Fixed point theory for nonlinear mappings in Banach spaces and applications

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.

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

Definition 1.1 Let E be a Banach space. Then a function ${\delta }_X:[0,2]\to [0,1]$ is said to be the modulus of convexity of E if

$${\delta }_E(ϵ)=inf\{1-\parallel \frac{x+y}{2}\parallel :\parallel x\parallel \le 1,\parallel y\parallel \le 1,\parallel x-y\parallel \ge ϵ\}.$$

If ${\delta }_E(ϵ)>0$ for all $ϵ\in (0,2]$, 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

$${\rho }_E(t)=sup\{\frac{\parallel x+y\parallel +\parallel x-y\parallel }{2}-1:\parallel x\parallel =1,\parallel y\parallel =t\},t\ge 0.$$

If ${lim}_{t\to 0}\frac{{\rho }_E(t)}{t}=0$, 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 ${\rho }_E(t)\le c{t}^q$. 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 $\eta \in (0,1)$ such that

$$〈Tx-Ty,j(x-y)〉\le {\parallel x-y\parallel }^2-\eta {\parallel (I-T)x-(I-T)y\parallel }^2$$

(1.1)

for every $x,y\in C$ and for some $j(x-y)\in J(x-y)$. It is clear that (1.1) is equivalent to the following:

$$〈(I-T)x-(I-T)y,j(x-y)〉\ge \eta {\parallel (I-T)x-(I-T)y\parallel }^2$$

(1.2)

for every $x,y\in C$ and for some $j(x-y)\in J(x-y)$. 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 [1]) provided $P(x+t(x-P(x)))=P(x)$ for all $x\in C$ and $t\ge 0$, whenever $x+t(x-P(x))\in C$. 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 [2]) 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 $j(x-y)\in J(x-y)$ such that

$$〈Ax-Ay,j(x-y)〉\ge 0,\mathrm{\forall }x,y\in C.$$

A mapping $A:C\to E$ is said to be α-inverse strongly accretive if there exist $j(x-y)\in J(x-y)$ and $\alpha >0$ such that

$$〈Ax-Ay,j(x-y)〉\ge \alpha {\parallel Ax-Ay\parallel }^2,\mathrm{\forall }x,y\in C.$$

A mapping $A:C\to E$ is called γ-strongly accretive if there exist $j(x-y)\in J(x-y)$ and a constant $\gamma >0$ such that

$$〈Ax-Ay,j(x-y)〉\ge \gamma {\parallel x-y\parallel }^2$$

for all $x,y\in C$.

In 2006, Aoyama et al. [3] studied the variational inequality problem in Banach spaces. Such a problem is to find a point $x^{\ast }\in C$ such that for some $j(x-x^{\ast })\in J(x-x^{\ast })$,

$$〈A{x}^{\ast },j(x^{\ast }-x)〉\ge 0,\mathrm{\forall }x\in C.$$

(1.3)

The set of solutions of (1.3) in Banach spaces is denoted by $S(C,A)$, that is,

$$S(C,A)=\{u\in C:〈Au,J(v-u)〉\ge 0,\mathrm{\forall }v\in C\}.$$

(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 $S(C,A)\ne \mathrm{\varnothing }$. Suppose $x_1=x\in C$ and $\{x_n\}$ is given by

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)Q_C(x_n-{\lambda }_{n}A{x}_n)$$

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

Many authors have studied the variational inequality problem; see, for example, [4–8]. 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 [11] 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 ${\{S_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strict pseudo-contractions of C into itself, and let ${\{T_i\}}_{i=1}^N$ be a finite family of nonexpansive mappings of C into itself with $\mathcal{F}={\bigcap }_{i=1}^{N}F(S_i)\cap {\bigcap }_{i=1}^{N}F(T_i)\cap S(C,A)\cap S(C,B)\ne \mathrm{\varnothing }$ and $\kappa =min\{{\kappa }_i:i=1,2,\dots ,N\}$ with $K^2\le \kappa$, where K is the 2-uniformly smooth constant of E. Let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j\in (0,1]$, ${\alpha }_2^j\in [0,1]$ and ${\alpha }_3^j\in (0,1)$ 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 $\{x_n\}$ be the sequence generated by $x_1$, $u\in C$ and

$$x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{Q}_C(I-aA)x_n+{\delta }_n{Q}_C(I-bB)x_n+{\eta }_n{S}^A{x}_n,\mathrm{\forall }n\ge 1,$$

(1.5)

where $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\},\{{\delta }_n\},\{{\eta }_n\}\in [0,1]$ and ${\alpha }_n+{\beta }_n+{\gamma }_n+{\delta }_n+{\eta }_n=1$ and satisfy the following conditions:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $\{{\gamma }_n\},\{{\delta }_n\},\{{\eta }_n\}\subseteq [c,d]\subset (0,1)$ for some $c,d>0$;

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

• (iv) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$;

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

Then $\{x_n\}$ converges strongly to $z_0=Q_{\mathcal{F}}u$, where $Q_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto ℱ.

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 $j(x-x^{\ast })\in J(x-x^{\ast })$,

$$〈\sum _{i=1}^N{a}_i{A}_i{x}^{\ast },j(x^{\ast }-x)〉\ge 0$$

(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 $S(C,{\sum }_{i=1}^N{a}_i{A}_i)$, that is,

$$S(C,\sum _{i=1}^N{a}_i{A}_i)=\{u\in C:〈\sum _{i=1}^N{a}_i{A}_{i}u,J(v-u)〉\ge 0,\mathrm{\forall }v\in C\}.$$

(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 [12])

Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,J(x)〉+2{\parallel Ky\parallel }^2$$

for any $x,y\in E$.

Lemma 2.2 (See [13])

Let X be a uniformly convex Banach space and $B_r=\{x\in X:\parallel x\parallel \le r\}$, $r>0$. Then there exists a continuous, strictly increasing and convex function $g:[0,\mathrm{\infty }]\to [0,\mathrm{\infty }]$, $g(0)=0$ such that

$${\parallel \alpha x+\beta y+\gamma z\parallel }^2\le \alpha {\parallel x\parallel }^2+\beta {\parallel y\parallel }^2+\gamma {\parallel z\parallel }^2-\alpha \beta g(\parallel x-y\parallel )$$

for all $x,y,z\in B_r$ and all $\alpha ,\beta ,\gamma \in [0,1]$ with $\alpha +\beta +\gamma =1$.

Remark 2.3 For every $i=1,2,\dots ,N$, if $x_i\in B_r(0)$, 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 $a_i\in [0,1]$ and ${\sum }_{i=1}^N{a}_i=1$.

Lemma 2.4 (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 $\lambda >0$,

$$S(C,A)=F(Q_C(I-\lambda A)).$$

Lemma 2.5 (See [12])

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

Lemma 2.6 (See [14])

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 $F(T)$. If $\{x_n\}\subset C$ 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 $Q_{F(T)}:C\to F(T)$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-Q_{F(T)}u,J(x_n-Q_{F(T)}u)〉\le 0$$

for any given $u\in C$.

Lemma 2.7 (See [15])

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

• (1) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$,

• (2) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\delta }_n}{{\alpha }_n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_n|<\mathrm{\infty }$.

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

Lemma 2.8 [11]

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 $F(T_1)\cap F(T_2)\cap F(T_3)\ne \mathrm{\varnothing }$. Define a mapping S by

$$Sx=\alpha T_{1}x+\beta T_{2}x+\gamma T_{3}x,\mathrm{\forall }x\in C,$$

where α, β, γ is a constant in $(0,1)$ and $\alpha +\beta +\gamma =1$. Then S is a nonexpansive mapping and $F(S)=F(T_1)\cap F(T_2)\cap F(T_3)$.

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 $\overline{\alpha }={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$ and ${\bigcap }_{i=1}^{N}S(C,A_i)\ne \mathrm{\varnothing }$. Then $S(C,{\sum }_{i=1}^N{a}_i{A}_i)={\bigcap }_{i=1}^{N}S(C,A_i)$, where $a_i\in [0,1]$ and ${\sum }_{i=1}^N{a}_i=1$.

Proof It is easy to see that ${\bigcap }_{i=1}^{N}S(C,A_i)\subseteq S(C,{\sum }_{i=1}^N{a}_i{A}_i)$. Let $x_0\in S(C,{\sum }_{i=1}^N{a}_i{A}_i)$ and $x^{\ast }\in {\bigcap }_{i=1}^{N}S(C,A_i)$. Then there exist $j(y-x^{\ast })\in J(y-x^{\ast })$ and $j(y-x_0)\in J(y-x_0)$ such that

$$〈\sum _{i=1}^N{a}_i{A}_i{x}_0,j(y-x_0)〉\ge 0,\mathrm{\forall }y\in C$$

(2.1)

and

$$〈\sum _{i=1}^N{a}_i{A}_i{x}^{\ast },j(y-x^{\ast })〉\ge 0,\mathrm{\forall }y\in C.$$

(2.2)

From (2.1), (2.2) and $x_0,x^{\ast }\in C$, we have

$$〈\sum _{i=1}^N{a}_i{A}_i{x}_0,j(x^{\ast }-x_0)〉\ge 0$$

(2.3)

and

$$〈\sum _{i=1}^N{a}_i{A}_i{x}^{\ast },j(x_0-x^{\ast })〉\ge 0.$$

(2.4)

From (2.3) and (2.4), we have

$$\begin{array}{rcl}0& \le & 〈\sum _{i=1}^N{a}_i{A}_i{x}_0-\sum _{i=1}^N{a}_i{A}_i{x}^{\ast },j(x^{\ast }-x_0)〉\\ =& -〈\sum _{i=1}^N{a}_i{A}_i{x}^{\ast }-\sum _{i=1}^N{a}_i{A}_i{x}_0,j(x^{\ast }-x_0)〉\\ =& -\sum _{i=1}^N{a}_i〈A_i{x}^{\ast }-A_i{x}_0,j(x^{\ast }-x_0)〉\le -\sum _{i=1}^N{a}_i{\alpha }_i{\parallel x^{\ast }-x_0\parallel }^2\\ \le & -\sum _{i=1}^N{a}_i\overline{\alpha }{\parallel x^{\ast }-x_0\parallel }^2=-\overline{\alpha }{\parallel x^{\ast }-x_0\parallel }^2.\end{array}$$

It implies that $x^{\ast }=x_0$, that is, $x_0\in {\bigcap }_{i=1}^{N}S(C,A_i)$. Therefore $S(C,{\sum }_{i=1}^N{a}_i{A}_i)\subseteq {\bigcap }_{i=1}^{N}S(C,A_i)$. □

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 $\mathcal{F}=F(T)\cap F(S)\cap {\bigcap }_{i=1}^{N}S(C,A_i)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be a sequence generated by u, $x_1\in C$ and

$$\{\begin{array}{c}{z}_n=c_n{x}_n+(1-c_n)S{x}_n,\hfill \\ y_n=b_n{x}_n+(1-b_n)T{z}_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{Q}_C(I-\lambda {\sum }_{i=1}^N{a}_i{A}_i)y_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where $a_i\in [0,1]$ for all $i=1,2,\dots ,N$ and $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\}\subseteq [0,1]$ with ${\alpha }_n+{\beta }_n+{\gamma }_n=1$ for all $n\in \mathbb{N}$ satisfy the following conditions:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $0<a\le {\beta }_n,{\gamma }_n,c_n,b_n\le b<1$ for some $a,b>0$, $\mathrm{\forall }n\in \mathbb{N}$ and ${\sum }_{i=1}^N{a}_i=1$;

• (iii) $0\le \lambda K^2\le \frac{\overline{\alpha }}{{\overline{L}}^2}$;

• (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 $\{x_n\}$ converges strongly to $z_0=Q_{\mathcal{F}}u$, where $Q_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto ℱ.

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 $j(x-y)\in J(x-y)$ and

$$\begin{array}{rcl}〈\sum _{i=1}^N{a}_i{A}_{i}x-\sum _{i=1}^N{a}_i{A}_{i}y,j(x-y)〉& =& \sum _{i=1}^N{a}_i〈A_{i}x-A_{i}y,j(x-y)〉\\ \ge & \sum _{i=1}^N{a}_i{\alpha }_i{\parallel x-y\parallel }^2\\ \ge & \sum _{i=1}^N{a}_i\frac{{\alpha }_i}{L_i^2}{\parallel A_{i}x-A_{i}y\parallel }^2\\ \ge & \frac{\overline{\alpha }}{{\overline{L}}^2}\sum _{i=1}^N{a}_i{\parallel A_{i}x-A_{i}y\parallel }^2\\ \ge & \frac{\overline{\alpha }}{{\overline{L}}^2}{\parallel \sum _{i=1}^N{a}_i{A}_{i}x-\sum _{i=1}^N{a}_i{A}_{i}y\parallel }^2.\end{array}$$

(3.2)

Next, we show that $Q_C(I-\lambda {\sum }_{i=1}^N{a}_i{A}_i)$ is a nonexpansive mapping. From (3.2), we have

$$\begin{array}{r}{\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)x-Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y\parallel }^2\\ \le {\parallel x-y-\lambda (\sum _{i=1}^N{a}_i{A}_{i}x-\sum _{i=1}^N{a}_i{A}_{i}y)\parallel }^2\\ \le {\parallel x-y\parallel }^2-2\lambda 〈\sum _{i=1}^N{a}_i{A}_{i}x-\sum _{i=1}^N{a}_i{A}_{i}y,j(x-y)〉\\ +2K^2{\lambda }^2{\parallel \sum _{i=1}^N{a}_i(A_{i}x-A_{i}y)\parallel }^2\\ \le {\parallel x-y\parallel }^2-2\lambda \frac{\overline{\alpha }}{{\overline{L}}^2}{\parallel \sum _{i=1}^N{a}_i{A}_{i}x-\sum _{i=1}^N{a}_i{A}_{i}y\parallel }^2\\ +2K^2{\lambda }^2{\parallel \sum _{i=1}^N{a}_i(A_{i}x-A_{i}y)\parallel }^2\\ ={\parallel x-y\parallel }^2\\ -2\lambda (\frac{\overline{\alpha }}{{\overline{L}}^2}-K^2\lambda ){\parallel \sum _{i=1}^N{a}_i{A}_{i}x-\sum _{i=1}^N{a}_i{A}_{i}y\parallel }^2\\ \le {\parallel x-y\parallel }^2\end{array}$$

for all $x,y\in C$. Let $x^{\ast }\in \mathcal{F}$, we have

$$\begin{array}{rcl}\parallel x_{n+1}-x^{\ast }\parallel & =& \parallel {\alpha }_n(u-x^{\ast })+{\beta }_n(x_n-x^{\ast })+{\gamma }_n(Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x^{\ast })\parallel \\ \le & {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n\parallel y_n-x^{\ast }\parallel \\ =& {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n\parallel b_n(x_n-p)+(1-b_n)(T{z}_n-x^{\ast })\parallel \\ \le & {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel \\ +{\gamma }_n(b_n\parallel x_n-p\parallel +(1-b_n)\parallel z_n-x^{\ast }\parallel )\\ =& {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n(b_n\parallel x_n-p\parallel \\ +(1-b_n)\parallel c_n(x_n-x^{\ast })+(1-c_n)(S{x}_n-x^{\ast })\parallel ).\end{array}$$

(3.3)

Since S is a strictly pseudo-contractive mapping, we have

$$\begin{array}{rcl}{\parallel c_n(x_n-x^{\ast })+(1-c_n)(S{x}_n-x^{\ast })\parallel }^2& =& {\parallel x_n-x^{\ast }+(1-c_n)(S{x}_n-x_n)\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2+2(1-c_n)〈S{x}_n-x_n,j(x_n-x^{\ast })〉\\ +2K^2{(1-c_n)}^2{\parallel S{x}_n-x_n\parallel }^2\\ =& {\parallel x_n-x^{\ast }\parallel }^2-2(1-c_n)〈(I-S)x_n,j(x_n-x^{\ast })〉\\ +2K^2{(1-c_n)}^2{\parallel (I-S)x_n\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-2(1-c_n)\eta {\parallel (I-S)x_n\parallel }^2\\ +2K^2{(1-c_n)}^2{\parallel (I-S)x_n\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-2{(1-c_n)}^2\eta {\parallel (I-S)x_n\parallel }^2\\ +2K^2{(1-c_n)}^2{\parallel (I-S)x_n\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-2{(1-c_n)}^2(\eta -K^2){\parallel (I-S)x_n\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2.\end{array}$$

(3.4)

From (3.3) and (3.4), we have

$$\begin{array}{rcl}\parallel x_{n+1}-x^{\ast }\parallel & \le & {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n(b_n\parallel x_n-p\parallel \\ +(1-b_n)\parallel c_n(x_n-x^{\ast })+(1-c_n)(S{x}_n-x^{\ast })\parallel )\\ \le & {\alpha }_n\parallel u-x^{\ast }\parallel +(1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel \\ \le & max\{\parallel u-x^{\ast }\parallel ,\parallel x_1-x^{\ast }\parallel \}.\end{array}$$

From induction we can conclude that $\{x_n\}$ is bounded and so are $\{y_n\}$, $\{z_n\}$.

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

$$\begin{array}{rcl}\parallel x_{n+1}-x_n\parallel & =& \parallel {\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{Q}_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n\\ -{\alpha }_{n-1}u-{\beta }_{n-1}{x}_{n-1}-{\gamma }_{n-1}{Q}_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_{n-1}\parallel \\ \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel u\parallel +{\beta }_n\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel x_{n-1}\parallel \\ +{\gamma }_n\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_{n-1}\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_{n-1}\parallel \\ \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel u\parallel +{\beta }_n\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel x_{n-1}\parallel \\ +{\gamma }_n\parallel y_n-y_{n-1}\parallel +|{\gamma }_n-{\gamma }_{n-1}|\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_{n-1}\parallel .\end{array}$$

(3.5)

From the definition of $y_n$, we have

$$\begin{array}{rcl}\parallel y_n-y_{n-1}\parallel & =& \parallel b_n{x}_n+(1-b_n)T{z}_n-b_{n-1}{x}_{n-1}-(1-b_{n-1})T{z}_{n-1}\parallel \\ \le & b_n\parallel x_n-x_{n-1}\parallel +|b_n-b_{n-1}|\parallel x_{n-1}\parallel +(1-b_n)\parallel T{z}_n-T{z}_{n-1}\parallel \\ +|b_n-b_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & b_n\parallel x_n-x_{n-1}\parallel +|b_n-b_{n-1}|\parallel x_{n-1}\parallel +(1-b_n)\parallel z_n-z_{n-1}\parallel \\ +|b_n-b_{n-1}|\parallel T{z}_{n-1}\parallel .\end{array}$$

(3.6)

From the definition of $z_n$, we have

$$\begin{array}{rcl}\parallel z_n-z_{n-1}\parallel & =& \parallel c_n{x}_n+(1-c_n)S{x}_n-c_{n-1}{x}_{n-1}-(1-c_{n-1})S{x}_{n-1}\parallel \\ =& \parallel c_n(x_n-x_{n-1})+(c_n-c_{n-1})x_{n-1}+(1-c_n)(S{x}_n-S{x}_{n-1})\\ +(c_{n-1}-c_n)S{x}_{n-1}\parallel \\ \le & \parallel c_n(x_n-x_{n-1})+(1-c_n)(S{x}_n-S{x}_{n-1})\parallel +|c_n-c_{n-1}|\parallel x_{n-1}\parallel \\ +|c_n-c_{n-1}|\parallel S{x}_{n-1}\parallel .\end{array}$$

(3.7)

Since S is an η-strictly pseudo-contractive mapping, we have

$$\begin{array}{r}{\parallel c_n(x_n-x_{n-1})+(1-c_n)(S{x}_n-S{x}_{n-1})\parallel }^2\\ ={\parallel x_n-x_{n-1}-(1-c_n)((I-S)x_n-(I-S)x_{n-1})\parallel }^2\\ \le {\parallel x_n-x_{n-1}\parallel }^2-2(1-c_n)〈(I-S)x_n-(I-S)x_{n-1},j(x_n-x_{n-1})〉\\ +2K^2{(1-c_n)}^2{\parallel (I-S)x_n-(I-S)x_{n-1}\parallel }^2\\ \le {\parallel x_n-x_{n-1}\parallel }^2\\ -2(1-c_n)\eta {\parallel (I-S)x_n-(I-S)x_{n-1}\parallel }^2\\ +2K^2{(1-c_n)}^2{\parallel (I-S)x_n-(I-S)x_{n-1}\parallel }^2\\ \le {\parallel x_n-x_{n-1}\parallel }^2\\ -2{(1-c_n)}^2(\eta -K^2){\parallel (I-S)x_n-(I-S)x_{n-1}\parallel }^2\\ \le {\parallel x_n-x_{n-1}\parallel }^2.\end{array}$$

(3.8)

From (3.6), (3.7) and (3.8), we have

$$\begin{array}{rcl}\parallel y_n-y_{n-1}\parallel & \le & b_n\parallel x_n-x_{n-1}\parallel +|b_n-b_{n-1}|\parallel x_{n-1}\parallel +(1-b_n)\parallel z_n-z_{n-1}\parallel \\ +|b_n-b_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & b_n\parallel x_n-x_{n-1}\parallel +|b_n-b_{n-1}|\parallel x_{n-1}\parallel +(1-b_n)(\parallel c_n(x_n-x_{n-1})\\ +(1-c_n)(S{x}_n-S{x}_{n-1})\parallel +|c_n-c_{n-1}|\parallel x_{n-1}\parallel \\ +|c_n-c_{n-1}|\parallel S{x}_{n-1}\parallel )+|b_n-b_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & b_n\parallel x_n-x_{n-1}\parallel +|b_n-b_{n-1}|\parallel x_{n-1}\parallel \\ +(1-b_n)(\parallel x_n-x_{n-1}\parallel +|c_n-c_{n-1}|\parallel x_{n-1}\parallel \\ +|c_n-c_{n-1}|\parallel S{x}_{n-1}\parallel )+|b_n-b_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & \parallel x_n-x_{n-1}\parallel +|b_n-b_{n-1}|\parallel x_{n-1}\parallel +|c_n-c_{n-1}|\parallel x_{n-1}\parallel \\ +|c_n-c_{n-1}|\parallel S{x}_{n-1}\parallel +|b_n-b_{n-1}|\parallel T{z}_{n-1}\parallel .\end{array}$$

(3.9)

From (3.9) and (3.5), we have

$$\begin{array}{rcl}\parallel x_{n+1}-x_n\parallel & \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel u\parallel +{\beta }_n\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel x_{n-1}\parallel \\ +{\gamma }_n\parallel y_n-y_{n-1}\parallel +|{\gamma }_n-{\gamma }_{n-1}|\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_{n-1}\parallel \\ \le & (1-{\alpha }_n)\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel u\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_{n-1}\parallel \\ +|b_n-b_{n-1}|\parallel x_{n-1}\parallel +|c_n-c_{n-1}|\parallel x_{n-1}\parallel \\ +|c_n-c_{n-1}|\parallel S{x}_{n-1}\parallel +|b_n-b_{n-1}|\parallel T{z}_{n-1}\parallel \\ \le & (1-{\alpha }_n)\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|M+|{\alpha }_n-{\alpha }_{n-1}|M\\ +|{\gamma }_n-{\gamma }_{n-1}|M+2|b_n-b_{n-1}|M+2|c_n-c_{n-1}|M,\end{array}$$

where $M={max}_{n\in \mathbb{N}}\{\parallel x_n\parallel ,\parallel u\parallel ,\parallel Q_C(I-\lambda {\sum }_{i=1}^N{a}_i{A}_i)y_n\parallel ,\parallel S{x}_n\parallel ,\parallel T{z}_n\parallel \}$. 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

$$\underset{n\to \mathrm{\infty }}{lim}\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)x_n-x_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T{x}_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-S{x}_n\parallel =0.$$

From the definition of $x_n$, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& =& {\parallel {\alpha }_n(u-x^{\ast })+{\beta }_n(x_n-x^{\ast })+{\gamma }_n(Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x^{\ast })\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(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x^{\ast }\parallel }^2\\ -{\beta }_n{\gamma }_n{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)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(I-\lambda \sum _{i=1}^N{a}_i{A}_i)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(x_n-x^{\ast })+(1-b_n)(T{z}_n-x^{\ast })\parallel }^2\\ -{\beta }_n{\gamma }_n{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)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(b_n{\parallel x_n-x^{\ast }\parallel }^2+(1-b_n){\parallel z_n-x^{\ast }\parallel }^2-b_n(1-b_n)g_2(\parallel x_n-T{z}_n\parallel ))\\ -{\beta }_n{\gamma }_n{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)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(b_n{\parallel x_n-x^{\ast }\parallel }^2+(1-b_n)({\parallel x_n-x^{\ast }\parallel }^2\\ -2{(1-c_n)}^2(\eta -K^2){\parallel S{x}_n-x_n\parallel }^2)-b_n(1-b_n)g_2(\parallel x_n-T{z}_n\parallel ))\\ -{\beta }_n{\gamma }_n{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)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(1-b_n){(1-c_n)}^2(\eta -K^2){\parallel S{x}_n-x_n\parallel }^2\\ -{\gamma }_n{b}_n(1-b_n)g_2(\parallel x_n-T{z}_n\parallel )\\ -{\beta }_n{\gamma }_n{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)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(1-b_n){(1-c_n)}^2(\eta -K^2){\parallel S{x}_n-x_n\parallel }^2\\ -{\gamma }_n{b}_n(1-b_n)g_2(\parallel x_n-T{z}_n\parallel )\\ -{\beta }_n{\gamma }_n{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x_n\parallel \right).\end{array}$$

It implies that

$$\begin{array}{r}2{\gamma }_n(1-b_n){(1-c_n)}^2(\eta -K^2){\parallel S{x}_n-x_n\parallel }^2+{\gamma }_n{b}_n(1-b_n)g_2(\parallel x_n-T{z}_n\parallel )\\ +{\beta }_n{\gamma }_n{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x_n\parallel \right)\\ \le {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2\\ \le {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+(\parallel x_n-x^{\ast }\parallel +\parallel x_{n+1}-x^{\ast }\parallel )\parallel x_{n+1}-x_n\parallel .\end{array}$$

From (3.10), conditions (i) and (ii), we have

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}\parallel S{x}_n-x_n\parallel & =\underset{n\to \mathrm{\infty }}{lim}{g}_1\left(\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x_n\parallel \right)\\ =\underset{n\to \mathrm{\infty }}{lim}{g}_2(\parallel x_n-T{z}_n\parallel )=0.\end{array}$$

(3.11)

From the properties of $g_1$ and $g_2$, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T{z}_n\parallel =0.$$

(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

$$\underset{n\to \mathrm{\infty }}{lim}\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-y_n\parallel =0.$$

(3.16)

Then

$$\begin{array}{rcl}\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)x_n-x_n\parallel & \le & \parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)x_n-Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n\parallel \\ +\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x_n\parallel \\ \le & \parallel x_n-y_n\parallel +\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-x_n\parallel .\end{array}$$

From (3.12) and (3.14), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)x_n-x_n\parallel =0.$$

(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 $Gx=\alpha Q_C(I-\lambda {\sum }_{i=1}^N{a}_i{A}_i)x+\beta Tx+\gamma Wx$, where $Wx=cx+(1-c)Sx$ for all $x\in C$ and $\alpha ,\beta ,\gamma ,c\in [0,1]$ with $\alpha +\beta +\gamma =1$. We show that W is a nonexpansive mapping. Let $x,y\in C$, we have

$$\begin{array}{rcl}{\parallel Wx-Wy\parallel }^2& =& {\parallel c(x-y)+(1-c)(Sx-Sy)\parallel }^2\\ =& {\parallel x-y-(1-c)((I-S)x-(I-S)y)\parallel }^2\\ \le & {\parallel x-y\parallel }^2-2(1-c)〈(I-S)x-(I-S)y,j(x-y)〉\\ +2K^2{(1-c)}^2{\parallel (I-S)x-(I-S)y\parallel }^2\\ \le & {\parallel x-y\parallel }^2-2(1-c)\eta {\parallel (I-S)x-(I-S)y\parallel }^2\\ +2K^2{(1-c)}^2{\parallel (I-S)x-(I-S)y\parallel }^2\\ \le & {\parallel x-y\parallel }^2-2{(1-c)}^2(\eta -K^2){\parallel (I-S)x-(I-S)y\parallel }^2\\ \le & {\parallel x-y\parallel }^2.\end{array}$$

Then W is a nonexpansive mapping. It is easy to see that the mapping G is nonexpansive. From the definition of W, we have

$$F(S)=F(W).$$

(3.19)

From (3.19) Lemmas 2.8, 2.9 and the definition of G, we have $F(G)=F(T)\cap F(S)\cap {\bigcap }_{i=1}^{N}S(C,A_i)=\mathcal{F}$. Since

$$\begin{array}{rcl}\parallel G{x}_n-x_n\parallel & \le & \alpha \parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)x_n-x_n\parallel +\beta \parallel T{x}_n-x_n\parallel +\gamma \parallel W{x}_n-x_n\parallel \\ =& \alpha \parallel Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)x_n-x_n\parallel +\beta \parallel T{x}_n-x_n\parallel +\gamma (1-c)\parallel S{x}_n-x_n\parallel ,\end{array}$$

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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-z_0,j(x_n-z_0)〉\le 0,$$

(3.21)

where $z_0=Q_{\mathcal{F}}u$.

Finally, we show that the sequence $\{x_n\}$ converges strongly to $z_0=Q_{\mathcal{F}}u$. From the definition of $x_n$, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-z_0\parallel }^2& =& {\parallel {\alpha }_n(u-z_0)+{\beta }_n(x_n-z_0)+{\gamma }_n(Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-z_0)\parallel }^2\\ \le & {\parallel {\beta }_n(x_n-z_0)+{\gamma }_n(Q_C(I-\lambda \sum _{i=1}^N{a}_i{A}_i)y_n-z_0)\parallel }^2\\ +2{\alpha }_n〈u-z_0,j(x_{n+1}-z_0)〉\\ \le & (1-{\alpha }_n){\parallel x_n-z_0\parallel }^2+2{\alpha }_n〈u-z_0,j(x_{n+1}-z_0)〉.\end{array}$$

From Lemma 2.7 and condition (i), we can conclude that the sequence $\{x_n\}$ 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 $\mathcal{F}=F(T)\cap F(S)\cap S(C,A)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by $u,x_1\in C$ and

$$\{\begin{array}{c}{z}_n=c_n{x}_n+(1-c_n)S{x}_n,\hfill \\ y_n=b_n{x}_n+(1-b_n)T{z}_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{Q}_C(I-\lambda A)y_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\}\subseteq [0,1]$ with ${\alpha }_n+{\beta }_n+{\gamma }_n=1$ for all $n\in \mathbb{N}$ satisfy the following conditions:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $0<a\le {\beta }_n,{\gamma }_n,c_n,b_n\le b<1$, for some $a,b>0$, $\mathrm{\forall }n\in \mathbb{N}$;

• (iii) $0\le \lambda K^2\le \frac{\alpha }{L^2}$;

• (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 $\{x_n\}$ converges strongly to $z_0=Q_{\mathcal{F}}u$, where $Q_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto ℱ.

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 [11]

Let C be a nonempty convex subset of a real Banach space. Let ${\{S_i\}}_{i=1}^N$ and ${\{T_i\}}_{i=1}^N$ be two finite families of the mappings of C into itself. For each $j=1,2,\dots ,N$, let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, where $I\in [0,1]$ and ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$. Define the mapping $S^A:C\to C$ as follows:

$$\begin{array}{c}{U}_0=T_1=I,\hfill \\ U_1=T_1({\alpha }_1^1{S}_1{U}_0+{\alpha }_2^1{U}_0+{\alpha }_3^{1}I),\hfill \\ U_2=T_2({\alpha }_1^2{S}_2{U}_1+{\alpha }_2^2{U}_1+{\alpha }_3^{2}I),\hfill \\ U_3=T_3({\alpha }_1^3{S}_3{U}_2+{\alpha }_2^3{U}_2+{\alpha }_3^{3}I),\hfill \\ ⋮\hfill \\ U_{N-1}=T_{N-1}({\alpha }_1^{N-1}{S}_{N-1}{U}_{N-2}+{\alpha }_2^{N-1}{U}_{N-2}+{\alpha }_3^{N-1}I),\hfill \\ S^A=U_N=T_N({\alpha }_1^N{S}_N{U}_{N-1}+{\alpha }_2^N{U}_{N-1}+{\alpha }_3^{N}I).\hfill \end{array}$$

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 [11]

Let C be a nonempty closed convex subset of a 2-uniformly smooth and uniformly convex Banach space. Let ${\{S_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strict pseudo-contractions of C into itself, and let ${\{T_i\}}_{i=1}^N$ be a finite family of nonexpansive mappings of C into itself with ${\bigcap }_{i=1}^{N}F(S_i)\cap {\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$ and $\kappa =min\{{\kappa }_i:i=1,2,\dots ,N\}$ with $K^2\le \kappa$, where K is the 2-uniformly smooth constant of E. Let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j\in (0,1]$, ${\alpha }_2^j\in [0,1]$ and ${\alpha }_3^j\in (0,1)$ 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 $F(S^A)={\bigcap }_{i=1}^{N}F(S_i)\cap {\bigcap }_{i=1}^{N}F(T_i)$ 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 ${\{S_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strict pseudo-contractions of C into itself, and let ${\{T_i\}}_{i=1}^N$ be a finite family of nonexpansive mappings of C into itself with $\kappa =min\{{\kappa }_i:i=1,2,\dots ,N\}$ with $K^2\le \kappa$. Let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j\in (0,1]$, ${\alpha }_2^j\in [0,1]$ and ${\alpha }_3^j\in (0,1)$ 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 $\mathcal{F}=F(S)\cap {\bigcap }_{i=1}^{N}S(C,A_i)\cap {\bigcap }_{i=1}^{N}F(S_i)\cap {\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by $u,x_1\in C$ and

$$\{\begin{array}{c}{z}_n=c_n{x}_n+(1-c_n)S{x}_n,\hfill \\ y_n=b_n{x}_n+(1-b_n)S^A{z}_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{Q}_C(I-\lambda {\sum }_{i=1}^N{a}_i{A}_i)y_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(4.1)

where $a_i\in [0,1]$ for all $i=1,2,\dots ,N$ and $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\}\subseteq [0,1]$ with ${\alpha }_n+{\beta }_n+{\gamma }_n=1$ for all $n\in \mathbb{N}$ and satisfy the following conditions:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $0<a\le {\beta }_n,{\gamma }_n,c_n,b_n\le b<1$ for some $a,b>0$, $\mathrm{\forall }n\in \mathbb{N}$ and ${\sum }_{i=1}^N{a}_i=1$;

• (iii) $0\le \lambda K^2\le \frac{\overline{\alpha }}{{\overline{L}}^2}$;

• (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 $\{x_n\}$ converges strongly to $z_0=Q_{\mathcal{F}}u$, where $Q_{\mathcal{F}}$ is the sunny nonexpansive retraction of C onto ℱ.

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