On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces

Abstract

In this paper, a monotone inclusion problem and a fixed point problem of nonexpansive mappings are investigated based on a Mann-type iterative algorithm with mixed errors. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.

MSC:47H05, 47H09, 47J25.

1 Introduction

Variational inclusion has become rich of inspiration in pure and applied mathematics. In recent years, classical variational inclusion problems have been extended and generalized to study a large variety of problems arising in image recovery, economics, and signal processing; for more details, see [1–14]. Based on the projection technique, it has been shown that the variational inclusion problems are equivalent to the fixed point problems. This alternative formulation has played a fundamental and significant part in developing several numerical methods for solving variational inclusion problems and related optimization problems.

The purposes of this paper is to study the zero point problem of the sum of a maximal monotone mapping and an inverse-strongly monotone mapping, and the fixed point problem of a nonexpansive mapping. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a Mann-type iterative algorithm with mixed errors is investigated. A weak convergence theorem is established. Applications of the main results are also discussed in this section.

2 Preliminaries

Throughout this paper, we always assume that H is a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$, respectively. Let C be a nonempty closed convex subset of H and let $P_C$ be the metric projection from H onto C.

Let $S:C\to C$ be a mapping. $F(S)$ stands for the fixed point set of S; that is, $F(S):=\{x\in C:x=Sx\}$.

Recall that S is said to be nonexpansive iff

$$\parallel Sx-Sy\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

If C is a bounded, closed, and convex subset of H, then $F(S)$ is not empty, closed, and convex; see [15].

Let $A:C\to H$ be a mapping. Recall that A is said to be monotone iff

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

A is said to be strongly monotone iff there exists a constant $\alpha >0$ such that

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

For such a case, A is also said to be α-strongly monotone. A is said to be inverse-strongly monotone iff there exists a constant $\alpha >0$ such that

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

For such a case, A is also said to be α-inverse-strongly monotone. It is not hard to see that inverse-strongly monotone mappings are monotone and Lipschitz continuous.

Recall that the classical variational inequality is to find an $x\in C$ such that

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

(2.1)

In this paper, we use $VI(C,A)$ to denote the solution set of (2.1). It is known that $\omega \in C$ is a solution to (2.1) iff ω is a fixed point of the mapping $P_C(I-\lambda A)$, where $\lambda >0$ is a constant, and I stands for the identity mapping. If A is α-inverse-strongly monotone and $\lambda \in (0,2\alpha ]$, then the mapping $P_C(I-\lambda A)$ is nonexpansive. Indeed, we have

$$\begin{array}{r}{\parallel (I-\lambda A)x-(I-\lambda A)y\parallel }^2\\ ={\parallel (x-y)-\lambda (Ax-Ay)\parallel }^2\\ ={\parallel x-y\parallel }^2-2\lambda 〈x-y,Ax-Ay〉+{\lambda }^2{\parallel Ax-Ay\parallel }^2\\ \le {\parallel x-y\parallel }^2-\lambda (2\alpha -\lambda ){\parallel Ax-Ay\parallel }^2.\end{array}$$

This shows that $P_C(I-\lambda A)$ is nonexpansive. It follows that $VI(C,A)$ is closed and convex.

A multivalued operator $T:H\to 2^H$ with the domain $D(T)=\{x\in H:Tx\ne \mathrm{\varnothing }\}$ and the range $R(T)=\{Tx:x\in D(T)\}$ is said to be monotone if for $x_1\in D(T)$, $x_2\in D(T)$, $y_1\in T{x}_1$, and $y_2\in T{x}_2$, we have $〈x_1-x_2,y_1-y_2〉\ge 0$. A monotone operator T is said to be maximal if its graph $G(T)=\{(x,y):y\in Tx\}$ is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and let $T:H\to 2^H$ be a maximal monotone operator. Then we can define, for each $\lambda >0$, a nonexpansive single-valued mapping $J_{\lambda }:H\to H$ by $J_{\lambda }={(I+\lambda T)}^{-1}$. It is called the resolvent of T. We know that $T^{-1}0=F(J_{\lambda })$ for all $\lambda >0$ and $J_{\lambda }$ is firmly nonexpansive, that is,

$${\parallel J_{\lambda }x-J_{\lambda }y\parallel }^2\le 〈J_{\lambda }x-J_{\lambda }y,x-y〉,\mathrm{\forall }x,y\in H;$$

for more details, see [16–22] and the references therein.

In [19], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator based on the following algorithm:

$$x_0\in H,x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)J_{{\lambda }_n}{x}_n,n=0,1,2,\dots ,$$

where $\{{\alpha }_n\}$ is a sequence in $(0,1)$, $\{{\lambda }_n\}$ is a positive sequence, $T:H\to 2^H$ is maximal monotone and $J_{{\lambda }_n}={(I+{\lambda }_{n}T)}^{-1}$. They showed that the sequence $\{x_n\}$ converges weakly to some $z\in T^{-1}(0)$ provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case that $T=\partial f$, where $f:H\to (-\mathrm{\infty },\mathrm{\infty }]$ is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces.

In [16], Takahashi an Toyoda investigated the problem of finding a common solutions of the variational inequality problem (2.1) and a fixed point problem of nonexpansive mappings based on the following algorithm:

$$x_0\in C,x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S{P}_C(x_n-{\lambda }_{n}A{x}_n),\mathrm{\forall }n\ge 0,$$

where $\{{\alpha }_n\}$ is a sequence in $(0,1)$, $\{{\lambda }_n\}$ is a positive sequence, $S:C\to C$ is a nonexpansive mapping and $A:C\to H$ is an inverse-strongly monotone mapping. They showed that the sequence $\{x_n\}$ converges weakly to some $z\in VI(C,A)\cap F(S)$ provided that the control sequence satisfies some restrictions.

In [23], Tada and Takahashi investigated the problem of finding common solutions of an equilibrium problem and a fixed point problem of nonexpansive mappings based on the following algorithm: $x_0\in H$ and

$$\{\begin{array}{c}{u}_n\in C\text{ such that }F(u_n,u)+\frac{1}{r_n}〈u-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }u\in C,\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S{u}_n,\mathrm{\forall }n\ge 0,\hfill \end{array}$$

where $\{{\alpha }_n\}$ is a sequence in $(0,1)$, $\{r_n\}$ is a positive sequence, $S:C\to C$ is a nonexpansive mapping and $F:C\times C\to R$ is a bifunction. They showed that the sequence $\{x_n\}$ converges weakly to some $z\in VI(C,A)\cap F(S)$ provided that the control sequence satisfies some restrictions.

Recently, fixed point and zero point problems have been studied by many authors based on iterative methods; see, for example, [23–34] and the references therein. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problem and the fixed point problem based on Mann-type iterative methods with errors. Weak convergence theorems are established in the framework of Hilbert spaces.

To obtain our main results in this paper, we need the following lemmas.

Recall that a space is said to satisfy Opial’s condition [35] if, for any sequence $\{x_n\}\subset H$ with $x_n\rightharpoonup x$, where ⇀ denotes the weak convergence, the inequality

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-x\parallel <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-y\parallel$$

holds for every $y\in H$ with $y\ne x$. Indeed, the above inequality is equivalent to the following:

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_n-x\parallel <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_n-y\parallel .$$

Lemma 2.1 [34]

Let C be a nonempty, closed, and convex subset of H, let $A:C\to H$ be a mapping, and let $B:H\rightrightarrows H$ be a maximal monotone operator. Then $F(J_{\lambda }(I-\lambda A))={(A+B)}^{-1}(0)$, where $J_{\lambda }(I-\lambda A)$ is the resolvent of B for $\lambda >0$.

Lemma 2.2 [36]

Let $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be three nonnegative sequences satisfying the following condition:

$$a_{n+1}\le (1+b_n)a_n+c_n,\mathrm{\forall }n\ge n_0,$$

where $n_0$ is some nonnegative integer, ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$. Then the limit ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists.

Lemma 2.3 [37]

Suppose that H is a real Hilbert space and $0<p\le t_n\le q<1$ for all $n\ge 1$. Suppose further that $\{x_n\}$ and $\{y_n\}$ are sequences of H such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_n\parallel \le r,\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel y_n\parallel \le r$$

and

$$\underset{n\to \mathrm{\infty }}{lim}\parallel t_n{x}_n+(1-t_n)y_n\parallel =r$$

hold for some $r\ge 0$. Then ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$.

Lemma 2.4 [15]

Let C be a nonempty, closed, and convex subset of H. Let $S:C\to C$ be a nonexpansive mapping. Then the mapping $I-S$ is demiclosed at zero, that is, if $\{x_n\}$ is a sequence in C such that $x_n\rightharpoonup \overline{x}$ and $x_n-S{x}_n\to 0$, then $\overline{x}\in F(S)$.

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, let $A:C\to H$ be an α-inverse-strongly monotone mapping, let $S:C\to C$ be a nonexpansive mapping and let B be a maximal monotone operator on H such that the domain of B is included in C. Assume that $\mathcal{F}=F(S)\cap {(A+B)}^{-1}(0)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be real number sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$. Let $\{{\lambda }_n\}$ be a positive real number sequence and let $\{e_n\}$ be a bounded error sequence in C. Let $\{x_n\}$ be a sequence in C generated in the following iterative process:

$$x_1\in C,x_{n+1}={\alpha }_{n}S{x}_n+{\beta }_n{J}_{{\lambda }_n}(x_n-{\lambda }_{n}A{x}_n)+{\gamma }_n{e}_n$$

(3.1)

for all $n\in \mathbb{N}$, where $J_{{\lambda }_n}={(I+{\lambda }_{n}B)}^{-1}$. Assume that the sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\lambda }_n\}$ satisfy the following restrictions:

• (a) $0<a\le {\beta }_n\le b<1$,

• (b) $0<c\le {\lambda }_n\le d<2\alpha$,

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

where a, b, c, and d are some real numbers. Then the sequence $\{x_n\}$ generated in (3.1) converges weakly to some point in ℱ.

Proof Notice that $I-{\lambda }_{n}A$ is nonexpansive. Indeed, we have

$$\begin{array}{r}{\parallel (I-{\lambda }_{n}A)x-(I-{\lambda }_{n}A)y\parallel }^2\\ ={\parallel (x-y)-{\lambda }_n(Ax-Ay)\parallel }^2\\ ={\parallel x-y\parallel }^2-2{\lambda }_n〈x-y,Ax-Ay〉+{{\lambda }_n}^2{\parallel Ax-Ay\parallel }^2\\ \le {\parallel x-y\parallel }^2-{\lambda }_n(2\alpha -{\lambda }_n){\parallel Ax-Ay\parallel }^2.\end{array}$$

In view of the restriction (b), we find that $I-{\lambda }_{n}A$ is nonexpansive. Fixing $p\in \mathcal{F}$, we find from Lemma 2.1 that

$$p=Sp=J_{{\lambda }_n}(p-{\lambda }_{n}Ap).$$

Put $y_n=J_{{\lambda }_n}(x_n-{\lambda }_{n}A{x}_n)$. Since $J_{{\lambda }_n}$ and $I-{\lambda }_{n}A$ are nonexpansive, we have

$$\begin{array}{rl}\parallel y_n-p\parallel & \le \parallel (x_n-{\lambda }_{n}A{x}_n)-(p-{\lambda }_{n}Ap)\parallel \\ \le \parallel x_n-p\parallel .\end{array}$$

(3.2)

On the other hand, we have

$$\begin{array}{rl}\parallel x_{n+1}-p\parallel & \le {\alpha }_n\parallel x_n-p\parallel +{\beta }_n\parallel y_n-p\parallel +{\gamma }_n\parallel e_n-p\parallel \\ \le \parallel x_n-p\parallel +{\gamma }_n\parallel e_n-p\parallel .\end{array}$$

(3.3)

We find that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists with the aid of Lemma 2.2. This in turn implies that $\{x_n\}$ and $\{y_n\}$ are bounded. Put ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel =L>0$. Notice that

$$\begin{array}{rl}\parallel S{x}_n-p+{\gamma }_n(e_n-S{x}_n)\parallel & \le \parallel S{x}_n-p\parallel +{\gamma }_n\parallel e_n-S{x}_n\parallel \\ \le \parallel x_n-p\parallel +{\gamma }_n\parallel e_n-S{x}_n\parallel .\end{array}$$

This implies from the restriction (c) that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel S{x}_n-p+{\gamma }_n(e_n-S{x}_n)\parallel \le L.$$

We also have

$$\begin{array}{rl}\parallel y_n-p+{\gamma }_n(e_n-S{x}_n)\parallel & \le \parallel y_n-p\parallel +{\gamma }_n\parallel e_n-S{x}_n\parallel \\ \le \parallel x_n-p\parallel +{\gamma }_n\parallel e_n-S{x}_n\parallel .\end{array}$$

This implies from the restriction (c) that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel y_n-p+{\gamma }_n(e_n-S{x}_n)\parallel \le L.$$

On the other hand, we have

$$\parallel x_{n+1}-p\parallel =\parallel (1-{\beta }_n)(S{x}_n-p+{\gamma }_n(e_n-S{x}_n))+{\beta }_n(y_n-p+{\gamma }_n(e_n-S{x}_n))\parallel .$$

It follows from Lemma 2.3 that

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

(3.4)

Notice that

$$\begin{array}{rl}{\parallel y_n-p\parallel }^2& \le {\parallel (x_n-{\lambda }_{n}A{x}_n)-(p-{\lambda }_{n}Ap)\parallel }^2\\ \le {\parallel x-p\parallel }^2-2\alpha {\lambda }_n{\parallel Ax-Ap\parallel }^2+{{\lambda }_n}^2{\parallel Ax-Ap\parallel }^2\\ ={\parallel x-p\parallel }^2-{\lambda }_n(2\alpha -{\lambda }_n){\parallel Ax-Ap\parallel }^2.\end{array}$$

(3.5)

This implies that

$$\begin{array}{rl}{\parallel x_{n+1}-p\parallel }^2& \le {\alpha }_n{\parallel x_n-p\parallel }^2+{\beta }_n{\parallel y_n-p\parallel }^2+{\gamma }_n{\parallel e_n-p\parallel }^2\\ \le {\alpha }_n{\parallel x_n-p\parallel }^2+{\beta }_n{\parallel x_n-p\parallel }^2-{\beta }_n{\lambda }_n(2\alpha -{\lambda }_n){\parallel A{x}_n-Ap\parallel }^2+{\gamma }_n{\parallel e_n-p\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-{\beta }_n{\lambda }_n(2\alpha -{\lambda }_n){\parallel A{x}_n-Ap\parallel }^2+{\gamma }_n{\parallel e_n-p\parallel }^2.\end{array}$$

(3.6)

It follows that

$${\beta }_n{\lambda }_n(2\alpha -{\lambda }_n){\parallel A{x}_n-Ap\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2+{\gamma }_n{\parallel e_n-p\parallel }^2.$$

In view of the restrictions (a), (b), and (c), we obtain that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel A{x}_n-Ap\parallel =0.$$

(3.7)

Notice that

$$\begin{array}{rl}{\parallel y_n-p\parallel }^2=& {\parallel J_{{\lambda }_n}(x_n-{\lambda }_{n}A{x}_n)-J_{{\lambda }_n}(p-{\lambda }_{n}Ap)\parallel }^2\\ \le & 〈(x_n-{\lambda }_{n}A{x}_n)-(p-{\lambda }_{n}Ap),y_n-p〉\\ =& \frac{1}{2}({\parallel (x_n-{\lambda }_{n}A{x}_n)-(p-{\lambda }_{n}Ap)\parallel }^2+{\parallel y_n-p\parallel }^2\\ -{\parallel (x_n-{\lambda }_{n}A{x}_n)-(p-{\lambda }_{n}Ap)-(y_n-p)\parallel }^2)\\ \le & \frac{1}{2}({\parallel x_n-p\parallel }^2+{\parallel y_n-p\parallel }^2-{\parallel x_n-y_n-{\lambda }_n(A{x}_n-Ap)\parallel }^2)\\ \le & \frac{1}{2}({\parallel x_n-p\parallel }^2+{\parallel y_n-p\parallel }^2-{\parallel x_n-y_n\parallel }^2-{\lambda }_n^2{\parallel A{x}_n-Ap\parallel }^2\\ +2{\lambda }_n\parallel x_n-y_n\parallel \parallel A{x}_n-Ap\parallel )\\ \le & \frac{1}{2}({\parallel x_n-p\parallel }^2+{\parallel y_n-p\parallel }^2-{\parallel x_n-y_n\parallel }^2+2{\lambda }_n\parallel x_n-y_n\parallel \parallel A{x}_n-Ap\parallel ).\end{array}$$

It follows that

$${\parallel y_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_n-y_n\parallel }^2+2{\lambda }_n\parallel x_n-y_n\parallel \parallel A{x}_n-Ap\parallel .$$

(3.8)

On the other hand, we have

$${\parallel x_{n+1}-p\parallel }^2\le {\alpha }_n{\parallel x_n-p\parallel }^2+{\beta }_n{\parallel y_n-p\parallel }^2+{\gamma }_n{\parallel e_n-p\parallel }^2.$$

(3.9)

Substituting (3.8) into (3.9), we arrive at

$$\begin{array}{rl}{\parallel x_{n+1}-p\parallel }^2\le & {\alpha }_n{\parallel x_n-p\parallel }^2+{\beta }_n{\parallel x_n-p\parallel }^2-{\beta }_n{\parallel x_n-y_n\parallel }^2\\ +2{\beta }_n{\lambda }_n\parallel x_n-y_n\parallel \parallel A{x}_n-Ap\parallel +{\gamma }_n{\parallel e_n-p\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-{\beta }_n{\parallel x_n-y_n\parallel }^2+2{\beta }_n{\lambda }_n\parallel x_n-y_n\parallel \parallel A{x}_n-Ap\parallel +{\gamma }_n{\parallel e_n-p\parallel }^2.\end{array}$$

It derives that

$${\beta }_n{\parallel x_n-y_n\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2+2{\beta }_n{\lambda }_n\parallel x_n-y_n\parallel \parallel A{x}_n-Ap\parallel +{\gamma }_n{\parallel e_n-p\parallel }^2.$$

In view of the restrictions (a) and (c), we find from (3.7) that

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

(3.10)

Notice that

$$\parallel S{x}_n-x_n\parallel \le \parallel S{x}_n-y_n\parallel +\parallel y_n-x_n\parallel .$$

It follows from (3.4) and (3.10) that

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

(3.11)

Since $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\rightharpoonup \omega \in C$, where ⇀ denotes the weak convergence. From Lemma 2.4, we find that $\omega \in F(S)$. In view of (3.10), we can choose a subsequence $\{y_{n_i}\}$ of $\{y_n\}$ such that $y_{n_i}\rightharpoonup \omega$. Notice that

$$y_n=J_{{\lambda }_n}(x_n-{\lambda }_{n}A{x}_n).$$

This implies that

$$x_n-{\lambda }_{n}A{x}_n\in (I+{\lambda }_{n}B)y_n.$$

That is,

$$\frac{x_n-y_n}{{\lambda }_n}-A{x}_n\in B{y}_n.$$

Since B is monotone, we get for any $(u,v)\in B$ that

$$〈y_n-u,\frac{x_n-y_n}{{\lambda }_n}-A{x}_n-v〉\ge 0.$$

(3.12)

Replacing n by $n_i$ and letting $i\to \mathrm{\infty }$, we obtain from (3.10) that

$$〈\omega -u,-A\omega -v〉\ge 0.$$

This means $-A\omega \in B\omega$, that is, $0\in (A+B)(\omega )$. Hence, we get $\omega \in {(A+B)}^{-1}(0)$. This completes the proof that $\omega \in \mathcal{F}$.

Suppose that there is another subsequence $\{x_{n_j}\}$ of $\{x_n\}$ such that $x_{n_j}\rightharpoonup {\omega }^{\ast }$. Then we can show that ${\omega }^{\ast }\in \mathcal{F}$ in exactly the same way. Assume that $\omega \ne {\omega }^{\ast }$ since ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exits for any $p\in F$. Put ${lim}_{n\to \mathrm{\infty }}\parallel x_n-\omega \parallel =d$. Since the space satisfies Opial’s condition, we see that

$$\begin{array}{rl}d& =\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-\omega \parallel \\ <\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-{\omega }^{\ast }\parallel \\ =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-{\omega }^{\ast }\parallel \\ =\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-{\omega }^{\ast }\parallel \\ <\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-\omega \parallel =d.\end{array}$$

This is a contradiction. This shows that $\omega ={\omega }^{\ast }$. This proves that the sequence $\{x_n\}$ converges weakly to $\omega \in F$. This completes the proof. □

We obtain from Theorem 3.1 the following inclusion problem.

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, let $A:C\to H$ be an α-inverse-strongly monotone mapping, and let B be a maximal monotone operator on H such that the domain of B is included in C. Assume that ${(A+B)}^{-1}(0)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be real number sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$. Let $\{{\lambda }_n\}$ be a positive real number sequence and let $\{e_n\}$ be a bounded error sequence in C. Let $\{x_n\}$ be a sequence in C generated in the following iterative process:

$$x_1\in C,x_{n+1}={\alpha }_n{x}_n+{\beta }_n{J}_{{\lambda }_n}(x_n-{\lambda }_{n}A{x}_n)+{\gamma }_n{e}_n$$

for all $n\in \mathbb{N}$, where $J_{{\lambda }_n}={(I+{\lambda }_{n}B)}^{-1}$. Assume that the sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\lambda }_n\}$ satisfy the following restrictions:

• (a) $0<a\le {\beta }_n\le b<1$,

• (b) $0<c\le {\lambda }_n\le d<2\alpha$,

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

where a, b, c, and d are some real numbers. Then the sequence $\{x_n\}$ converges weakly to some point in ${(A+B)}^{-1}(0)$.

Let $f:H\to (-\mathrm{\infty },\mathrm{\infty }]$ be a proper lower semicontinuous convex function. Define the subdifferential

$$\partial f(x)=\{z\in H:f(x)+〈y-x,z〉\le f(y),\mathrm{\forall }y\in H\}$$

for all $x\in H$. Then ∂f is a maximal monotone operator of H into itself; for more details, see [38]. Let C be a nonempty closed convex subset of H and let $i_C$ be the indicator function of C, that is,

$$i_{C}x=\{\begin{array}{cc}0,\hfill & x\in C,\hfill \\ \mathrm{\infty },\hfill & x\notin C.\hfill \end{array}$$

Furthermore, we define the normal cone $N_C(v)$ of C at v as follows:

$$N_{C}v=\{z\in H:〈z,y-v〉\le 0,\mathrm{\forall }y\in H\}$$

for any $v\in C$. Then $i_C:H\to (-\mathrm{\infty },\mathrm{\infty }]$ is a proper lower semicontinuous convex function on H and $\partial i_C$ is a maximal monotone operator. Let $J_{\lambda }x={(I+\lambda \partial i_C)}^{-1}x$ for any $\lambda >0$ and $x\in H$. From $\partial i_{C}x=N_{C}x$ and $x\in C$, we get

$$\begin{array}{rl}v=J_{\lambda }x& \iff x\in v+\lambda N_{C}v\\ \iff 〈x-v,y-v〉\le 0,\mathrm{\forall }y\in C,\\ \iff v=P_{C}x,\end{array}$$

where $P_C$ is the metric projection from H into C. Similarly, we can get that $x\in {(A+\partial i_C)}^{-1}(0)\iff x\in VI(A,C)$. Putting $B=\partial i_C$ in Theorem 3.1, we find that $J_{{\lambda }_n}=P_C$. The following results are not hard to derive.

Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, let $A:C\to H$ be an α-inverse-strongly monotone mapping and let $S:C\to C$ be a nonexpansive mapping. Assume that $\mathcal{F}=F(S)\cap VI(C,A)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be real number sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$. Let $\{{\lambda }_n\}$ be a positive real number sequence and let $\{e_n\}$ be a bounded error sequence in C. Let $\{x_n\}$ be a sequence in C generated in the following iterative process:

$$x_1\in C,x_{n+1}={\alpha }_{n}S{x}_n+{\beta }_n{P}_C(x_n-{\lambda }_{n}A{x}_n)+{\gamma }_n{e}_n$$

for all $n\in \mathbb{N}$. Assume that the sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\lambda }_n\}$ satisfy the following restrictions:

• (a) $0<a\le {\beta }_n\le b<1$,

• (b) $0<c\le {\lambda }_n\le d<2\alpha$,

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

where a, b, c, and d are some real numbers. Then the sequence $\{x_n\}$ converges weakly to some point in ℱ.

In view of Theorem 3.3, we have the following result.

Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let $A:C\to H$ be an α-inverse-strongly monotone mapping such that $VI(C,A)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be real number sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$. Let $\{{\lambda }_n\}$ be a positive real number sequence and let $\{e_n\}$ be a bounded error sequence in C. Let $\{x_n\}$ be a sequence in C generated in the following iterative process:

$$x_1\in C,x_{n+1}={\alpha }_n{x}_n+{\beta }_n{P}_C(x_n-{\lambda }_{n}A{x}_n)+{\gamma }_n{e}_n$$

for all $n\in \mathbb{N}$. Assume that the sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\lambda }_n\}$ satisfy the following restrictions:

• (a) $0<a\le {\beta }_n\le b<1$,

• (b) $0<c\le {\lambda }_n\le d<2\alpha$,

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

where a, b, c, and d are some real numbers. Then the sequence $\{x_n\}$ converges weakly to some point in $VI(C,A)$.

Let F be a bifunction of $C\times C$ into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem.

$$\text{Find }x\in C\text{ such that }F(x,y)\ge 0,\mathrm{\forall }y\in C.$$

In this paper, we use $\mathit{EP}(F)$ to denote the solution set of the equilibrium problem.

To study the equilibrium problems, we may assume that F satisfies the following conditions:

• (A1) $F(x,x)=0$ for all $x\in C$;

• (A2) F is monotone, i.e., $F(x,y)+F(y,x)\le 0$ for all $x,y\in C$;

• (A3) for each $x,y,z\in C$,

  $$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}F(tz+(1-t)x,y)\le F(x,y);$$

• (A4) for each $x\in C$, $y\mapsto F(x,y)$ is convex and weakly lower semi-continuous.

Putting $F(x,y)=〈Ax,y-x〉$ for every $x,y\in C$, we see that the equilibrium problem is reduced to the variational inequality (2.1).

The following lemma can be found in [39].

Lemma 3.5 Let C be a nonempty closed convex subset of H and let $F:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A4). Then, for any $r>0$ and $x\in H$, there exists $z\in C$ such that

$$F(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C.$$

Further, define

$$T_{r}x=\{z\in C:F(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C\}$$

(3.13)

for all $r>0$ and $x\in H$. Then the following hold:

• (a) $T_r$ is single-valued,

• (b) $T_r$ is firmly nonexpansive, i.e., for any $x,y\in H$,

  $${\parallel T_{r}x-T_{r}y\parallel }^2\le 〈T_{r}x-T_{r}y,x-y〉,$$

• (c) $F(T_r)=\mathit{EP}(F)$,

• (d) $\mathit{EP}(F)$ is closed and convex.

Lemma 3.6 [5]

Let C be a nonempty closed convex subset of a real Hilbert space H, let F a bifunction from $C\times C$ to ℝ which satisfies (A1)-(A4) and let $A_F$ be a multivalued mapping of H into itself defined by

$$A_{F}x=\{\begin{array}{cc}\{z\in H:F(x,y)\ge 〈y-x,z〉,\mathrm{\forall }y\in C\},\hfill & x\in C,\hfill \\ \mathrm{\varnothing },\hfill & x\notin C.\hfill \end{array}$$

Then $A_F$ is a maximal monotone operator with the domain $D(A_F)\subset C$, $\mathit{EP}(F)=A_F^{-1}(0)$ and

$$T_{r}x={(I+r{A}_F)}^{-1}x,\mathrm{\forall }x\in H,r>0,$$

where $T_r$ is defined as in (3.13).

Theorem 3.7 Let C be a nonempty closed convex subset of a real Hilbert space H, let $S:C\to C$ be a nonexpansive mapping and let F be a bifunction from $C\times C$ to ℝ which satisfies (A1)-(A4). Assume that $\mathcal{F}=F(S)\cap \mathit{EP}(F)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be real number sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$. Let $\{{\lambda }_n\}$ be a positive real number sequence and let $\{e_n\}$ be a bounded error sequence in C. Let $\{x_n\}$ be a sequence in C generated in the following iterative process:

$$x_1\in C,x_{n+1}={\alpha }_{n}S{x}_n+{\beta }_n{y}_n+{\gamma }_n{e}_n$$

for all $n\in \mathbb{N}$, where $y_n\in C$ such that

$$F(y_n,u)+\frac{1}{{\lambda }_n}〈u-y_n,y_n-x_n〉\ge 0,\mathrm{\forall }u\in C.$$

Assume that the sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\lambda }_n\}$ satisfy the following restrictions:

• (a) $0<a\le {\beta }_n\le b<1$,

• (b) $0<c\le {\lambda }_n\le d<\mathrm{\infty }$,

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

where a, b, c, and d are some real numbers. Then the sequence $\{x_n\}$ converges weakly to some point in ℱ.