Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings

Abstract

In this paper, we investigate the problem of finding some common element in the set of common fixed points of an infinite family of nonexpansive mappings and in the set of solutions of variational inequalities based on an extragradient-like iterative algorithm. Strong convergence of the purposed iterative algorithm is obtained.

MSC:47H05, 47H09, 47J25, 90C33.

1 Introduction

Iterative algorithms have been playing an important role in the approximation solvability, especially of nonlinear variational inequalities as well as of nonlinear equations in several fields such as mechanics, traffic, economics, information, medicine, and many others. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings; see, for example, [1–11]. The Mann iterative algorithm is an efficient method to study the class of nonexpansive mappings. Indeed, Picard cannot converge even that the fixed point set of nonexpansive mappings is nonempty.

It is known that Mann iterative algorithm only has weak convergence for nonexpansive mappings in infinite-dimensional Hilbert spaces; see [12] for more details and the references therein. In many disciplines, including economics [13], image recovery [14], quantum physics [15–20], and control theory [21], problems arise in infinite dimension spaces. To improve the weak convergence of the Mann iterative algorithm, many authors considered using contractions to approximate nonexpansive mappings; for more details, see [22] and [23] and the references therein.

In this paper, we focus on the problem of finding some common element in the set of common fixed points of an infinite family of nonexpansive mappings and in the set of solutions of variational inequalities based on an extragradient-like iterative algorithm. Some deduced sub-results and applications are obtained.

2 Preliminaries

Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$, respectively. Let K be a nonempty, closed, and convex subset of H. Let $P_K$ be the metric projection from H onto K.

Recall that a mapping $B:K\to H$ is said to be inverse-strongly monotone iff there exists a positive real number μ such that

$$〈Bx-By,x-y〉\ge \mu {\parallel Bx-By\parallel }^2,\mathrm{\forall }x,y\in K.$$

For such a case, B is also said to be μ-inverse-strongly monotone.

Recall that a mapping $T:K\to K$ is said to be nonexpansive iff

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in K.$$

In this paper, we use $F(T)$ to denote the fixed point set of the mapping T.

Recall that a mapping $f:K\to K$ is said to be a contraction iff there exists a coefficient $\alpha \in (0,1)$ such that

$$\parallel f(x)-f(y)\parallel \le \alpha \parallel x-y\parallel ,\mathrm{\forall }x,y\in K.$$

For such a case, f is also said to be an α-contraction.

Recall that a linear bounded operator $A:K\to K$ is strongly positive iff there exists a constant $\overline{\gamma }>0$ such that

$$〈Ax,x〉\ge \overline{\gamma }{\parallel x\parallel }^2,\mathrm{\forall }x\in K.$$

Recall that a set-valued mapping $S:H\to 2^H$ is said to be monotone iff $f\in Sx$ and $g\in Sy$ imply

$$〈x-y,f-g〉\ge 0,\mathrm{\forall }x,y\in H.$$

A monotone mapping $S:H\to 2^H$ is maximal iff the graph of $G(S)$ of S is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping S is maximal iff for $(x,f)\in H\times H$, $〈x-y,f-g〉\ge 0$ for every $(y,g)\in G(S)$ implies $f\in Sx$. Let $Q:C\to H$ be a monotone mapping and $N_{K}v$ be the normal cone to K at $v\in K$, i.e., $N_{K}v=\{w\in H:〈v-u,w〉\ge 0,\mathrm{\forall }u\in K\}$, and define

$$Sv=\{\begin{array}{cc}Qv+N_{K}v,\hfill & v\in K,\hfill \\ \mathrm{\varnothing },\hfill & v\notin K.\hfill \end{array}$$

Then S is maximal monotone and $0\in Sv$ iff $v\in VI(K,A)$; see [24] for more details.

Recall that the classical variational inequality is to find a $u\in K$ such that

$$〈Bu,v-u〉\ge 0,\mathrm{\forall }v\in K,$$

(2.1)

where $B:K\to H$ is a monotone mapping. It is known that $u\in K$ is a solution to (2.1) iff u is a fixed point of the mapping $P_K(I-\lambda B)$, where $\lambda >0$ is a constant and I stands for the identity mapping. In this paper, we use $VI(K,B)$ to denote the solution set of the variational inequality (2.1).

Iterative algorithms for nonexpansive mappings have recently been applied to solve convex minimization problems. A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping T on a real Hilbert space H,

$$\underset{x\in F(T)}{min}\frac{1}{2}〈Ax,x〉-〈x,u〉,$$

(2.2)

where A is a linear bounded self-adjoint operator on H and u is a given point in H. In [25], it is proved that the sequence $\{x_n\}$ defined by the iterative algorithm

$$x_0\in H,x_{n+1}=(I-{\alpha }_{n}A)T{x}_n+{\alpha }_{n}u,n\ge 0,$$

strongly converges to the unique solution of the minimization problem (2.2) provided that the sequence $\{{\alpha }_n\}$ satisfies certain restriction.

Recently, Marino and Xu [26] reconsidered the problem by viscosity approximation method. They investigated the following iterative algorithm:

$$x_0\in H,x_{n+1}=(I-{\alpha }_{n}A)T{x}_n+{\alpha }_n\gamma f(x_n),n\ge 0,$$

where A is a linear bounded self-adjoint operator on H, $T:H\to H$ is a nonexpansive mapping, and $f:H\to H$ is a contraction. They proved that the sequence $\{x_n\}$ generated in the above iterative process converges strongly to the unique solution of the following variational inequality:

$$〈(A-\gamma f)x^{\ast },x-x^{\ast }〉\ge 0,x\in K,$$

which is the optimality condition for the minimization problem

$$\underset{x\in F(T)}{min}\frac{1}{2}〈Ax,x〉-h(x),$$

where h is a potential function for γf, that is, $h^{\prime }(x)=\gamma f(x)$ for $x\in H$.

Recently, the problem of finding a common element in the fixed point set of a nonexpansive mapping and in the solution set of a variational inequality has been considered by many authors; see, for example, [27–40] and the references therein. In 2003, Takahashi and Toyoda [35] considered the following iterative algorithm:

$$x_1\in K,x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{P}_K(x_n-{\lambda }_{n}B{x}_n),n\ge 1,$$

(2.3)

where $T:K\to K$ is a nonexpansive mapping, $B:K\to H$ is a μ-inverse-strongly monotone mapping, $\{{\alpha }_n\}$ is a sequence in $(0,1)$, and $\{{\lambda }_n\}$ is a sequence in $(0,2\mu )$. They showed that the sequence $\{x_n\}$ generated in (2.3) weakly converges to some point $z\in F(T)\cap VI(K,B)$.

Iiduka and Takahashi [36] reconsidered the common element problem via the following iterative algorithm:

$$x_1=x\in K,x_{n+1}={\alpha }_{n}x+(1-{\alpha }_n)T{P}_K(x_n-{\lambda }_{n}B{x}_n),n\ge 1,$$

(2.4)

where $T:K\to K$ is a nonexpansive mapping, $B:K\to H$ is a μ-inverse-strongly monotone mapping, $\{{\alpha }_n\}$ is a sequence in $(0,1)$, and $\{{\lambda }_n\}$ is a sequence in $(0,2\mu )$. They proved that the sequence $\{x_n\}$ strongly converges to some point $z\in F(T)\cap VI(K,B)$.

In this paper, we will consider an infinite family of nonexpansive mappings. More precisely, we consider the mapping $W_n$ defined by

(2.5)

where ${\gamma }_1,{\gamma }_2,\dots$ are real numbers such that $0\le {\gamma }_n\le 1$, $T_1,T_2,\dots$ is an infinite family of mappings of K into itself. Nonexpansivity of each $T_i$ ensures the nonexpansivity of $W_n$.

Regarding $W_n$, we have the following lemmas which are important to prove our main results.

Lemma 2.1 [41]

Let K be a nonempty, closed, and convex subset of a strictly convex Banach space E. Let $T_1,T_2,\dots$ be nonexpansive mappings of K into itself such that ${\bigcap }_{n=1}^{\mathrm{\infty }}F(T_n)$ is nonempty, and let ${\gamma }_1,{\gamma }_2,\dots$ be real numbers such that $0<{\gamma }_n\le b<1$ for any $n\ge 1$. Then, for every $x\in K$ and $k\in N$, the limit ${lim}_{n\to \mathrm{\infty }}{U}_{n,k}x$ exists.

Using Lemma 2.1, one can define the mapping W as follows:

$$Wx=\underset{n\to \mathrm{\infty }}{lim}{W}_{n}x=\underset{n\to \mathrm{\infty }}{lim}{U}_{n,1}x,\mathrm{\forall }x\in K.$$

(2.6)

Such a mapping W is called W-mapping generated by $T_1,T_2,\dots$ and ${\gamma }_1,{\gamma }_2,\dots$ .

Throughout this paper, we will assume that $0<{\gamma }_n\le b<1$ for each $n\ge 1$.

Lemma 2.2 [41]

Let K be a nonempty, closed, and convex subset of a strictly convex Banach space E. Let $T_1,T_2,\dots$ be nonexpansive mappings of K into itself such that ${\bigcap }_{n=1}^{\mathrm{\infty }}F(T_n)$ is nonempty, and let ${\gamma }_1,{\gamma }_2,\dots$ be real numbers such that $0<{\gamma }_n\le b<1$ for each $n\ge 1$. Then $F(W)={\bigcap }_{n=1}^{\mathrm{\infty }}F(T_n)$.

In this paper, motivated by the above results, we investigate the problem of approximating a common element in the solution set of variational inequalities and in the common fixed point set of a family of nonexpansive mappings based on an extragradient-like iterative algorithm. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.

In order to prove our main results, we also need the following lemmas.

Lemma 2.3 In a real Hilbert space H, the following inequality holds:

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉,\mathrm{\forall }x,y\in H.$$

Lemma 2.4 [26]

Assume A is a strongly positive linear bounded self-adjoint operator on a Hilbert space H with the coefficient $\overline{\gamma }>0$ and $0<\rho \le {\parallel A\parallel }^{-1}$. Then $\parallel I-\rho A\parallel \le 1-\rho \overline{\gamma }$.

Lemma 2.5 [26]

Let H be a Hilbert space. Let A be a strongly positive linear bounded self-adjoint operator with the coefficient $\overline{\gamma }>0$. Assume that $0<\gamma <\overline{\gamma }/\alpha$. Let T be a nonexpansive mapping with a fixed point $x_t\in H$ of the contraction $x\mapsto t\gamma f(x)+(I-tA)Tx$. Then $\{x_t\}$ converges strongly as $t\to 0$ to a fixed point $\overline{x}$ of T, which solves the variational inequality

$$〈(A-\gamma f)\overline{x},z-\overline{x}〉\le 0,\mathrm{\forall }z\in F(T).$$

Equivalently, we have $P_{F(T)}(I-A+\gamma f)\overline{x}=\overline{x}$.

Lemma 2.6 [42]

Assume that $\{{\alpha }_n\}$ is a sequence of nonnegative real numbers such that

$${\alpha }_{n+1}\le (1-{\gamma }_n){\alpha }_n+{\delta }_n,$$

where $\{{\gamma }_n\}$ is a sequence in $(0,1)$ and $\{{\delta }_n\}$ is a sequence such that

• (a) ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$;

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

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

Lemma 2.7 [39]

Let K be a nonempty closed convex subset of a Hilbert space H, $\{T_i:C\to C\}$ be a family of infinitely nonexpansive mappings with ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\ne \mathrm{\varnothing }$, $\{{\gamma }_n\}$ be a real sequence such that $0<{\gamma }_n\le b<1$ for each $n\ge 1$. If C is any bounded subset of K, then ${lim}_{n\to \mathrm{\infty }}{sup}_{x\in C}\parallel Wx-W_{n}x\parallel =0$.

3 Main results

Theorem 3.1 Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Let $B_i:K\to H$ be ${\mu }_i$-inverse-strongly monotone mappings for each $i=1,2$, and $f:K\to K$ be an α-contraction. Let $A:K\to K$ be a strongly positive linear bounded self-adjoint operator with the coefficient $\overline{\gamma }>0$. Let $\{x_n\}$ be a sequence generated in the following extragradient-like iterative algorithm:

$$\{\begin{array}{c}{x}_1\in K,\hfill \\ y_n=P_K(x_n-{\eta }_n{B}_2{x}_n),\hfill \\ x_{n+1}=P_K({\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)W_n{P}_K(I-{\lambda }_n{B}_1)y_n),n\ge 1,\hfill \end{array}$$

(3.1)

where $P_K$ is the metric projection from H onto K, $W_n$ is a mapping defined by (2.5), $\{{\alpha }_n\}$ is a real number sequence in $(0,1)$, and $\{{\lambda }_n\}$, $\{{\eta }_n\}$ are two positive real number sequences. Assume that $F={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\cap VI(K,B_1)\cap VI(K,B_2)\ne \mathrm{\varnothing }$, $0<\gamma <\overline{\gamma }/\alpha$ and the following restrictions are satisfied:

• (a) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, and ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|\le \mathrm{\infty }$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}|{\eta }_{n+1}-{\eta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n+1}-{\lambda }_n|<\mathrm{\infty }$;

• (c) $\{{\eta }_n\},\{{\lambda }_n\}\subset [u,v]$, where $0<u<v<2min\{{\mu }_1,{\mu }_2\}$.

Then the sequence $\{x_n\}$ strongly converges to $x^{\ast }\in F$, where $x^{\ast }=P_F(\gamma f+(I-A))x^{\ast }$.

Proof First, we show that $I-{\lambda }_n{B}_1$ and $I-{\eta }_n{B}_2$ are nonexpansive. Indeed, we see from the restriction (c) that

$$\begin{array}{rl}{\parallel (I-{\lambda }_n{B}_1)x-(I-{\lambda }_n{B}_1)y\parallel }^2& ={\parallel x-y-{\lambda }_n(B_{1}x-B_{1}y)\parallel }^2\\ ={\parallel x-y\parallel }^2-2{\lambda }_n〈x-y,B_{1}x-B_{1}y〉+{\lambda }_n^2{\parallel B_{1}x-B_{1}y\parallel }^2\\ \le {\parallel x-y\parallel }^2+{\lambda }_n({\lambda }_n-2{\mu }_1){\parallel B_{1}x-B_{1}y\parallel }^2\\ \le {\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in C.\end{array}$$

This shows that $I-{\lambda }_n{B}_1$ is nonexpansive, so is $I-{\eta }_n{B}_2$. Noticing the condition (a), we may assume, with no loss of generality, that ${\alpha }_n\le {\parallel A\parallel }^{-1}$ for each $n\ge 1$. It follows from Lemma 2.4 that $\parallel I-{\alpha }_{n}A\parallel \le 1-{\alpha }_n\overline{\gamma }$.

Next, we show that the sequence $\{x_n\}$ is bounded. Letting $p\in F$, we see that

$$\parallel y_n-p\parallel =\parallel P_K(I-{\eta }_n{B}_2)x_n-P_K(I-{\eta }_n{B}_2)p\parallel \le \parallel x_n-p\parallel .$$

It follows that

$$\begin{array}{rl}\parallel x_{n+1}-p\parallel & \le \parallel {\alpha }_n(\gamma f(x_n)-Ap)+(I-{\alpha }_{n}A)(W_n{P}_C(I-{\lambda }_{n}B)y_n-p)\parallel \\ \le {\alpha }_n\parallel \gamma f(x_n)-Ap\parallel +(1-{\alpha }_n\overline{\gamma })\parallel W_n{P}_C(I-{\lambda }_{n}B)y_n-p\parallel \\ \le {\alpha }_n\gamma \parallel f(x_n)-f(p)\parallel +{\alpha }_n\parallel \gamma f(p)-Ap\parallel +(1-{\alpha }_n\overline{\gamma })\parallel y_n-p\parallel \\ =(1-{\alpha }_n(\overline{\gamma }-\gamma \alpha ))\parallel x_n-p\parallel +{\alpha }_n\parallel \gamma f(p)-Ap\parallel .\end{array}$$

By simple induction, we have

$$\parallel x_n-p\parallel \le max\{\parallel x_1-p\parallel ,\frac{\parallel Ap-\gamma f(p)\parallel }{\overline{\gamma }-\gamma \alpha }\},$$

which yields that the sequence $\{x_n\}$ is bounded, so is $\{y_n\}$. Notice that

(3.2)

Putting ${\rho }_n=P_K(I-{\lambda }_n{B}_1)y_n$, we have

(3.3)

Substituting (3.2) into (3.3), we arrive at

$$\parallel {\rho }_{n+1}-{\rho }_n\parallel \le \parallel x_{n+1}-x_n\parallel +M_1(|{\eta }_{n+1}-{\eta }_n|+|{\lambda }_{n+1}-{\lambda }_n|),$$

(3.4)

where $M_1$ is an appropriate constant such that

$$M_1\ge max\{\underset{n\ge 1}{sup}\parallel B_1{y}_n\parallel ,\underset{n\ge 1}{sup}\parallel B_2{x}_n\parallel \}.$$

Notice that

(3.5)

Since $T_i$ and $U_{n,i}$ are nonexpansive, we have from (2.6) that

$$\begin{array}{rcl}\parallel W_{n+1}{\rho }_n-W_n{\rho }_n\parallel & =& \parallel {\gamma }_1{T}_1{U}_{n+1,2}{\rho }_n-{\gamma }_1{T}_1{U}_{n,2}{\rho }_n\parallel \\ \le & {\gamma }_1\parallel U_{n+1,2}{\rho }_n-U_{n,2}{\rho }_n\parallel \\ =& {\gamma }_1\parallel {\gamma }_2{T}_2{U}_{u+1,3}{\rho }_n-{\gamma }_2{T}_2{U}_{n,3}{\rho }_n\parallel \\ \le & {\gamma }_1{\gamma }_2\parallel U_{u+1,3}{\rho }_n-U_{n,3}{\rho }_n\parallel \\ \le & \cdots \\ \le & {\gamma }_1{\gamma }_2\cdots {\gamma }_n\parallel U_{n+1,n+1}{\rho }_n-U_{n,n+1}{\rho }_n\parallel \\ \le & M_2\prod _{i=1}^n{\gamma }_i,\end{array}$$

(3.6)

where $M_2\ge 0$ is an appropriate constant such that $\parallel U_{n+1,n+1}{\rho }_n-U_{n,n+1}{\rho }_n\parallel \le M_2$ for each $n\ge 1$. Substituting (3.4) and (3.6) into (3.5), we arrive at

$$\begin{array}{rcl}\parallel x_{n+2}-x_{n+1}\parallel & \le & (1-{\alpha }_{n+1}(\overline{\gamma }-\alpha \gamma ))\parallel x_{n+1}-x_n\parallel \\ +M_3(\prod _{i=1}^n{\gamma }_i+|{\alpha }_{n+1}-{\alpha }_n|+|{\lambda }_{n+1}-{\lambda }_n|+|{\eta }_{n+1}-{\eta }_n|),\end{array}$$

(3.7)

where $M_3$ is an appropriate constant such that

$$M_3=max\{M_1,M_2,\gamma \underset{n\ge 1}{sup}\{\parallel f(x_n)\parallel +\parallel A{W}_n{\rho }_n\parallel \}\}.$$

From the restrictions (a) and (b), we obtain from Lemma 2.6 that

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

(3.8)

Notice that

$$\begin{array}{rcl}\parallel x_{n+1}-W_n{\rho }_n\parallel & =& \parallel P_K({\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)W_n{P}_K(I-{\lambda }_n{B}_1)y_n)-P_K(W_n{\rho }_n)\parallel \\ \le & {\alpha }_n\parallel \gamma f(x_n)-A{W}_n{\rho }_n\parallel .\end{array}$$

It follows from the restriction (a) that

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

(3.9)

Notice that

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\parallel (x_n-p)-{\eta }_n(B_2{x}_n-B_{2}p)\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-2{\eta }_n{\mu }_2{\parallel B_2{x}_n-B_{2}p\parallel }^2+{\eta }_n^2{\parallel B_2{x}_n-B_{2}p\parallel }^2\\ =& {\parallel x_n-p\parallel }^2+({\eta }_n^2-2{\eta }_n{\mu }_2){\parallel B_2{x}_n-B_{2}p\parallel }^2.\end{array}$$

(3.10)

In a similar way, we find that

$${\parallel {\rho }_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2+({\lambda }_n^2-2\lambda {\mu }_1){\parallel B_1{y}_n-B_{1}p\parallel }^2.$$

(3.11)

On the other hand, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & {\parallel {\alpha }_n(\gamma f(x_n)-Ap)+(I-{\alpha }_{n}A)(W_n{\rho }_n-p)\parallel }^2\\ \le & {({\alpha }_n\parallel \gamma f(x_n)-Ap\parallel +(1-{\alpha }_n\overline{\gamma })\parallel {\rho }_n-p\parallel )}^2\\ \le & {\alpha }_n{\parallel \gamma f(x_n)-Ap\parallel }^2+{\parallel {\rho }_n-p\parallel }^2+2{\alpha }_n\parallel \gamma f(x_n)-Ap\parallel \parallel {\rho }_n-p\parallel .\end{array}$$

(3.12)

Substituting (3.11) into (3.12) gives

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & {\alpha }_n{\parallel \gamma f(x_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2+({\lambda }_n^2-2{\lambda }_n{\mu }_1){\parallel B_1{y}_n-B_{1}p\parallel }^2\\ +2{\alpha }_n\parallel \gamma f(x_n)-Ap\parallel \parallel {\rho }_n-p\parallel .\end{array}$$

It follows from the restriction (c) that

In view of the restriction (a), we obtain from (3.8) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel B_1{y}_n-B_{1}p\parallel =0.$$

(3.13)

From (3.12), we also have

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & {\alpha }_n{\parallel \gamma f(x_n)-Ap\parallel }^2+{\parallel y_n-p\parallel }^2\\ +2{\alpha }_n\parallel \gamma f(x_n)-Ap\parallel \parallel {\rho }_n-p\parallel .\end{array}$$

(3.14)

Combining (3.10) with (3.14), we arrive at

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & {\alpha }_n{\parallel \gamma f(x_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2+({\eta }_n^2-2{\eta }_n{\mu }_2){\parallel B_2{x}_n-B_{2}p\parallel }^2\\ +2{\alpha }_n\parallel \gamma f(x_n)-Ap\parallel \parallel {\rho }_n-p\parallel ,\end{array}$$

which implies from the restriction (c) that

In view of the restriction (a), we obtain from (3.8) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel B_2{x}_n-B_{2}p\parallel =0.$$

(3.15)

On the other hand, we see from the firm expansivity of $P_K$ that

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& =& {\parallel P_K(I-{\eta }_n{B}_2)x_n-P_K(I-{\eta }_n{B}_2)p\parallel }^2\\ \le & 〈(I-{\eta }_n{B}_2)x_n-(I-{\eta }_n{B}_2)p,y_n-p〉\\ =& \frac{1}{2}({\parallel (I-{\eta }_n{B}_2)x_n-(I-{\eta }_n{B}_2)p\parallel }^2+{\parallel y_n-p\parallel }^2\\ -{\parallel (I-{\eta }_n{B}_2)x_n-(I-{\eta }_n{B}_2)p-(y_n-p)\parallel }^2)\\ \le & \frac{1}{2}({\parallel x_n-p\parallel }^2+{\parallel y_n-p\parallel }^2-{\parallel (x_n-y_n)-{\eta }_n(B_2{x}_n-B_{2}p)\parallel }^2)\\ =& \frac{1}{2}({\parallel x_n-p\parallel }^2+{\parallel y_n-p\parallel }^2-{\parallel x_n-y_n\parallel }^2-{\eta }_n^2{\parallel B_2{x}_n-B_{2}p\parallel }^2\\ +2{\eta }_n〈x_n-y_n,B_2{x}_n-B_{2}p〉),\end{array}$$

which yields

$${\parallel y_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_n-y_n\parallel }^2+2{\eta }_n\parallel x_n-y_n\parallel \parallel B_2{x}_n-B_{2}p\parallel .$$

(3.16)

In the same way, we can obtain that

$${\parallel {\rho }_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel {\rho }_n-y_n\parallel }^2+2{\lambda }_n\parallel {\rho }_n-y_n\parallel \parallel B_1{y}_n-B_{1}p\parallel .$$

(3.17)

Substituting (3.16) into (3.14) yields

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & {\alpha }_n{\parallel \gamma f(x_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2-{\parallel x_n-y_n\parallel }^2\\ +2{\eta }_n\parallel x_n-y_n\parallel \parallel B_2{x}_n-B_{2}p\parallel +2{\alpha }_n\parallel \gamma f(x_n)-Ap\parallel \parallel {\rho }_n-p\parallel .\end{array}$$

It follows that

$$\begin{array}{rcl}{\parallel x_n-y_n\parallel }^2& \le & {\alpha }_n{\parallel \gamma f(x_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\\ +2{\eta }_n\parallel x_n-y_n\parallel \parallel B_2{x}_n-B_{2}p\parallel +2{\alpha }_n\parallel \gamma f(x_n)-Ap\parallel \parallel {\rho }_n-p\parallel \\ \le & {\alpha }_n{\parallel \gamma f(x_n)-Ap\parallel }^2+(\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_{n+1}-x_n\parallel \\ +2{\eta }_n\parallel x_n-y_n\parallel \parallel B_2{x}_n-B_{2}p\parallel +2{\alpha }_n\parallel \gamma f(x_n)-Ap\parallel \parallel {\rho }_n-p\parallel .\end{array}$$

In view of (3.7) and (3.15), we see from the restriction (a) that

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

(3.18)

Similarly, we can obtain that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel {\rho }_n-y_n\parallel =0.$$

(3.19)

Observe that

$$\parallel {\rho }_n-W_n{\rho }_n\parallel \le \parallel y_n-{\rho }_n\parallel +\parallel x_n-y_n\parallel +\parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-W_n{\rho }_n\parallel .$$

It follows from (3.7), (3.9), (3.18) and (3.19) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel W_n{\rho }_n-{\rho }_n\parallel =0.$$

(3.20)

From Lemma 2.7, we have $\parallel W{\rho }_n-W_n{\rho }_n\parallel \to 0$ as $n\to \mathrm{\infty }$. This in turn implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel W{\rho }_n-{\rho }_n\parallel =0.$$

(3.21)

Next, we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_n-x^{\ast }〉\le 0.$$

(3.22)

To show it, we choose a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_n-x^{\ast }〉=\underset{i\to \mathrm{\infty }}{lim}〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n_i}-x^{\ast }〉.$$

As $\{x_{n_i}\}$ is bounded, we have that a subsequence $\{x_{n_{i_j}}\}$ of $\{x_{n_i}\}$ converges weakly to p. We may assume, without loss of generality, that $x_{n_i}\rightharpoonup p$. From (3.18) and (3.19), we also have $y_{n_i}\rightharpoonup p$ and $z_{n_i}\rightharpoonup p$, respectively. Notice that $p\in F$. Indeed, let us first show that $p\in VI(K,B_1)$. Put

$$Sw=\{\begin{array}{cc}{B}_{1}v+N_{K}v,\hfill & v\in K,\hfill \\ \mathrm{\varnothing },\hfill & v\notin K.\hfill \end{array}$$

Then S is maximal monotone. Let $(v,w)\in G(S)$. Since $w-B_{1}v\in N_{K}v$ and ${\rho }_n\in K$, we have

$$〈v-{\rho }_n,w-B_{1}v〉\ge 0.$$

On the other hand, we have from ${\rho }_n=P_K(I-{\lambda }_n{B}_1)y_n$ that

$$〈v-{\rho }_n,{\rho }_n-(I-{\lambda }_n{B}_1)y_n〉\ge 0$$

and hence

$$〈v-{\rho }_n,\frac{{\rho }_n-y_n}{{\lambda }_n}+B_1{y}_n〉\ge 0.$$

It follows that

$$\begin{array}{rcl}〈v-{\rho }_{n_i},w〉& \ge & 〈v-{\rho }_{n_i},B_{1}v〉\\ \ge & 〈v-{\rho }_{n_i},B_{1}v〉-〈v-{\rho }_{n_i},\frac{{\rho }_{n_i}-y_{n_i}}{{\lambda }_{n_i}}+B_1{y}_{n_i}〉\\ \ge & 〈v-{\rho }_{n_i},B_{1}v-\frac{{\rho }_{n_i}-y_{n_i}}{{\lambda }_{n_i}}-B_1{y}_{n_i}〉\\ =& 〈v-{\rho }_{n_i},B_{1}v-B_1{\rho }_{n_i}〉+〈v-{\rho }_{n_i},B_1{\rho }_{n_i}-B_1{y}_{n_i}〉-〈v-{\rho }_{n_i},\frac{{\rho }_{n_i}-y_{n_i}}{{\lambda }_{n_i}}〉\\ \ge & 〈v-{\rho }_{n_i},B_1{\rho }_{n_i}-B_1{y}_{n_i}〉-〈v-{\rho }_{n_i},\frac{{\rho }_{n_i}-y_{n_i}}{{\lambda }_{n_i}}〉,\end{array}$$

which implies that $〈v-p,w〉\ge 0$. We have $p\in B_1^{-1}0$ and hence $p\in VI(K,B_1)$. In a similar way, we can show $p\in VI(K,B_2)$. Next, let us show $p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$. Since Hilbert spaces satisfy Opial’s condition, we see from (3.21) that

$$\begin{array}{rcl}\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel {\rho }_{n_i}-p\parallel & <& \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel {\rho }_{n_i}-Wp\parallel \\ =& \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel {\rho }_{n_i}-W{\rho }_{n_i}+W{\rho }_{n_i}-Wp\parallel \\ \le & \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel W{\rho }_{n_i}-Wp\parallel \\ \le & \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel {\rho }_{n_i}-p\parallel ,\end{array}$$

which derives a contradiction. Thus, we have $p\in F(W)={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$. From Lemma 2.5, we see that there exists a unique $x^{\ast }$ such that $x^{\ast }=P_F(\gamma f+(I-A))x^{\ast }$. It follows that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_n-x^{\ast }〉=〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },p-x^{\ast }〉\le 0.$$

That is, (3.22) holds. It follows from Lemma 2.3 that

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& \le & {\parallel {\alpha }_n(\gamma f(x_n)-A{x}^{\ast })+(I-{\alpha }_{n}A)(W_n{\rho }_n-x^{\ast })\parallel }^2\\ \le & {(1-{\alpha }_n\overline{\gamma })}^2{\parallel W_n{\rho }_n-x^{\ast }\parallel }^2+2{\alpha }_n〈\gamma f(x_n)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\\ \le & {(1-{\alpha }_n\overline{\gamma })}^2{\parallel x_n-x^{\ast }\parallel }^2+\alpha \gamma {\alpha }_n({\parallel x_n-x^{\ast }\parallel }^2+{\parallel x_{n+1}-x^{\ast }\parallel }^2)\\ +2{\alpha }_n〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉.\end{array}$$

It follows that

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& \le & \frac{{(1-{\alpha }_n\overline{\gamma })}^2+{\alpha }_n\gamma \alpha }{1-{\alpha }_n\gamma \alpha }{\parallel x_n-x^{\ast }\parallel }^2+\frac{2{\alpha }_n}{1-{\alpha }_n\gamma \alpha }〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\\ =& \frac{(1-2{\alpha }_n\overline{\gamma }+{\alpha }_n\alpha \gamma )}{1-{\alpha }_n\gamma \alpha }{\parallel x_n-x^{\ast }\parallel }^2+\frac{{\alpha }_n^2{\overline{\gamma }}^2}{1-{\alpha }_n\gamma \alpha }{\parallel x_n-x^{\ast }\parallel }^2\\ +\frac{2{\alpha }_n}{1-{\alpha }_n\gamma \alpha }〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\\ \le & [1-\frac{2{\alpha }_n(\overline{\gamma }-\alpha \gamma )}{1-{\alpha }_n\gamma \alpha }]{\parallel x_n-x^{\ast }\parallel }^2\\ +\frac{2{\alpha }_n(\overline{\gamma }-\alpha \gamma )}{1-{\alpha }_n\gamma \alpha }(\frac{1}{\overline{\gamma }-\alpha \gamma }〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉+\frac{{\alpha }_n{\overline{\gamma }}^2}{2(\overline{\gamma }-\alpha \gamma )}{M}_4),\end{array}$$

where $M_4$ is an appropriate constant such that $M_4\ge {sup}_{n\ge 1}{\parallel x_n-x^{\ast }\parallel }^2$. Put $b_n=\frac{2{\alpha }_n(\overline{\gamma }-\alpha \gamma )}{1-{\alpha }_n\alpha \gamma }$ and $c_n=\frac{1}{\overline{\gamma }-\alpha \gamma }〈\gamma f(x^{\ast })-A{x}^{\ast },x_{n+1}-x^{\ast }〉+\frac{{\alpha }_n{\overline{\gamma }}^2}{2(\overline{\gamma }-\alpha \gamma )}{M}_4$. That is,

$${\parallel x_{n+1}-x^{\ast }\parallel }^2\le (1-b_n)\parallel x_n-x^{\ast }\parallel +b_n{c}_n.$$

(3.23)

In view of the restrictions (a) and (b), we see from (3.22) that

$$\underset{n\to \mathrm{\infty }}{lim}{b}_n=0,\sum _{n=1}^{\mathrm{\infty }}{b}_n=\mathrm{\infty }\text{and}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{c}_n\le 0.$$

Apply Lemma 2.6 to (3.23) to conclude that $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$. This completes the proof. □

If $B_2=0$, the zero mapping, then Theorem 3.1 is reduced to the following.

Corollary 3.2 Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Let $B_1:K\to H$ be ${\mu }_1$-inverse-strongly monotone mappings and $f:K\to K$ be an α-contraction. Let $A:K\to K$ be a strongly positive linear bounded self-adjoint operator with the coefficient $\overline{\gamma }>0$. Assume that $0<\gamma <\overline{\gamma }/\alpha$. Let $\{x_n\}$ be a sequence generated in the following iterative algorithm:

$$\{\begin{array}{c}{x}_1\in K,\hfill \\ x_{n+1}=P_K({\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)W_n{P}_K(I-{\lambda }_n{B}_1)x_n),n\ge 1,\hfill \end{array}$$

where $P_K$ is the metric projection from H onto K, $W_n$ is a mapping defined by (2.5), $\{{\alpha }_n\}$ is a real number sequence in $(0,1)$, and $\{{\lambda }_n\}$ is a positive real number sequence. Assume that $F={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\cap VI(K,B_1)\ne \mathrm{\varnothing }$ and the following restrictions are satisfied:

• (a) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, and ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|\le \mathrm{\infty }$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n+1}-{\lambda }_n|<\mathrm{\infty }$;

• (c) $\{{\lambda }_n\}\subset [u,v]$, where $0<u<v<2{\mu }_1$.

Then the sequence $\{x_n\}$ strongly converges to $x^{\ast }\in F$, where $x^{\ast }=P_F(\gamma f+(I-A))x^{\ast }$.

Remark 3.3 Corollary 3.2 includes the corresponding results in Iiduka and Takahashi [36] as a special case.

As an application of our main results, we consider another class of important nonlinear operators: strict pseudocontractions.

Recall that a mapping $S:K\to K$ is said to be a κ-strict pseudocontraction if there exists a constant $\kappa \in [0,1)$ such that

$${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\kappa {\parallel (I-S)x-(I-S)y\parallel }^2,\mathrm{\forall }x,y\in K.$$

It is easy to see that the class of κ-strict pseudocontractions strictly includes the class of nonexpansive mappings as a special case.

Putting $B=I-S$, where $S:K\to K$ is a κ-strict pseudocontraction, we know that B is $\frac{1-\kappa }{2}$-inverse-strongly monotone; see [43] and the references therein.

Corollary 3.4 Let H be a real Hilbert space and K be a nonempty closed convex subset of H. Let $S_i:K\to K$ be ${\kappa }_i$-inverse-strongly monotone mappings for each $i=1,2$ and $f:K\to K$ be an α-contraction. Let $A:K\to K$ be a strongly positive linear bounded self-adjoint operator with the coefficient $\overline{\gamma }>0$. Assume that $0<\gamma <\overline{\gamma }/\alpha$. Let $\{x_n\}$ be a sequence generated in the following iterative process:

$$\{\begin{array}{c}{x}_1\in K,\hfill \\ y_n=(1-{\eta }_n)x_n+{\eta }_n{S}_2{x}_n,\hfill \\ x_{n+1}=P_K({\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)W_n((1-{\lambda }_n)y_n+{\lambda }_n{S}_1{y}_n)),n\ge 1,\hfill \end{array}$$

where $P_K$ is the metric projection from H onto K, $W_n$ is a mapping defined by (2.5), $\{{\alpha }_n\}$ is a real number sequence in $(0,1)$, and $\{{\lambda }_n\}$, $\{{\eta }_n\}$ are two positive real number sequences. Assume that $F={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\cap F(S_1)\cap F(S_2)\ne \mathrm{\varnothing }$ and the following restrictions are satisfied:

• (a) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, and ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|\le \mathrm{\infty }$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}|{\eta }_{n+1}-{\eta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n+1}-{\lambda }_n|<\mathrm{\infty }$;

• (c) $\{{\eta }_n\},\{{\lambda }_n\}\subset [u,v]$, where $0<u<v<2min\{{\mu }_1,{\mu }_2\}$.

Then the sequence $\{x_n\}$ strongly converges to $x^{\ast }\in F$, where $x^{\ast }=P_F(\gamma f+(I-A))x^{\ast }$.

Proof Put $B_1=I-S_1$ and $B_2=I-S_2$. Then $B_1$ is $(1-{\kappa }_1)/2$-inverse-strongly monotone and $B_2$ is $(1-{\kappa }_2)/2$-inverse-strongly monotone, respectively. We have $F(S_1)=VI(K,B_1)$, $F(S_2)=VI(K,B_2)$, $P_K(I-{\lambda }_n{B}_1)y_n=(1-{\lambda }_n)y_n+{\lambda }_n{T}_1{y}_n$ and $P_K(I-{\eta }_n{B}_2)x_n=(1-{\eta }_n)x_n+{\eta }_n{T}_2{x}_n$. The desired conclusion can be immediately obtained from Theorem 3.1. □