An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and of the set of solutions for a modified system of variational inequalities

Abstract

In this paper, we introduce a new iterative algorithm for finding a common element of the set of fixed points of a finite family of ${\kappa }_i$-strictly pseudo-contractive mappings and the set of solutions of new variational inequalities problems in Hilbert space. By using our main results, we obtain an interesting theorem involving a finite family of κ-strictly pseudo-contractive mappings and two sets of solutions of the variational inequalities problem.

1 Introduction

Let H be a real Hilbert space whose inner product and norm are denoted by $\parallel \cdot \parallel$ and $〈\cdot ,\cdot 〉$, respectively. Let C be a nonempty closed convex subset of H. A mapping $S:C\to C$ is called nonexpansive if

$$\parallel Sx-Sy\parallel \le \parallel x-y\parallel ,$$

for all $x,y\in C$.

A mapping S is called a κ-strictly pseudo-contractive mapping if there exists $\kappa \in [0,1)$ such that

$${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\kappa {\parallel (I-T)x-(I-T)y\parallel }^2,$$

for all $x,y\in C$.

It is easy to see that every noexpansive mapping is a κ-strictly pseudo-contractive mapping.

Let $A:C\to H$. The variational inequality problem is to find a point $u\in C$ such that

$$〈Au,v-u〉\ge 0$$

(1.1)

for all $v\in C$. The set of solutions of (1.1) is denoted by $VI(C,A)$.

Variational inequalities were initially studied by Kinderlehrer and Stampacchia [1] and Lions and Stampacchia [2]. Such a problem has been studied by many researchers, and it is connected with a wide range of applications in industry, finance, economics, social sciences, ecology, regional, pure and applied sciences; see, e.g., [3–9].

A mapping A of C into H is called α-inverse-strongly monotone, see [10], if there exists a positive real number α such that

$$〈x-y,Ax-Ay〉\ge \alpha {\parallel Ax-Ay\parallel }^2$$

for all $x,y\in C$.

Let $D_1,D_2:C\to H$ be two mappings. In 2008, Ceng et al. [11] introduced a problem for finding $(x^{\ast },z^{\ast })\in C\times C$ such that

$$\{\begin{array}{c}〈{\lambda }_1{D}_1{z}^{\ast }+x^{\ast }-z^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ 〈{\lambda }_2{D}_2{x}^{\ast }+z^{\ast }-x^{\ast },x-z^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\hfill \end{array}$$

(1.2)

which is called a system of variational inequalities where ${\lambda }_1,{\lambda }_2>0$. By a modification of (1.2), we consider the problem for finding $(x^{\ast },z^{\ast })\in C\times C$ such that

$$\{\begin{array}{c}〈x^{\ast }-(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)z^{\ast }),x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ 〈z^{\ast }-(I-{\lambda }_2{D}_2)x^{\ast },x-z^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\hfill \end{array}$$

(1.3)

which is called a modification of system of variational inequalities, for every ${\lambda }_1,{\lambda }_2>0$ and $a\in [0,1]$. If $a=0$, (1.3) reduce to (1.2).

In 2008, Ceng et al. [11] introduce and studied a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space as follows.

Theorem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mappings $A,B:C\to H$ be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let $S:C\to C$ be a nonexpansive mapping such that $F(S)\cap \mathrm{\Omega }$, where Ω is the set of fixed points of the mapping $G:C\to C$, defined by $G(x)=P_C(P_C(x-\mu Bx)-\lambda A{P}_C(x-\mu Bx))$, for all $x\in C$. Suppose that $x_1=u\in C$ and $\{x_n\}$ is generated by

$$\{\begin{array}{c}{y}_n=P_C(x_n-\mu B{x}_n),\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{P}_C(x_n-\lambda A{x}_n),\hfill \end{array}$$

(1.4)

where $\lambda \in (0,2\alpha )$, $\mu \in (0,2\beta )$ and $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$ are three sequences in $[0,1]$ such that

$$\begin{array}{r}\text{(i)}{\alpha }_n+{\beta }_n+{\gamma }_n=1,\mathrm{\forall }n\ge 1,\\ \text{(ii)}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=0\mathit{\text{and}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },\\ \text{(iii)}0<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\beta }_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_n<1.\end{array}$$

Then $\{x_n\}$ converges strongly to $\tilde{x}=P_{F(S)\cap \mathrm{\Omega }}u$ and $(\tilde{x},\tilde{y})$ is a solution of problem (1.2), where $\tilde{y}=P_C(\tilde{x}-\mu B\tilde{x})$.

In the last decade, many author studied the problem for finding an element of the set of fixed points of a nonlinear mapping; see, for instance, [12–14].

From the motivation of [11] and the research in the same direction, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of ${\kappa }_i$-strictly pseudo-contractive mappings and the set of solutions of a modified general system of variational inequalities problems. Moreover, in the last section, we prove an interesting theorem involving the set of a finite family of ${\kappa }_i$-strictly pseudo-contractive mappings and two sets of solutions of variational inequalities problems by using our main results.

2 Preliminaries

In this section, we collect and give some useful lemmas that will be used for our main result in the next section.

Let C be a closed convex subset of a real Hilbert space H, let $P_C$ be the metric projection of H onto C, i.e., for $x\in H$, $P_{C}x$ satisfies the property

$$\parallel x-P_{C}x\parallel =\underset{y\in C}{min}\parallel x-y\parallel .$$

It is well known that $P_C$ is a nonexpansive mapping and satisfies

$$〈x-y,P_{C}x-P_{C}y〉\ge {\parallel P_{C}x-P_{C}y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Obviously, this immediately implies that

$${\parallel (x-y)-(P_{C}x-P_{C}y)\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel P_{C}x-P_{C}y\parallel }^2,\mathrm{\forall }x,y\in H.$$

The following characterizes the projection $P_C$.

Lemma 2.1 (See [15])

Given $x\in H$ and $y\in C$. Then $P_{C}x=y$ if and only if the following inequality holds:

$$〈x-y,y-z〉\ge 0,\mathrm{\forall }z\in C.$$

Lemma 2.2 (See [16])

Let $\{s_n\}$ be a sequence of nonnegative real numbers satisfying

$$s_{n+1}=(1-{\alpha }_n)s_n+{\alpha }_n{\beta }_n,\mathrm{\forall }n\ge 0,$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$ satisfy the conditions

$$\begin{array}{r}(1)\{{\alpha }_n\}\subset [0,1],\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },\\ (2)\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_n\le 0\mathit{\text{or}}\sum _{n=1}^{\mathrm{\infty }}|{\alpha }_n{\beta }_n|<\mathrm{\infty }.\end{array}$$

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

Lemma 2.3 (See [17])

Let $\{x_n\}$ and $\{z_n\}$ be bounded sequences in a Banach space X and let $\{{\beta }_n\}$ be a sequence in $[0,1]$ with $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$. Suppose that

$$x_{n+1}={\beta }_n{x}_n+(1-{\beta }_n)z_n$$

for all integer $n\ge 0$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel z_{n+1}-z_n\parallel -\parallel x_{n+1}-x_n\parallel )\le 0.$$

Then ${lim}_{n\to \mathrm{\infty }}\parallel x_n-z_n\parallel =0$.

Definition 2.1 (See [18])

Let C be a nonempty convex subset of a real Hilbert space. Let ${\{T_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strict pseudo-contractions 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 $S:C\to C$ as follows:

$$\begin{array}{r}{U}_0=I,\\ U_1={\alpha }_1^1{T}_1{U}_0+{\alpha }_2^1{U}_0+{\alpha }_3^{1}I,\\ U_2={\alpha }_1^2{T}_2{U}_1+{\alpha }_2^2{U}_1+{\alpha }_3^{2}I,\\ U_3={\alpha }_1^3{T}_3{U}_2+{\alpha }_2^3{U}_2+{\alpha }_3^{3}I,\\ ⋮\\ U_{N-1}={\alpha }_1^{N-1}{T}_{N-1}{U}_{N-2}+{\alpha }_2^{N-1}{U}_{N-2}+{\alpha }_3^{N-1}I,\\ S=U_N={\alpha }_1^N{T}_N{U}_{N-1}+{\alpha }_2^N{U}_{N-1}+{\alpha }_3^{N}I.\end{array}$$

(2.1)

This mapping is called S-mapping generated by $T_1,T_2,\dots ,T_N$ and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N$.

Lemma 2.4 (See [18])

Let C be a nonempty closed convex subset of a real Hilbert space. Let ${\{T_i\}}_{i=1}^N$ be a finite family of κ-strict pseudo-contractive mappings of C into C with ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$ and $\kappa =max\{{\kappa }_i:i=1,2,\dots ,N\}$ and let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, $j=1,2,3,\dots ,N$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j,{\alpha }_3^j\in (\kappa ,1)$ for all $j=1,2,\dots ,N-1$ and ${\alpha }_1^N\in (\kappa ,1]$, ${\alpha }_3^N\in [\kappa ,1)$, ${\alpha }_2^j\in [\kappa ,1)$ for all $j=1,2,\dots ,N$. Let S be a mapping generated by $T_1,T_2,\dots ,T_N$ and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N$. Then $F(S)={\bigcap }_{i=1}^{N}F(T_i)$ and S is a nonexpansive mapping.

Lemma 2.5 (See [19])

Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and let $S:C\to C$ be a nonexpansive mapping. Then $I-S$ is demi-closed at zero.

Lemma 2.6 In a real Hilbert space H, the following inequality holds:

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉$$

for all $x,y\in H$.

Lemma 2.7 Let C be a nonempty closed convex subset of a Hilbert space H and let $D_1,D_2:C\to H$ be mappings. For every ${\lambda }_1,{\lambda }_2>0$ and $a\in [0,1]$, the following statements are equivalent:

• (a) $(x^{\ast },z^{\ast })\in C\times C$ is a solution of problem (1.3),

• (b) $x^{\ast }$ is a fixed point of the mapping $G:C\to C$, i.e., $x^{\ast }\in F(G)$, defined by

  $$G(x)=P_C(I-{\lambda }_1{D}_1)(ax+(1-a)P_C(I-{\lambda }_2{D}_2)x),$$

where $z^{\ast }=P_C(I-{\lambda }_2{D}_2)x^{\ast }$.

Proof (a) ⇒ (b) Let $(x^{\ast },z^{\ast })\in C\times C$ be a solution of problem (1.3). For every ${\lambda }_1,{\lambda }_2>0$ and $a\in [0,1]$, we have

$$\{\begin{array}{c}〈x^{\ast }-(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)z^{\ast }),x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ 〈z^{\ast }-(I-{\lambda }_2{D}_2)x^{\ast },x-z^{\ast }〉\ge 0,\mathrm{\forall }x\in C.\hfill \end{array}$$

From the properties of $P_C$, we have

$$\{\begin{array}{c}{x}^{\ast }=P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)z^{\ast }),\hfill \\ z^{\ast }=P_C(I-{\lambda }_2{D}_2)x^{\ast }.\hfill \end{array}$$

It implies that

$$x^{\ast }=P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)P_C(I-{\lambda }_2{D}_2)x^{\ast })=G\left(x^{\ast }\right).$$

Hence, we have $x^{\ast }\in F(G)$, where $z^{\ast }=P_C(I-{\lambda }_2{D}_2)x^{\ast }$.

(b) ⇒ (a) Let $x^{\ast }\in F(G)$ and $z^{\ast }=P_C(I-{\lambda }_2{D}_2)x^{\ast }$. Then, we have

$$\begin{array}{rcl}{x}^{\ast }& =& G\left(x^{\ast }\right)=P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)P_C(I-{\lambda }_2{D}_2)x^{\ast })\\ =& P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)z^{\ast }).\end{array}$$

From the properties of $P_C$, we have

$$\{\begin{array}{c}〈x^{\ast }-(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)z^{\ast }),x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ 〈z^{\ast }-(I-{\lambda }_2{D}_2)x^{\ast },x-z^{\ast }〉\ge 0,\mathrm{\forall }x\in C.\hfill \end{array}$$

Hence, we have $(x^{\ast },z^{\ast })\in C\times C$ is a solution of (1.3). □

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let $D_1,D_2:C\to H$ be $d_1,d_2$-inverse strongly monotone mappings, respectively. Define the mapping $G:C\to C$ by $G(x)=P_C(I-{\lambda }_1{D}_1)(ax+(1-a)P_C(I-{\lambda }_2{D}_2)x)$ for all $x\in C$, ${\lambda }_1,{\lambda }_2>0$ and $a\in [0,1)$. Let ${\{T_i\}}_{i=1}^N$ be a finite family of κ-strict pseudo-contractive mappings of C into C with $\mathcal{F}={\bigcap }_{i=1}^{N}F(T_i)\cap F(G)\ne \mathrm{\varnothing }$ and $\kappa =max\{{\kappa }_i:i=1,2,\dots ,N\}$ and let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, $j=1,2,3,\dots ,N$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j,{\alpha }_3^j\in (\kappa ,1)$ for all $j=1,2,\dots ,N-1$ and ${\alpha }_1^N\in (\kappa ,1]$, ${\alpha }_3^N\in [\kappa ,1)$, ${\alpha }_2^j\in [\kappa ,1)$ for all $j=1,2,\dots ,N$. Let S be a mapping generated by $T_1,T_2,\dots ,T_N$ and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N$. Suppose that $x_1,u\in C$ and let $\{x_n\}$ be the sequence generated by

$$\{\begin{array}{c}{y}_n=P_C(I-{\lambda }_2{D}_2)x_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_{n}S{P}_C(a{x}_n+(1-a)y_n-{\lambda }_1{D}_1(a{x}_n+(1-a)y_n)),\hfill \\ \phantom{x_{n+1}=}\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where ${\lambda }_1\in (0,2d_1)$, ${\lambda }_2\in (0,2d_2)$ and $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$ are sequences in $[0,1]$. Assume that the following conditions hold:

$$\begin{array}{r}\text{(i)}{\alpha }_n+{\beta }_n+{\gamma }_n=1,\\ \text{(ii)}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=0\mathit{\text{and}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },\\ \text{(iii)}0<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\beta }_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_n<1.\end{array}$$

Then $\{x_n\}$ converges strongly to $x_0=P_{\mathcal{F}}u$ and $(x_0,y_0)$ is a solution of (1.3), where $y_0=P_C(I-{\lambda }_2{D}_2)x_0$.

Proof First, we show that $P_C(I-{\lambda }_1{D}_1)$ and $P_C(I-{\lambda }_2{D}_2)$ are nonexpansive mappings for every ${\lambda }_1\in (0,2d_1)$, ${\lambda }_2\in (0,2d_2)$. Let $x,y\in C$. Since $D_1$ is $d_1$-inverse strongly monotone and ${\lambda }_1<2d_1$, we have

$$\begin{array}{rcl}{\parallel (I-{\lambda }_1{D}_1)x-(I-{\lambda }_1{D}_1)y\parallel }^2& =& {\parallel x-y-{\lambda }_1(D_{1}x-D_{1}y)\parallel }^2\\ =& {\parallel x-y\parallel }^2-2{\lambda }_1〈x-y,D_{1}x-D_{1}y〉+{\lambda }_1^2{\parallel D_{1}x-D_{1}y\parallel }^2\\ \le & {\parallel x-y\parallel }^2-2d_1{\lambda }_1{\parallel D_{1}x-D_{1}y\parallel }^2+{\lambda }_1^2{\parallel D_{1}x-D_{1}y\parallel }^2\\ =& {\parallel x-y\parallel }^2+{\lambda }_1({\lambda }_1-2d_1){\parallel D_{1}x-D_{1}y\parallel }^2\\ \le & {\parallel x-y\parallel }^2.\end{array}$$

(3.2)

Thus $(I-{\lambda }_1{D}_1)$ is a nonexpansive mapping. By using the same method as (3.2), we have $(I-{\lambda }_2{D}_2)$ is a nonexpansive mapping. Hence, $P_C(I-{\lambda }_1{D}_1)$, $P_C(I-{\lambda }_2{D}_2)$ are nonexpansive mappings. It is easy to see that the mapping G is a nonexpansive mapping. Let $x^{\ast }\in \mathcal{F}$. Then we have $x^{\ast }=S{x}^{\ast }$ and

$$x^{\ast }=G\left(x^{\ast }\right)=P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)P_C(I-{\lambda }_2{D}_2)x^{\ast }).$$

Put $w_n=P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)$ and $y^{\ast }=P_C(I-{\lambda }_2{D}_2)x^{\ast }$, we can rewrite (3.1) by

$$x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_{n}S{w}_n,\mathrm{\forall }n\ge 1,$$

and $x^{\ast }=P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)y^{\ast })$.

From the definition of $x_n$, we have

$$\begin{array}{rcl}\parallel x_{n+1}-x^{\ast }\parallel & =& \parallel {\alpha }_n(u-x^{\ast })+{\beta }_n(x_n-x^{\ast })+{\gamma }_n(S{w}_n-x^{\ast })\parallel \\ \le & {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n\parallel w_n-x^{\ast }\parallel \\ =& {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n\parallel P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)\\ -P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)P_C(I-{\lambda }_2{D}_2)x^{\ast })\parallel \\ \le & {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n\parallel a(x_n-x^{\ast })\\ +(1-a)(P_C(I-{\lambda }_2{D}_2)x_n-P_C(I-{\lambda }_2{D}_2)x^{\ast })\parallel \\ \le & {\alpha }_n\parallel u-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel +{\gamma }_n(a\parallel x_n-x^{\ast }\parallel +(1-a)\parallel x_n-x^{\ast }\parallel )\\ =& {\alpha }_n\parallel u-x^{\ast }\parallel +(1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel \\ \le & max\{\parallel u-x^{\ast }\parallel ,\parallel x_1-x^{\ast }\parallel \}.\end{array}$$

By induction we can conclude that $\parallel x_n-x^{\ast }\parallel \le max\{\parallel u-x^{\ast }\parallel ,\parallel x_1-x^{\ast }\parallel \}$ for all $n\in \mathbb{N}$. It implies that $\{x_n\}$ is bounded and so are $\{y_n\}$ and $\{w_n\}$.

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

Let

$$x_{n+1}=(1-{\beta }_n)z_n+{\beta }_n{x}_n,$$

(3.3)

where $z_n=\frac{x_{n+1}-{\beta }_n{x}_n}{1-{\beta }_n}$.

Since $x_{n+1}-{\beta }_n{x}_n={\alpha }_{n}u+{\gamma }_{n}S{w}_n$ and (3.3), we have

$$\begin{array}{rcl}{z}_{n+1}-z_n& =& \frac{x_{n+2}-{\beta }_{n+1}{x}_{n+1}}{1-{\beta }_{n+1}}-\frac{x_{n+1}-{\beta }_n{x}_n}{1-{\beta }_n}\\ =& \frac{{\alpha }_{n+1}u+{\gamma }_{n+1}S{w}_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_{n}u+{\gamma }_{n}S{w}_n}{1-{\beta }_n}\\ -\frac{{\gamma }_{n+1}S{w}_n}{1-{\beta }_{n+1}}+\frac{{\gamma }_{n+1}S{w}_n}{1-{\beta }_{n+1}}\\ =& (\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n})u+\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}(S{w}_{n+1}-S{w}_n)\\ +(\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n})S{w}_n\\ =& (\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n})u+\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}(S{w}_{n+1}-S{w}_n)\\ +(\frac{{\alpha }_n}{1-{\beta }_n}-\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}})S{w}_n.\end{array}$$

It follows that

$$\begin{array}{rcl}\parallel z_{n+1}-z_n\parallel & \le & |\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}|\parallel u\parallel +\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}\parallel S{w}_{n+1}-S{w}_n\parallel \\ +|\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}|\parallel S{w}_n\parallel \\ =& |\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}|(\parallel u\parallel +\parallel S{w}_n\parallel )+\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}\parallel w_{n+1}-w_n\parallel \\ =& |\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}|(\parallel u\parallel +\parallel S{w}_n\parallel )\\ +\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}\parallel P_C(I-{\lambda }_1{D}_1)(a{x}_{n+1}+(1-a)y_{n+1})\\ -P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)\parallel \\ \le & |\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}|(\parallel u\parallel +\parallel S{w}_n\parallel )\\ +\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}\parallel a(x_{n+1}-x_n)+(1-a)(y_{n+1}-y_n)\parallel \\ \le & |\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}|(\parallel u\parallel +\parallel S{w}_n\parallel )\\ +\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}(a\parallel x_{n+1}-x_n\parallel +(1-a)\parallel P_C(I-{\lambda }_2{D}_2)x_{n+1}-P_C(I-{\lambda }_2{D}_2)x_n\parallel )\\ \le & |\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}|(\parallel u\parallel +\parallel S{w}_n\parallel )\\ +\parallel x_{n+1}-x_n\parallel .\end{array}$$

From conditions (ii) and (iii), we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel z_{n+1}-z_n\parallel -\parallel x_{n+1}-x_n\parallel )\le 0.$$

From Lemma 2.3 and (3.3) we have ${lim}_{n\to \mathrm{\infty }}\parallel z_n-x_n\parallel =0$. Since $x_{n+1}-x_n=(1-{\beta }_n)(z_n-x_n)$, then we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0.$$

(3.4)

From the definition of $w_n$, we have

$$\begin{array}{rcl}\parallel w_{n+1}-w_n\parallel & \le & \parallel P_C(I-{\lambda }_1{D}_1)(a{x}_{n+1}+(1-a)y_{n+1})-P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)\parallel \\ \le & a\parallel x_{n+1}-x_n\parallel +(1-a)\parallel y_{n+1}-y_n\parallel \\ =& a\parallel x_{n+1}-x_n\parallel +(1-a)\parallel P_C(I-{\lambda }_2{D}_2)x_{n+1}-P_C(I-{\lambda }_2{D}_2)x_n\parallel \\ \le & a\parallel x_{n+1}-x_n\parallel +(1-a)\parallel x_{n+1}-x_n\parallel \\ =& \parallel x_{n+1}-x_n\parallel .\end{array}$$

From (3.4), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel w_{n+1}-w_n\parallel =0.$$

(3.5)

From the definition of $x_n$, we have

$$x_{n+1}-x_n={\alpha }_n(u-x_n)+{\gamma }_n(S{w}_n-x_n).$$

From (3.4), conditions (ii) and (iii), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S{w}_n-x_n\parallel =0.$$

(3.6)

From the definition of $y_n$, we have

$$\parallel y_{n+1}-y_n\parallel =\parallel P_C(I-{\lambda }_2{D}_2)x_{n+1}-P_C(I-{\lambda }_2{D}_2)x_n\parallel \le \parallel x_{n+1}-x_n\parallel .$$

(3.7)

From (3.4) and (3.7), we derive

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_{n+1}-y_n\parallel =0.$$

(3.8)

From the nonexpansiveness of $P_C(I-{\lambda }_1{D}_1)$ and $P_C(I-{\lambda }_2{D}_2)$, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-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 S{w}_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 w_n-x^{\ast }\parallel }^2\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n{\parallel P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)-P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)y^{\ast })\parallel }^2\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a){\parallel y_n-y^{\ast }\parallel }^2)\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a){\parallel P_C(I-{\lambda }_2{D}_2)x_n-P_C(I-{\lambda }_2{D}_2)x^{\ast }\parallel }^2)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a){\parallel (I-{\lambda }_2{D}_2)x_n-(I-{\lambda }_2{D}_2)x^{\ast }\parallel }^2)\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a){\parallel (x_n-x^{\ast })-{\lambda }_2(D_2{x}_n-D_2{x}^{\ast })\parallel }^2)\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a)({\parallel x_n-x^{\ast }\parallel }^2-2{\lambda }_2〈x_n-x^{\ast },D_2{x}_n-D_2{x}^{\ast }〉\\ +{\lambda }_2^2{\parallel D{x}_n-D{x}^{\ast }\parallel }^2\left)\right)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a)({\parallel x_n-x^{\ast }\parallel }^2-2{\lambda }_2{d}_2{\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2\\ +{\lambda }_2^2{\parallel D{x}_n-D{x}^{\ast }\parallel }^2\left)\right)\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a)({\parallel x_n-x^{\ast }\parallel }^2\\ -{\lambda }_2(2d_2-{\lambda }_2){\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2\left)\right)\\ =& {\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-{\lambda }_2(1-a)(2d_2-{\lambda }_2){\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2-{\lambda }_2{\gamma }_n(1-a)(2d_2-{\lambda }_2){\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2.\end{array}$$

It implies that

$$\begin{array}{rcl}{\lambda }_2{\gamma }_n(1-a)(2d_2-{\lambda }_2){\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2& \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+(\parallel x_n-x^{\ast }\parallel +\parallel x_{n+1}-x^{\ast }\parallel )\\ \times \parallel x_{n+1}-x_n\parallel .\end{array}$$

(3.9)

From (3.4), (3.9) conditions (ii) and (iii), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel D_2{x}_n-D_2{x}^{\ast }\parallel =0.$$

(3.10)

Put $h^{\ast }=a{x}^{\ast }+(1-a)y^{\ast }$ and $h_n=a{x}_n+(1-a)y_n$. From the definition of $x_n$, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-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 w_n-x^{\ast }\parallel }^2\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n{\parallel P_C(I-{\lambda }_1{D}_1)h_n-P_C(I-{\lambda }_1{D}_1)h^{\ast }\parallel }^2\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n{\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel }^2\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n{\parallel (h_n-h^{\ast })-{\lambda }_1(D_1{h}_n-D_1{h}^{\ast })\parallel }^2\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n({\parallel h_n-h^{\ast }\parallel }^2-2{\lambda }_1〈h_n-h^{\ast },D_1{h}_n-D_1{h}^{\ast }〉+{\lambda }_1^2{\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n({\parallel h_n-h^{\ast }\parallel }^2-2{\lambda }_1{d}_1{\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2\\ +{\lambda }_1^2{\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2)\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n({\parallel h_n-h^{\ast }\parallel }^2-{\lambda }_1(2d_1-{\lambda }_1){\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2)\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n({\parallel a(x_n-x^{\ast })+(1-a)(y_n-y^{\ast })\parallel }^2\\ -{\lambda }_1(2d_1-{\lambda }_1){\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2\\ +(1-a){\parallel P_C(I-{\lambda }_2{D}_2)x_n-P_C(I-{\lambda }_2{D}_2)x^{\ast }\parallel }^2\\ -{\lambda }_1(2d_1-{\lambda }_1){\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2-{\lambda }_1{\gamma }_n(2d_1-{\lambda }_1){\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2,\end{array}$$

which implies that

$$\begin{array}{rl}{\lambda }_1{\gamma }_n(2d_1-{\lambda }_1){\parallel D_1{h}_n-D_1{h}^{\ast }\parallel }^2\le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+(\parallel x_n-x^{\ast }\parallel +\parallel x_{n+1}-x^{\ast }\parallel )\\ \times \parallel x_{n+1}-x_n\parallel .\end{array}$$

(3.11)

From (3.4), (3.11), conditions (ii) and (iii), we can conclude

$$\underset{n\to \mathrm{\infty }}{lim}\parallel D_1{h}_n-D_1{h}^{\ast }\parallel =0.$$

(3.12)

Next, we show that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S{w}_n-w_n\parallel =0.$$

(3.13)

From the definition of $y_n$, we have

$$\begin{array}{rcl}{\parallel y_n-y^{\ast }\parallel }^2& =& {\parallel P_C(I-{\lambda }_2{D}_2)x_n-P_C(I-{\lambda }_2{D}_2)x^{\ast }\parallel }^2\\ \le & 〈x_n-{\lambda }_2{D}_2{x}_n-(x^{\ast }-{\lambda }_2{D}_2{x}^{\ast }),y_n-y^{\ast }〉\\ =& \frac{1}{2}({\parallel x_n-{\lambda }_2{D}_2{x}_n-(x^{\ast }-{\lambda }_2{D}_2{x}^{\ast })\parallel }^2+{\parallel y_n-y^{\ast }\parallel }^2\\ -{\parallel x_n-{\lambda }_2{D}_2{x}_n-(x^{\ast }-{\lambda }_2{D}_2{x}^{\ast })-(y_n-y^{\ast })\parallel }^2)\\ =& \frac{1}{2}({\parallel x_n-{\lambda }_2{D}_2{x}_n-(x^{\ast }-{\lambda }_2{D}_2{x}^{\ast })\parallel }^2+{\parallel y_n-y^{\ast }\parallel }^2\\ -{\parallel x_n-y_n-(x^{\ast }-y^{\ast })-{\lambda }_2(D_2{x}_n-D_2{x}^{\ast })\parallel }^2)\\ =& \frac{1}{2}({\parallel x_n-{\lambda }_2{D}_2{x}_n-(x^{\ast }-{\lambda }_2{D}_2{x}^{\ast })\parallel }^2+{\parallel y_n-y^{\ast }\parallel }^2\\ -{\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel }^2+2{\lambda }_2〈x_n-y_n-(x^{\ast }-y^{\ast }),D_2{x}_n-D_2{x}^{\ast }〉\\ -{\lambda }_1^2{\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2).\end{array}$$

It implies that

$$\begin{array}{rcl}\parallel y_n-y^{\ast }\parallel & \le & {\parallel x_n-{\lambda }_2{D}_2{x}_n-(x^{\ast }-{\lambda }_2{D}_2{x}^{\ast })\parallel }^2-{\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel }^2\\ +2{\lambda }_2〈x_n-y_n-(x^{\ast }-y^{\ast }),D_2{x}_n-D_2{x}^{\ast }〉-{\lambda }_1^2{\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel }^2\\ +2{\lambda }_2〈x_n-y_n-(x^{\ast }-y^{\ast }),D_2{x}_n-D_2{x}^{\ast }〉\\ -{\lambda }_1^2{\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2.\end{array}$$

(3.14)

From the nonexpansiveness of $P_C(I-{\lambda }_1{D}_1)$ and (3.14), we have

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n{\parallel S{w}_n-x^{\ast }\parallel }^2\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n{\parallel w_n-x^{\ast }\parallel }^2\\ =& {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n{\parallel P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)-P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)y^{\ast })\parallel }^2\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2+{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a){\parallel y_n-y^{\ast }\parallel }^2)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a)({\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel }^2\\ +2{\lambda }_2〈x_n-y_n-(x^{\ast }-y^{\ast }),D_2{x}_n-D_2{x}^{\ast }〉-{\lambda }_1^2{\parallel D_2{x}_n-D_2{x}^{\ast }\parallel }^2\left)\right)\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\beta }_n{\parallel x_n-x^{\ast }\parallel }^2\\ +{\gamma }_n(a{\parallel x_n-x^{\ast }\parallel }^2+(1-a){\parallel x_n-x^{\ast }\parallel }^2-(1-a){\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel }^2\\ +2{\lambda }_2\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel \parallel D_2{x}_n-D_2{x}^{\ast }\parallel )\\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2-{\gamma }_n(1-a){\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel }^2\\ +2{\lambda }_2\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel \parallel D_2{x}_n-D_2{x}^{\ast }\parallel .\end{array}$$

It follows that

$$\begin{array}{rcl}{\gamma }_n(1-a){\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel }^2& \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2\\ +2{\lambda }_2\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel \parallel D_2{x}_n-D_2{x}^{\ast }\parallel \\ \le & {\alpha }_n{\parallel u-x^{\ast }\parallel }^2+(\parallel x_n-x^{\ast }\parallel +\parallel x_{n+1}-x^{\ast }\parallel )\parallel x_{n+1}-x_n\parallel \\ +2{\lambda }_2\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel \parallel D_2{x}_n-D_2{x}^{\ast }\parallel .\end{array}$$

From condition (ii), (3.4) and (3.10), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel =0.$$

(3.15)

From the definition of $w_n$, $x^{\ast }$, $h_n$, $h^{\ast }$, we have

$$w_n=P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)=P_C(I-{\lambda }_1{D}_1)h_n$$

and

$$x^{\ast }=P_C(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)y^{\ast })=P_C(I-{\lambda }_1{D}_1)h^{\ast }.$$

From the properties of $P_C$, we have

$$\begin{array}{rcl}{\parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel }^2& =& {\parallel y_n-y^{\ast }-(w_n-x^{\ast })\parallel }^2\\ =& \parallel y_n-a{x}_n+a{x}_n-a{y}_n+a{y}_n-{\lambda }_1{D}_1(a{x}_n+(1-a)y_n)\\ +{\lambda }_1{D}_1(a{x}_n+(1-a)y_n-y^{\ast }+a{x}^{\ast }-a{x}^{\ast }+a{y}^{\ast }-a{y}^{\ast }\\ +{\lambda }_1{D}_1(a{x}^{\ast }+(1-a)y^{\ast })\\ -{\lambda }_1{D}_1(a{x}^{\ast }+(1-a)y^{\ast })-(w_n-x^{\ast }){\parallel }^2\\ =& \parallel a{x}_n+(1-a)y_n-{\lambda }_1{D}_1(a{x}_n+(1-a)y_n)\\ -(a{x}^{\ast }+(1-a)y^{\ast }-{\lambda }_1{D}_1(a{x}^{\ast }+(1-a)y^{\ast }))-(w_n-x^{\ast })\\ +{\lambda }_1(D_1(a{x}_n+(1-a)y_n)-D_1(a{x}^{\ast }+(1-a)y^{\ast }))\\ +a(y_n-x_n-y^{\ast }+x^{\ast }){\parallel }^2\\ =& \parallel (I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n)-(I-{\lambda }_1{D}_1)(a{x}^{\ast }+(1-a)y^{\ast })\\ -(w_n-x^{\ast })+{\lambda }_1(D_1(a{x}_n+(1-a)y_n)-D_1(a{x}^{\ast }+(1-a)y^{\ast }))\\ +a(y_n-x_n-y^{\ast }+x^{\ast }){\parallel }^2\\ =& \parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\\ -(P_C(I-{\lambda }_1{D}_1)h_n-P_C(I-{\lambda }_1{D}_1)h^{\ast })+{\lambda }_1(D_1{h}_n-D_1{h}^{\ast })\\ +a(y_n-x_n-y^{\ast }+x^{\ast }){\parallel }^2\\ \le & \parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }-(P_C(I-{\lambda }_1{D}_1)h_n\\ -P_C(I-{\lambda }_1{D}_1)h^{\ast }){\parallel }^2\\ +2〈{\lambda }_1(D_1{h}_n-D_1{h}^{\ast })+a(y_n-x_n-y^{\ast }+x^{\ast }),\\ y_n-w_n+(x^{\ast }-y^{\ast })〉\\ \le & {\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel }^2\\ -{\parallel P_C(I-{\lambda }_1{D}_1)h_n-P_C(I-{\lambda }_1{D}_1)h^{\ast }\parallel }^2\\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel \\ =& {\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel }^2-{\parallel w_n-x^{\ast }\parallel }^2\\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel \\ \le & {\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel }^2-{\parallel S{w}_n-S{x}^{\ast }\parallel }^2\\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel \\ \le & (\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel +\parallel S{w}_n-S{x}^{\ast }\parallel )\\ \times \parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }-(S{w}_n-x^{\ast })\parallel \\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel \\ =& (\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel +\parallel S{w}_n-S{x}^{\ast }\parallel )\\ \times \parallel h_n-h^{\ast }-{\lambda }_1(D_1{h}_n-D_1{h}^{\ast })-(S{w}_n-x^{\ast })\parallel \\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel \\ =& (\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel +\parallel S{w}_n-S{x}^{\ast }\parallel )\\ \times \parallel x_n-S{w}_n+(x^{\ast }-h^{\ast })-(x_n-h_n)-{\lambda }_1(D_1{h}_n-D_1{h}^{\ast }))\parallel \\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel \\ \le & (\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel +\parallel S{w}_n-S{x}^{\ast }\parallel )\\ \times (\parallel x_n-S{w}_n\parallel +\parallel (x^{\ast }-h^{\ast })-(x_n-h_n)\parallel \\ +{\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel )\\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel \\ =& (\parallel (I-{\lambda }_1{D}_1)h_n-(I-{\lambda }_1{D}_1)h^{\ast }\parallel +\parallel S{w}_n-S{x}^{\ast }\parallel )\\ \times (\parallel x_n-S{w}_n\parallel +(1-a)\parallel x^{\ast }-y^{\ast }-x_n+y_n\parallel \\ +{\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast })\parallel \\ +2({\lambda }_1\parallel D_1{h}_n-D_1{h}^{\ast }\parallel +a\parallel y_n-x_n-y^{\ast }+x^{\ast }\parallel )\\ \times \parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel .\end{array}$$

From (3.6), (3.12) and (3.15), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-w_n+(x^{\ast }-y^{\ast })\parallel =0.$$

(3.16)

Since

$$\parallel x_n-w_n\parallel \le \parallel x_n-y_n-(x^{\ast }-y^{\ast })\parallel +\parallel y_n+(x^{\ast }-y^{\ast })-w_n\parallel$$

and (3.15), (3.16), then we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-w_n\parallel =0.$$

(3.17)

From (3.6) and (3.17), we can conclude that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S{w}_n-w_n\parallel =0.$$

Next we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-x_0,x_n-x_0〉\le 0,$$

(3.18)

where $x_0=P_{\mathcal{F}}u$. To show this inequality, take a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-x_0,x_n-x_0〉=\underset{k\to \mathrm{\infty }}{lim}〈u-x_0,x_{n_k}-x_0〉.$$

Without loss of generality, we may assume that $x_{n_k}\rightharpoonup \omega$ as $k\to \mathrm{\infty }$, where $\omega \in C$. From (3.17), we have $w_{n_k}\rightharpoonup \omega$ as $k\to \mathrm{\infty }$. From Lemma 2.5 and (3.13), we have

$$\omega \in F(S).$$

From Lemma 2.4, we have $F(S)={\bigcap }_{i=1}^{N}F(T_i)$. Then we obtain

$$\omega \in \bigcap _{i=1}^{N}F(T_i).$$

From the nonexpansiveness of the mapping G and the definition of $w_n$, we have

$$\begin{array}{rl}\parallel w_n-G{w}_n\parallel & =\parallel P_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)P_C(I-{\lambda }_2{D}_2)x_n)-G(w_n)\parallel \\ =\parallel G{x}_n-G{w}_n\parallel \\ \le \parallel x_n-w_n\parallel .\end{array}$$

From (3.17), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel w_n-G{w}_n\parallel =0.$$

(3.19)

From $w_{n_k}\rightharpoonup \omega$ as $k\to \mathrm{\infty }$, (3.19) and Lemma 2.5, we have

$$\omega \in F(G).$$

Hence, we can conclude that $\omega \in \mathcal{F}$.

Since $x_{n_k}\rightharpoonup \omega$ as $k\to \mathrm{\infty }$ and $\omega \in \mathcal{F}$, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-x_0,x_n-x_0〉=\underset{k\to \mathrm{\infty }}{lim}〈u-x_0,x_{n_k}-x_0〉=〈u-x_0,\omega -x_0〉\le 0.$$

(3.20)

From the definition of $x_n$ and $x_0=P_{\mathcal{F}}u$, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-x_0\parallel }^2& =& {\parallel {\alpha }_n(u-x_0)+{\beta }_n(x_n-x_0)+{\gamma }_n(S{w}_n-x_0)\parallel }^2\\ \le & {\parallel {\beta }_n(x_n-x_0)+{\gamma }_n(S{w}_n-x_0)\parallel }^2+2{\alpha }_n〈u-x_0,x_{n+1}-x_0〉\\ \le & {\beta }_n{\parallel x_n-x_0\parallel }^2+{\gamma }_n{\parallel G{x}_n-x_0\parallel }^2+2{\alpha }_n〈u-x_0,x_{n+1}-x_0〉\\ \le & {\beta }_n{\parallel x_n-x_0\parallel }^2+{\gamma }_n{\parallel x_n-x_0\parallel }^2+2{\alpha }_n〈u-x_0,x_{n+1}-x_0〉\\ \le & (1-{\alpha }_n){\parallel x_n-x_0\parallel }^2+2{\alpha }_n〈u-x_0,x_{n+1}-x_0〉.\end{array}$$

From condition (ii), (3.18) and Lemma 2.2, we can conclude that the sequence $\{x_n\}$ converges strongly to $x_0=P_{\mathcal{F}}u$. This completes the proof. □

Remark 3.2 (1) If we take $a=0$, then the iterative scheme (3.1) reduces to the following scheme:

$$\{\begin{array}{c}{x}_1,u\in C,\hfill \\ y_n=P_C(I-{\lambda }_2{D}_2)x_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_{n}S{P}_C(I-{\lambda }_1{D}_1)y_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.21)

which is an improvement to (1.4). From Theorem 3.1, we obtain that the sequence $\{x_n\}$ generated by (3.21) converges strongly to $x_0=P_{{\bigcap }_{i=1}^{N}F(T_i)\cap F(G)}u$, where the mapping $G:C\to C$ defined by $Gx=P_C(I-{\lambda }_1{D}_1)P_C(I-{\lambda }_2{D}_2)x$ for all $x\in C$ and $(x_0,y_0)$ is a solution of (1.2) where $y_0=P_C(I-{\lambda }_2{D}_2)x_0$.

(2) If we take $N=1$, ${\alpha }_1^1=1$ and $T_1=T$, then the iterative scheme (3.1) reduces to the following scheme:

$$\{\begin{array}{c}{x}_1,u\in C,\hfill \\ y_n=P_C(I-{\lambda }_2{D}_2)x_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_{n}T{P}_C(I-{\lambda }_1{D}_1)(a{x}_n+(1-a)y_n),\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.22)

From Theorem 3.1, we obtain that the sequence $\{x_n\}$ generated by (3.22) converges strongly to $x_0=P_{F(T)\cap F(G)}u$, where the mapping $G:C\to C$ defined by $G(x)=P_C(I-{\lambda }_1{D}_1)(ax+(1-a)P_C(I-{\lambda }_2{D}_2)x)$ for all $x\in C$ and $(x_0,y_0)$ is a solution of (1.3) where $y_0=P_C(I-{\lambda }_2{D}_2)x_0$.

4 Applications

In this section we prove a strong convergence theorem involving variational inequalities problems by using our main result. We need the following lemmas to prove the desired results.

Lemma 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T,S:C\to C$ be nonexpansive mappings. Define a mapping $B^A:C\to C$ by $B^{A}x=T(\alpha I+(1-\alpha )S)x$ for every $x\in C$ and $\alpha \in (0,1)$. Then $F(B^A)=F(T)\cap F(S)$ and $B^A$ is a nonexpansive mapping.

Proof It is easy to see that $F(T)\cap F(S)\subseteq F(B^A)$. Let $x_0\in F(B^A)$ and $x^{\ast }\in F(T)\cap F(S)$. By the definition of $B^A$, we have

$$\begin{array}{rcl}{\parallel x_0-x^{\ast }\parallel }^2& =& {\parallel B{x}_0-x^{\ast }\parallel }^2={\parallel T(\alpha I+(1-\alpha )S)x_0-x^{\ast }\parallel }^2\\ \le & {\parallel \alpha x_0+(1-\alpha )S{x}_0-x^{\ast }\parallel }^2\\ =& \alpha {\parallel x_0-x^{\ast }\parallel }^2+(1-\alpha ){\parallel S{x}_0-x^{\ast }\parallel }^2-\alpha (1-\alpha ){\parallel x_0-S{x}_0\parallel }^2\\ \le & \alpha {\parallel x_0-x^{\ast }\parallel }^2+(1-\alpha ){\parallel x_0-x^{\ast }\parallel }^2-\alpha (1-\alpha ){\parallel x_0-S{x}_0\parallel }^2\\ =& {\parallel x_0-x^{\ast }\parallel }^2-\alpha (1-\alpha ){\parallel x_0-S{x}_0\parallel }^2.\end{array}$$

(4.1)

From (4.1), it implies that

$$\alpha (1-\alpha ){\parallel x_0-S{x}_0\parallel }^2\le 0.$$

Then we have $x_0=S{x}_0$, that is, $x_0\in F(S)$. By the definition of $B^A$, we have

$$x_0=B^A{x}_0=T(\alpha x_0+(1-\alpha )S{x}_0)=T{x}_0.$$

It follows that $x_0\in F(T)$. Then we have $x_0\in F(T)\cap F(S)$. Hence $F(B^A)\subseteq F(T)\cap F(S)$.

Next, we show that $B^A$ is a nonexpansive mapping. Let $x,y\in C$, since

$$\begin{array}{rcl}{\parallel B^{A}x-B^{A}y\parallel }^2& =& {\parallel T(\alpha I+(1-\alpha )S)x-T(\alpha I+(1-\alpha )S)y\parallel }^2\\ \le & {\parallel (\alpha I+(1-\alpha )S)x-(\alpha I+(1-\alpha )S)y\parallel }^2\\ =& {\parallel \alpha (x-y)+(1-\alpha )(Sx-Sy)\parallel }^2\\ \le & \alpha {\parallel x-y\parallel }^2+(1-\alpha ){\parallel Sx-Sy\parallel }^2\\ \le & {\parallel x-y\parallel }^2.\end{array}$$

(4.2)

Then we have $B^A$ is a nonexpansive mapping. □

Lemma 4.2 (See [15])

Let H be a real Hibert space, let C be a nonempty closed convex subset of H and let A be a mapping of C into H. Let $u\in C$. Then for $\lambda >0$,

$$u=P_C(I-\lambda A)u\iff u\in VI(C,A),$$

where $P_C$ is the metric projection of H onto C.

Lemma 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let $D_1,D_2:C\to H$ be $d_1,d_2$-inverse strongly monotone mappings, respectively, which $VI(C,D_1)\cap VI(C,D_2)\ne \mathrm{\varnothing }$. Define a mapping $G:C\to C$ as in Lemma 2.7 for every ${\lambda }_1\in (0,2d_1)$, ${\lambda }_2\in (0,2d_2)$ and $a\in (0,1)$. Then $F(G)=VI(C,D_1)\cap VI(C,D_2)$.

Proof First, we show that $(I-{\lambda }_1{D}_1)$, $(I-{\lambda }_{2}D2)$ are nonexpansive. Let $x,y\in C$. Since $D_1$ is $d_1$-inverse strongly monotone and ${\lambda }_1<2d_1$, we have

$$\begin{array}{r}{\parallel (I-{\lambda }_1{D}_1)x-(I-{\lambda }_1{D}_1)y\parallel }^2\\ ={\parallel x-y-{\lambda }_1(D_{1}x-D_{1}y)\parallel }^2\\ ={\parallel x-y\parallel }^2-2{\lambda }_1〈x-y,D_{1}x-D_{1}y〉+{\lambda }_1^2{\parallel D_{1}x-D_{1}y\parallel }^2\\ \le {\parallel x-y\parallel }^2-2d_1{\lambda }_1{\parallel D_{1}x-D_{1}y\parallel }^2+{\lambda }_1^2{\parallel D_{1}x-D_{1}y\parallel }^2\\ ={\parallel x-y\parallel }^2+{\lambda }_1({\lambda }_1-2d_1){\parallel D_{1}x-D_{1}y\parallel }^2\\ \le {\parallel x-y\parallel }^2.\end{array}$$

(4.3)

Thus $(I-{\lambda }_1{D}_1)$ is nonexpansive. By using the same method as (4.3), we have $(I-{\lambda }_2{D}_2)$ is a nonexpansive mapping. Hence $P_C(I-{\lambda }_1{D}_1)$, $P_C(I-{\lambda }_2{D}_2)$ are nonexpansive mappings. From

$$G(x)=P_C(I-{\lambda }_1{D}_1)(ax+(1-a)P_C(I-{\lambda }_2{D}_2)x),$$

for every $x\in C$ and Lemma 4.1, we have

$$F(G)=F(P_C(I-{\lambda }_1{D}_1))\cap F(P_C(I-{\lambda }_2{D}_2)).$$

(4.4)

From Lemma 4.2, we have

$$F(G)=VI(C,D_1)\cap VI(C,D_2).$$

 □

Theorem 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let $D_1,D_2:C\to H$ be $d_1,d_2$-inverse strongly monotone mappings, respectively. Define the mapping $G:C\to C$ by $G(x)=P_C(I-{\lambda }_1{D}_1)(ax+(1-a)P_C(I-{\lambda }_2{D}_2)x)$ for all $x\in C$, ${\lambda }_1,{\lambda }_2>0$ and $a\in (0,1)$. Let ${\{T_i\}}_{i=1}^N$ be a finite family of κ-strict pseudo-contractive mappings of C into C with $\mathcal{F}={\bigcap }_{i=1}^{N}F(T_i)\cap VI(C,D_1)\cap VI(C,D_2)\ne \mathrm{\varnothing }$ and $\kappa =max\{{\kappa }_i:i=1,2,\dots ,N\}$ and let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, $j=1,2,3,\dots ,N$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j,{\alpha }_3^j\in (\kappa ,1)$ for all $j=1,2,\dots ,N-1$ and ${\alpha }_1^N\in (\kappa ,1]$, ${\alpha }_3^N\in [\kappa ,1)$, ${\alpha }_2^j\in [\kappa ,1)$ for all $j=1,2,\dots ,N$. Let S be a mapping generated by $T_1,T_2,\dots ,T_N$ and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N$. Suppose that $x_1,u\in C$ and let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{y}_n=P_C(I-{\lambda }_2{D}_2)x_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_{n}S{P}_C(a{x}_n+(1-a)y_n-{\lambda }_1{D}_1(a{x}_n+(1-a)y_n)),\hfill \\ \phantom{x_{n+1}=}\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(4.5)

where ${\lambda }_1\in (0,2d_1)$, ${\lambda }_2\in (0,2d_2)$ and $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$ are sequences in $[0,1]$. Assume that the following conditions hold:

$$\begin{array}{r}\text{(i)}{\alpha }_n+{\beta }_n+{\gamma }_n=1,\\ \text{(ii)}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=0\mathit{\text{and}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },\\ \text{(iii)}0<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\beta }_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_n<1.\end{array}$$

Then $\{x_n\}$ converges strongly to $x_0=P_{\mathcal{F}}u$.

Proof From Lemma 4.3 and Theorem 3.1 we can conclude the desired conclusion. □