{0}\right)〉\\ \le & \left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{z}_{0}\parallel }^{2}+2{\alpha }_{n}〈u-{x}_{0},j\left({x}_{n+1}-{z}_{0}\right)〉.\end{array}$

Applying Lemma 2.6 and condition (i), we have ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{z}_{0}\parallel =0$. This completes the proof. □

## 4 Applications

From our main results, we obtain strong convergence theorems in a Banach space. Before proving these theorem, we need the following lemma which is the result from Lemma 2.7 and Definition 1.4. Therefore, we omit the proof.

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 with ${\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\ne \mathrm{\varnothing }$ 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 be the S-mapping generated by ${S}_{1},{S}_{2},\dots ,{S}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$. Then $F\left(S\right)={\bigcap }_{i=1}^{N}F\left({S}_{i}\right)$ and S 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 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 with $\mathcal{F}={\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap S\left(C,A\right)\cap S\left(C,B\right)\ne \mathrm{\varnothing }$ 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 be the S-mapping generated by ${S}_{1},{S}_{2},\dots ,{S}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$. Let $\left\{{x}_{n}\right\}$ be the sequence generated by ${x}_{1},u\in C$ and
${x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-aA\right){x}_{n}+{\delta }_{n}{Q}_{C}\left(I-bB\right){x}_{n}+{\eta }_{n}S{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,$
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:
$\begin{array}{r}\text{(i)}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}=0,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty },\\ \text{(ii)}\phantom{\rule{1em}{0ex}}\left\{{\gamma }_{n}\right\},\left\{{\delta }_{n}\right\},\left\{{\eta }_{n}\right\}\subseteq \left[c,d\right]\subset \left(0,1\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for some}}c,d>0,\phantom{\rule{0.25em}{0ex}}\mathrm{\forall }n\ge 1,\\ \text{(iii)}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_{n}|,\\ \phantom{\text{(iii)}\phantom{\rule{1em}{0ex}}}\sum _{n=1}^{\mathrm{\infty }}|{\eta }_{n+1}-{\eta }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|<\mathrm{\infty },\\ \text{(iv)}\phantom{\rule{1em}{0ex}}0<\underset{n\to \mathrm{\infty }}{lim inf}{\beta }_{n}\le \underset{n\to \mathrm{\infty }}{lim sup}{\beta }_{n}<1,\\ \text{(v)}\phantom{\rule{1em}{0ex}}a\in \left(0,\frac{\alpha }{{K}^{2}}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}b\in \left(0,\frac{\beta }{{K}^{2}}\right).\end{array}$

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 Put $I={T}_{1}={T}_{2}=\cdots ={T}_{N}$ in Theorem 3.1. From Lemma 4.1 and Theorem 3.1 we can conclude the desired result. □

Theorem 4.3 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 ${A}_{i}$, A, B be ${\alpha }_{i}$-, α- and β-inverse strongly accretive mappings of C into E, respectively. Define a mapping ${G}_{i}:C\to C$ by ${Q}_{C}\left(I-{\lambda }_{i}{A}_{i}\right)x={G}_{i}x$, where ${\lambda }_{i}\in \left(0,\frac{{\alpha }_{i}}{{K}^{2}}\right)$, K is the 2-uniformly smooth constant of E, for all $x\in C$ and $i=1,2,\dots ,N$. Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strict pseudo-contractions of C into itself and with $\mathcal{F}={\bigcap }_{i=1}^{N}F\left({S}_{i}\right)\cap {\bigcap }_{i=1}^{N}S\left(C,{A}_{i}\right)\cap S\left(C,A\right)\cap S\left(C,B\right)\ne \mathrm{\varnothing }$ and $\kappa =min\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$ with ${K}^{2}\le \kappa$. 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 ${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${G}_{1},{G}_{2},\dots ,{G}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$. Let $\left\{{x}_{n}\right\}$ be the sequence generated by ${x}_{1},u\in C$ and
${x}_{n+1}={\alpha }_{n}u+{\beta }_{n}{x}_{n}+{\gamma }_{n}{Q}_{C}\left(I-aA\right){x}_{n}+{\delta }_{n}{Q}_{C}\left(I-bB\right){x}_{n}+{\eta }_{n}{S}^{A}{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,$
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:
$\begin{array}{r}\text{(i)}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}=0,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty },\\ \text{(ii)}\phantom{\rule{1em}{0ex}}\left\{{\gamma }_{n}\right\},\left\{{\delta }_{n}\right\},\left\{{\eta }_{n}\right\}\subseteq \left[c,d\right]\subset \left(0,1\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for some}}c,d>0,\mathrm{\forall }n\ge 1,\\ \text{(iii)}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_{n}|,\\ \phantom{\text{(iii)}\phantom{\rule{1em}{0ex}}}\sum _{n=1}^{\mathrm{\infty }}|{\eta }_{n+1}-{\eta }_{n}|,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\alpha }_{n}\end{array}$