Hybrid extragradient method for generalized mixed equilibrium problems and fixed point problems in Hilbert space

Abstract

In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of a nonexpansive mapping, and the set of solutions of a variational inequality problem for inverse strongly monotone mapping. We obtain some strong convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known convergence theorems in literature.

MSC:47H09, 47J05, 47J20, 47J25.

1 Introduction

Let H be a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$, and let C be a nonempty closed convex subset of H. Let $B:C\to H$ be a nonlinear mapping and let $\phi :C\to R\cup \{+\mathrm{\infty }\}$ be a function and F be a bifunction from $C\times C$ to R, where R is the set of real numbers. Peng and Yao [1] considered the following generalized mixed equilibrium problem:

$$\text{Finding }x\in C\text{ such that }F(x,y)+\phi (y)+〈Bx,y-x〉\ge \phi (x).$$

(1.1)

The set of solutions of (1.1) is denoted by $\mathit{GMEP}(F,\phi ,B)$. It is easy to see that x is a solution of problem (1.1) implying that $x\in dom\phi =\{x\in C:\phi (x)<+\mathrm{\infty }\}$.

If $B=0$, then the generalized mixed equilibrium problem (1.1) becomes the following mixed equilibrium problem:

$$\text{Finding }x\in C\text{ such that }F(x,y)+\phi (y)\ge \phi (x).$$

(1.2)

Problem (1.2) was studied by Ceng and Yao [2] and Peng and Yao [3, 4]. The set of solutions of (1.2) is denoted by $\mathit{MEP}(F,\phi )$.

If $\phi =0$, then the generalized mixed equilibrium problem (1.1) becomes the following generalized equilibrium problem:

$$\text{Finding }x\in C\text{ such that }F(x,y)+〈Bx,y-x〉\ge 0.$$

(1.3)

Problem (1.3) was studied by Takahashi and Takahashi [5]. The set of solutions of (1.3) is denoted by $\mathit{GEP}(F,B)$.

If $\phi =0$ and $B=0$, then the generalized mixed equilibrium problem (1.1) becomes the following equilibrium problem:

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

(1.4)

The set of solutions of (1.4) is denoted by $\mathit{EP}(F)$.

If $F(x,y)=0$ for all $x,y\in C$, the generalized mixed equilibrium problem (1.1) becomes the following generalized variational inequality problem:

$$\text{Finding }x\in C\text{ such that }\phi (y)+〈Bx,y-x〉\ge \phi (x).$$

(1.5)

The set of solutions of (1.5) is denoted by $\mathit{GVI}(C,\phi ,B)$.

If $\phi =0$ and $F(x,y)=0$ for all $x,y\in C$, the generalized mixed equilibrium problem (1.1) becomes the following variational inequality problem:

$$\text{Finding }x\in C\text{ such that }〈Bx,y-x〉\ge 0.$$

(1.6)

The set of solutions of (1.6) is denoted by $\mathit{VI}(C,B)$.

If $B=0$ and $F(x,y)=0$ for all $x,y\in C$, the generalized mixed equilibrium problem (1.1) becomes the following minimization problem:

$$\text{Finding }x\in C\text{ such that }\phi (y)\ge \phi (x).$$

(1.7)

Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others, see for instance, [1–7].

For solving the variational inequality problem in the finite-dimensional Euclidean spaces, in 1976, Korpelevich [8] introduced the following so-called extragradient method:

$$\{\begin{array}{c}{x}_1=x\in C,\hfill \\ y_n=P_C(x_n-\lambda B{x}_n),\hfill \\ x_{n+1}=P_C(x_n-\lambda B{y}_n)\hfill \end{array}$$

(1.8)

for every $n=0,1,2,\dots$ , $\lambda \in (0,\frac{1}{k})$, where C is a closed convex subset of $R^n$, $B:C\to R^n$ is a monotone and k-Lipschitz continuous mapping, and $P_C$ is the metric projection of $R^n$ into C. She showed that if $\mathit{VI}(C,B)$ is nonempty, then the sequences $\{x_n\}$ and $\{y_n\}$, generated by (1.8), converge to the same point $x\in \mathit{VI}(C,A)$. The idea of the extragradient iterative process introduced by Korpelevich was successfully generalized and extended not only in Euclidean but also in Hilbert and Banach spaces, see, e.g., the recent papers of He et at. [9], Gárciga Otero and Iuzem [10], Solodov and Svaiter [11], Solodov [12]. Moreover, Zeng and Yao [13] and Nadezhkina and Takahashi [14] introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. Yao and Yao [15] introduced an iterative process based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of a variational inequality problem for a k-inverse strongly monotone mapping. Plubtieng and Punpaeng [16] introduced an iterative process, based on the extragradient method, for finding the common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of a variational inequality problem for α-inverse strongly monotone mappings.

In 2003, Takahashi and Toyoda [17], introduced the following iterative scheme:

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S{P}_C(x_n-{\lambda }_{n}T{x}_n),$$

(1.9)

where $\{{\alpha }_n\}$ is a sequence in $(0,1)$, and $\{{\lambda }_n\}$ is a sequence in $(0,2\alpha )$. They proved that if $F(S)\cap \mathit{VI}(A)\ne \mathrm{\varnothing }$, then the sequence $\{x_n\}$ generated by (1.9) converges weakly to some $z\in F(S)\cap \mathit{VI}(A)$. Recently, Zeng and Yao [18] introduced the following iterative scheme:

$$\{\begin{array}{c}{x}_0=x\in C,\hfill \\ y_n=P_C(x_n-{\lambda }_n{x}_n),\hfill \\ x_{n+1}={\alpha }_n{x}_0+(1-{\alpha }_n)S{P}_C(x_n-{\lambda }_{n}A{y}_n),\hfill \end{array}$$

(1.10)

where $\{{\lambda }_n\}$ and $\{{\alpha }_n\}$ satisfy the following conditions: (i) ${\lambda }_{n}k\subset (0,1-\delta )$ for some $\delta \in (0,1)$ and (ii) ${\alpha }_n\subset (0,1)$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$. They proved that the sequence $\{x_n\}$ and $\{y_n\}$ generated by (1.10) converges strongly to the same point $P_{F(S)\cap \mathit{VI}(C,A)}{x}_0$ provided that ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$.

In 2006, Nadezhkina and Takahashi [19] also considered the extragradient method (1.9) for finding a common element of a fixed point of nonexpansive mapping and a set of solutions of variational inequalities, but the convergence result was still the weak convergence. The question posed by Takahashi and Toyoda [17] on whether the strong convergence result can be proved by the same iteration scheme Algorithm (1.9) remains open.

In 2010, with the techniques adopted by Noor and Rassias [20], Huang, Noor and Al-Said [21] set the projected residual function by

$$R_{\lambda }(x)=x-P_C(x-\lambda Ax),$$

(1.11)

it is well known that $x\in C$ is a solution of variational inequality (1.6) if and only if $x\in C$ is a zero of the projected residual function (1.11). They proved the strong convergence result of the iteration scheme (1.9) using the error analysis technique.

In this paper, inspired and motivated by the above researches and Huang, Noor and Al-Said [21], we introduce a new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of nonexpansive mappings and the set of solutions of an inverse strongly monotone mapping, as follows:

$$\{\begin{array}{c}{x}_1=x\in C,\hfill \\ F(u_n,y)+〈B{x}_n,y-u_n〉+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ y_n=P_C(u_n-{\lambda }_{n}A{u}_n),\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S[{\beta }_n{x}_n+(1-{\beta }_n)P_C(y_n-{\lambda }_{n}A{y}_n)],\hfill \end{array}$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{r_n\}$, $\{{\lambda }_n\}$ satisfy some parameters controlling conditions. We will obtain some strong convergence theorems using the error analysis technique as in [21]. The results in this paper generalize, extend and unify some well-known convergence theorems in the literature.

2 Preliminaries

Let C be a closed convex subset of a Hilbert space H for every point $x\in H$. There exists a unique nearest point in C, denoted by $P_{C}x$, such that

$$\parallel x-P_{C}x\parallel \le \parallel x-y\parallel \text{for all }y\in C.$$

$P_C$ is called the metric projection of H onto C. It is well known that $P_C$ is a nonexpansive mapping of H onto C and satisfies

$$〈x-y,P_{C}x-P_{C}y〉\ge {\parallel P_{C}x-P_{C}y\parallel }^2$$

for every $x,y\in H$. Moreover, $P_{C}x$ is characterized by the following properties: $P_{C}x\in C$ and

$$\begin{array}{r}〈x-P_{C}x,y-P_{C}y〉\le 0,\\ {\parallel x-y\parallel }^2\ge {\parallel x-P_{C}x\parallel }^2+{\parallel y-P_{C}x\parallel }^2\end{array}$$

(2.1)

for all $x\in H$, $y\in C$.

A mapping A of C into H is called monotone if

$$〈Ax-Ay,x-y〉\ge 0$$

for all $x,y\in C$. A mapping A of C into H is called inverse strongly monotone with a modulus α (in short, α-inverse strongly monotone) if there exists a positive real number α such that

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

for all $x,y\in C$.

Recall that a mapping S of C into itself is nonexpansive if

$$\parallel Sx-Sy\parallel \le \parallel x-y\parallel \text{for all }x,y\in C.$$

A mapping T of C into itself is pseudocontractive if

$$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2$$

for all $x,y\in C$. Obviously, the class of pseudocontractive mappings is more general than the class of nonexpansive mappings.

Let A be a monotone mapping from C into H. In the context of the variational inequality problem, the characterization of projection (2.1) implies the following:

$$u\in \mathit{VI}(A,C)⟺u=P_C(u-\lambda Au),\lambda >0.$$

It is also known that H satisfies Opial’s condition; for any sequence $\{x_n\}$ with $x_n\rightharpoonup x$, 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 $y\ne x$.

For solving the generalized mixed equilibrium problem, let us give the following assumptions for the bifunction F, the function φ and the set C:

• (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 any $x,y\in C$;

• (A3) for each $y\in C$, $x⊢F(x,y)$ is weakly upper semicontinuous;

• (A4) for each $x\in C$, $y⊢F(x,y)$ is convex;

• (A5) for each $x\in C$, $y⊢F(x,y)$ is lower semicontinuous;

• (B1) for each $x\in H$ and $r>0$, there exist a bounded subset $D(x)\subset C$ and $y_x\in C\cap dom(\phi )$ such that for any $z\in C-D_x$,

  $$F(z,y_x)+\phi (y_x)+〈Bz,y_x-z〉+\frac{1}{r}〈y_x-z,z-x〉\le \phi (z);$$

• (B2) C is a bounded set.

Lemma 2.1 [1]

Let C be a nonempty closed convex subset of a Hilbert space H. Let F be a bifunction from $C\times C$ to R satisfying (A1)-(A5) and let $\phi :C\to R\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For $r>0$ and $x\in H$, define a mapping $T_r:H\to C$ as follows:

$$T_r=\{z\in C:F(z,y)+\phi (y)+〈Bz,y-z〉+\frac{1}{r}〈y-z,z-x〉\le \phi (z),\mathrm{\forall }y\in C\}$$

for all $x\in H$. Then the following conclusions hold:

• (1) For each $x\in H$, $T_r(x)\ne \mathrm{\varnothing }$;

• (2) $T_r$ is single-valued;

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

  $${\parallel T_r(x)-T_r(y)\parallel }^2\le 〈T_r(x)-T_r(y),x-y〉;$$

• (4) $Fix(T_r(I-rB))=\mathit{GMEP}(F,\phi ,B)$;

• (5) $\mathit{GMEP}(F,\phi ,B)$ is closed and convex.

Lemma 2.2 [22]

For any $x^{\ast }\in \mathit{VI}(C,A)$, if $A:C\to H$ is α-inverse strongly monotone, then $R_{\lambda }(x)$ is $(1-\frac{\lambda }{4\alpha })$-inverse strongly monotone for any $\lambda \in [0,4\alpha ]$ and

$$〈x-x^{\ast },R_{\lambda }(x)〉\ge (1-\frac{\lambda }{4\alpha }){\parallel R_{\lambda }(x)\parallel }^2,$$

where $R_{\lambda }(x)=x-P_C(x-\lambda Ax)$.

Lemma 2.3 [21]

For all $x\in H$ and ${\lambda }^{\prime }\ge \lambda >0$, it holds that

$$\parallel R_{{\lambda }^{\prime }}(x)\parallel \ge \parallel R_{\lambda }(x)\parallel ,$$

where $R_{\lambda }(x)=x-P_C(x-\lambda Ax)$.

Lemma 2.4 [23]

Let $\{a_n\}$ and $\{b_n\}$ be two sequences of non-negative numbers, such that $a_{n+1}\le a_n+b_n$ for all $n\in N$. If ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<+\mathrm{\infty }$, and if $\{a_n\}$ has a subsequence $\{a_{n_k}\}$ converging to 0, then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Lemma 2.5 [24]

Let H be a real Hilbert space, and let C be a nonempty, closed and convex subset of H. Let $\{x_n\}$ be a sequence in H. Suppose that, for any $x^{\ast }\in C$,

$$\parallel x_{n+1}-x^{\ast }\parallel \le \parallel x_n-x^{\ast }\parallel (n\in N).$$

Then ${lim}_{n\to \mathrm{\infty }}{P}_C(x_n)=z$ for some $z\in C$.

3 Main results

Theorem 3.1 Let C be a closed and convex subset of a real Hilbert space H. Let F be a bifunction from $C\times C\to R$ satisfying (A1)-(A5) and $\phi :C\to R\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function. Let A be an α-inverse strongly monotone mapping from C into H and B be an β-inverse strongly monotone mapping from C into H. Let S be a nonexpansive mapping of C into itself, such that $\mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\cap \mathit{GMEP}(F,\phi ,B)\ne \mathrm{\varnothing }$. Assume that either (B1) or (B2) holds. Let $\{x_n\}$, $\{y_n\}$ and $\{u_n\}$ be sequences generated by

$$\{\begin{array}{c}{x}_1=x\in C,\hfill \\ F(u_n,y)+〈B{x}_n,y-u_n〉+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ y_n=P_C(u_n-{\lambda }_{n}A{u}_n),\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S[{\beta }_n{x}_n+(1-{\beta }_n)P_C(y_n-{\lambda }_{n}A{y}_n)]\hfill \end{array}$$

(3.1)

for every $n=1,2,\dots$ , where $\{{\lambda }_n\}$, $\{r_n\}$, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ satisfy the following conditions: (i) $0<r_n<2\beta$, $\{{\lambda }_n\}\subset [a,b]$ for some $a,b\in (0,2\alpha )$ and (ii) $\{{\alpha }_n\}\subset [c,d]$, $\{{\beta }_n\}\subset [e,f]$ for some $c,d,e,f\in (0,1)$, then $\{x_n\}$ converges strongly to $p^{\ast }\in \mathrm{\Omega }$, where $p^{\ast }={lim}_{n\to \mathrm{\infty }}{P}_{\mathrm{\Omega }}(x_n)$.

Proof We divide the proof into five steps.

Step 1. We claim that $\{x_n\}$ is bounded and ${lim}_{n\to \mathrm{\infty }}{R}_a(u_n)={lim}_{n\to \mathrm{\infty }}{R}_{{\lambda }_n}(u_n)=0$.

Put

$$\begin{array}{c}{v}_n=P_C(y_n-{\lambda }_{n}A{y}_n),w_n={\beta }_n{x}_n+(1-{\beta }_n)v_n,\hfill \\ R_{{\lambda }_n}(u_n)=u_n-P_C(u_n-{\lambda }_{n}A{u}_n),R_{{\lambda }_n}(y_n)=y_n-P_C(y_n-{\lambda }_{n}A{y}_n)\hfill \end{array}$$

for every $n=1,2,\dots$ . Take any $p\in \mathrm{\Omega }$ and let $\{T_{r_n}\}$ be a sequence of mappings defined as in Lemma 2.1, then $p=P_C(p-{\lambda }_{n}Ap)=T_{r_n}(p-r_{n}Bp)$. From $u_n=T_{r_n}(x_n-r_{n}B{x}_n)\in C$, the β-inverse strongly monotonicity of B and $0<r_n<2\beta$, we have

$$\begin{array}{rl}{\parallel u_n-p\parallel }^2& ={\parallel T_{r_n}(x_n-r_{n}B{x}_n)-T_{r_n}(p-r_{n}Bp)\parallel }^2\\ \le {\parallel x_n-r_{n}B{x}_n-(p-r_{n}Bp)\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-2r_n〈x_n-p,B{x}_n-Bp〉+r_n^2{\parallel B{x}_n-Bp\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-2r_n\beta {\parallel B{x}_n-Bp\parallel }^2+r_n^2{\parallel B{x}_n-Bp\parallel }^2\\ ={\parallel x_n-p\parallel }^2+r_n(r_n-2\beta ){\parallel B{x}_n-Bp\parallel }^2\\ \le {\parallel x_n-p\parallel }^2,\end{array}$$

(3.2)

and from Lemma 2.2, we have

$$\begin{array}{rl}{\parallel y_n-p\parallel }^2& ={\parallel u_n-R_{{\lambda }_n}(u_n)-p\parallel }^2\\ ={\parallel u_n-p\parallel }^2-2〈u_n-p,R_{{\lambda }_n}(u_n)〉+{\parallel R_{{\lambda }_n}(u_n)\parallel }^2\\ \le {\parallel u_n-p\parallel }^2-2(1-\frac{{\lambda }_n}{4\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2+{\parallel R_{{\lambda }_n}(u_n)\parallel }^2\\ ={\parallel u_n-p\parallel }^2-(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2,\end{array}$$

(3.3)

which implies from (3.2) that

$${\parallel y_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2-(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2.$$

(3.4)

By the same process as in (3.3), we also have from (3.4) that

$$\begin{array}{rl}{\parallel v_n-p\parallel }^2& \le {\parallel y_n-p\parallel }^2-(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(y_n)\parallel }^2\\ \le {\parallel y_n-p\parallel }^2-(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(y_n)\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2-(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(y_n)\parallel }^2.\end{array}$$

(3.5)

Further, from (3.1) and (3.5), we get

$$\begin{array}{rcl}{\parallel w_n-p\parallel }^2& =& {\beta }_n^2{\parallel x_n-p\parallel }^2+2{\beta }_n(1-{\beta }_n)〈x_n-p,v_n-p〉+{(1-{\beta }_n)}^2{\parallel v_n-p\parallel }^2\\ \le & {\beta }_n^2{\parallel x_n-p\parallel }^2+2{\beta }_n(1-{\beta }_n){\parallel x_n-p\parallel }^2+{(1-{\beta }_n)}^2{\parallel x_n-p\parallel }^2\\ -{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2\\ -{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(y_n)\parallel }^2.\end{array}$$

(3.6)

Hence, from (3.6), the nonexpansive property of the mapping S and $0<{\lambda }_n<2\alpha$, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& =& {\alpha }_n^2{\parallel x_n-p\parallel }^2+{(1-{\alpha }_n)}^2{\parallel S{w}_n-p\parallel }^2+2{\alpha }_n(1-{\alpha }_n)〈S{w}_n-Sp,x_n-p〉\\ \le & {\alpha }_n^2{\parallel x_n-p\parallel }^2+{(1-{\alpha }_n)}^2{\parallel w_n-p\parallel }^2+2{\alpha }_n(1-{\alpha }_n){\parallel x_n-p\parallel }^2\\ \le & {\alpha }_n^2{\parallel x_n-p\parallel }^2+{(1-{\alpha }_n)}^2{\parallel x_n-p\parallel }^2+2{\alpha }_n(1-{\alpha }_n){\parallel x_n-p\parallel }^2\\ -{(1-{\alpha }_n)}^2{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2\\ =& {\parallel x_n-p\parallel }^2-{(1-{\alpha }_n)}^2{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2\\ -{(1-{\alpha }_n)}^2{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(y_n)\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2.\end{array}$$

(3.7)

Since the sequence $\{\parallel x_n-p\parallel \}$ is a bounded and nonincreasing sequence, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists. Hence $\{x_n\}$ is bounded. Consequently, the sets $\{u_n\}$, $\{v_n\}$, $\{w_n\}$, $\{y_n\}$ are also bounded. By (3.7), we have

$${(1-{\alpha }_n)}^2{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2.$$

From the conditions (i) and (ii), there must exist a constant $M_1>0$ such that

$$M_1{\parallel R_{{\lambda }_n}(u_n)\parallel }^2\le {(1-{\alpha }_n)}^2{(1-{\beta }_n)}^2(1-\frac{{\lambda }_n}{2\alpha }){\parallel R_{{\lambda }_n}(u_n)\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2,$$

from which it follows that

$$M_1\sum _{n=1}^{\mathrm{\infty }}{\parallel R_{{\lambda }_n}(u_n)\parallel }^2\le \sum _{n=1}^{\mathrm{\infty }}[{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2]={\parallel x_1-p\parallel }^2<\mathrm{\infty }.$$

Hence, ${lim}_{n\to \mathrm{\infty }}{R}_{{\lambda }_n}(u_n)={lim}_{n\to \mathrm{\infty }}\parallel R_{{\lambda }_n}(u_n)\parallel =0$. Since $R_{{\lambda }_n}(u_n)=u_n-P_C(u_n-{\lambda }_{n}A{u}_n)=u_n-y_n$, ${lim}_{n\to \mathrm{\infty }}\parallel u_n-y_n\parallel =0$. Notice that ${\lambda }_n\ge a$, then by Lemma 2.3, $\parallel R_a(u_n)\parallel \le \parallel R_{{\lambda }_n}(u_n)\parallel$. Therefore,

$$\underset{n\to \mathrm{\infty }}{lim}{R}_a(u_n)=\underset{n\to \mathrm{\infty }}{lim}{R}_{{\lambda }_n}(u_n)=0.$$

(3.8)

By the same way, we also get that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel R_{{\lambda }_n}(y_n)\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-v_n\parallel =0,$$

and thus

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-v_n\parallel =0.$$

(3.9)

Step 2. We show that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u_n\parallel ={lim}_{n\to \mathrm{\infty }}\parallel S{x}_n-x_n\parallel =0$.

Indeed, for any $p\in \mathrm{\Omega }$, it follows from (3.1) and (3.5) that

$$\begin{array}{rcl}{\parallel w_n-p\parallel }^2& =& {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel x_n-v_n\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel x_n-v_n\parallel }^2,\end{array}$$

which implies that

$$\begin{array}{rl}{\parallel x_{n+1}-p\parallel }^2& ={\alpha }_n{\parallel x_n-p\parallel }^2+(1-{\alpha }_n){\parallel w_n-p\parallel }^2-{\alpha }_n(1-{\alpha }_n){\parallel S{w}_n-x_n\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-{\alpha }_n(1-{\alpha }_n){\parallel S{w}_n-x_n\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel x_n-v_n\parallel }^2.\end{array}$$

(3.10)

Thus, it follows from (3.10) that

$${\alpha }_n(1-{\alpha }_n){\parallel S{w}_n-x_n\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2.$$

From the condition (ii), there exists a constant $M_2>0$ such that

$$M_2{\parallel S{w}_n-x_n\parallel }^2\le {\alpha }_n(1-{\alpha }_n){\parallel S{w}_n-x_n\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2,$$

from which it follows that

$$M_2\sum _{n=1}^{\mathrm{\infty }}{\parallel S{w}_n-x_n\parallel }^2\le \sum _{n=1}^{\mathrm{\infty }}[{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2]={\parallel x_1-p\parallel }^2<\mathrm{\infty }.$$

Hence

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

(3.11)

From (3.10), we also get that

$${\beta }_n(1-{\beta }_n){\parallel x_n-v_n\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2.$$

By the same way, we obtain that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-v_n\parallel =0,$$

(3.12)

which combining (3.9) implies that

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

(3.13)

Since

$$\begin{array}{rcl}\parallel S{x}_n-x_n\parallel & \le & \parallel S{x}_n-S{v}_n\parallel +\parallel S{v}_n-S{w}_n\parallel +\parallel S{w}_n-x_n\parallel \\ \le & \parallel x_n-v_n\parallel +\parallel v_n-w_n\parallel +\parallel S{w}_n-x_n\parallel \\ \le & \parallel x_n-v_n\parallel +{\beta }_n\parallel x_n-v_n\parallel +\parallel S{w}_n-x_n\parallel ,\end{array}$$

which implies from (3.11), (3.12) that

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

(3.14)

Further, it follows from (3.1) and (3.11) that

$$\parallel x_{n+1}-x_n\parallel =(1-{\alpha }_n)\parallel S{w}_n-x_n\parallel \le (1-c)\parallel S{w}_n-x_n\parallel \to 0(n\to \mathrm{\infty }).$$

(3.15)

Step 3. We claim that $\{x_n\}$ must have a convergent subsequence $\{x_{n_k}\}$ such that ${lim}_{k\to \mathrm{\infty }}{x}_{n_k}=p^{\ast }$ for some $p^{\ast }\in C$. Moreover, $p^{\ast }\in \mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\cap \mathit{GMEP}(F,\phi ,B)$.

Since $\{x_n\}$ is a bounded sequence generated by Algorithm (3.1), then $\{x_n\}$ must have a weakly convergent subsequence $\{x_{n_k}\}$ such that $x_{n_k}\rightharpoonup p^{\ast }$ ($k\to \mathrm{\infty }$), which implies from (3.11) and (3.13) that $S{w}_{n_k}\rightharpoonup p^{\ast }$ ($k\to \mathrm{\infty }$) and $u_{n_k}\rightharpoonup p^{\ast }$ ($k\to \mathrm{\infty }$). Next we will show that $p^{\ast }\in \mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\cap \mathit{GMEP}(F,\phi ,B)$.

Since A is inverse strongly monotone with the positive constant $\alpha >0$, so A is $\frac{1}{\alpha }$-Lipschitz continuous. Indeed, it yields that $\parallel Ax-Ay\parallel \le \frac{1}{\alpha }\parallel x-y\parallel$ from the definition of the inverse strongly monotonicity of A, such that

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

From the $\frac{1}{\alpha }$-Lipschitz continuity of A and the continuity of $P_C$, it follows that $R_a(x)=x-P_C[x-aAx]$ is also continuous. Notice that ${\rho }_n\ge a$, then by Lemma 2.3, $\parallel R_x(x_n)\parallel \le \parallel R_{{\rho }_n}(x_n)\parallel$. Then from Step 1,

$$\underset{k\to \mathrm{\infty }}{lim}\parallel R_x(x_{n_k})\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel R_{{\rho }_n}(x_{n_k})\parallel =0.$$

Therefore from the continuity of $R_a(x)$,

$$R_a\left(p^{\ast }\right)=\underset{n\to \mathrm{\infty }}{lim}{R}_a(x_{n_k})=0.$$

This shows that $p^{\ast }$ is a solution of the variational inequality (1.6), that is $p^{\ast }\in \mathit{VI}(C,A)$. From (3.12), ${lim}_{n\to \mathrm{\infty }}\parallel x_{n_k}-p^{\ast }\parallel =0$ and the property of the nonexpansive mapping S, it follows that $p^{\ast }=S{p}^{\ast }$, that is $p^{\ast }\in Fix(S)$. Finally, by the same argument as in the proof of [[7], Theorem 3.1], we prove that $p^{\ast }\in \mathit{GMEP}(F,\phi ,B)$. Thus $p^{\ast }\in \mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\cap \mathit{GMEP}(F,\phi ,B)$.

Next, we will prove that $x_{n_k}\to p^{\ast }$ ($k\to \mathrm{\infty }$).

From (3.1), (3.6) and (3.7) we can calculate

$$\begin{array}{rcl}{\parallel x_{n+1}-p^{\ast }\parallel }^2& =& 〈{\alpha }_n{x}_n+(1-{\alpha }_n)S{w}_n-p^{\ast },x_{n+1}-p^{\ast }〉\\ =& {\alpha }_n〈x_n-p^{\ast },x_{n+1}-p^{\ast }〉+(1-{\alpha }_n)〈S{w}_n-p^{\ast },x_{n+1}-p^{\ast }〉\\ \le & {\alpha }_n{\parallel x_n-p^{\ast }\parallel }^2+(1-{\alpha }_n)〈S{w}_n-p^{\ast },x_{n+1}-p^{\ast }〉\\ \le & {\alpha }_n{\parallel x_n-p^{\ast }\parallel }^2+(1-{\alpha }_n)〈S{w}_n-p^{\ast },x_{n+1}-x_n〉\\ +(1-{\alpha }_n)〈S{w}_n-p^{\ast },x_n-p^{\ast }〉\\ \le & {\alpha }_n{\parallel x_n-p^{\ast }\parallel }^2+(1-{\alpha }_n){\parallel x_n-p^{\ast }\parallel }^2+(1-{\alpha }_n)〈S{w}_n-p^{\ast },x_{n+1}-x_n〉\\ =& {\parallel x_n-p^{\ast }\parallel }^2+(1-{\alpha }_n)〈S{w}_n-p^{\ast },x_{n+1}-x_n〉,\end{array}$$

which implies

$$\begin{array}{rl}{\parallel x_{n+1}-p^{\ast }\parallel }^2-{\parallel x_n-p^{\ast }\parallel }^2& \le (1-{\alpha }_n)〈S{w}_n-p^{\ast },x_{n+1}-x_n〉\\ \le (1-c)〈S{w}_n-p^{\ast },x_{n+1}-x_n〉.\end{array}$$

(3.16)

From $S{w}_{n_k}\rightharpoonup p^{\ast }$ and $x_{n_{k+1}}-x_{n_k}\to 0$ as $k\to \mathrm{\infty }$, it follows from (3.16) that

$$\parallel x_{n_{k+1}}-p^{\ast }\parallel \to \parallel x_{n_k}-p^{\ast }\parallel (k\to \mathrm{\infty }).$$

Using the Kadec-Klee property of H, we obtain that ${lim}_{k\to \mathrm{\infty }}{x}_{n_k}=p^{\ast }$.

Step 4. We claim that the sequence $\{x_n\}$ generated by Algorithm (3.1) converges strongly to $p^{\ast }\in \mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\cap \mathit{GMEP}(F,\phi ,B)$.

In fact, from the result of Step 3, $p^{\ast }\in \mathrm{\Omega }$. Let $p=p^{\ast }$ in (3.7). Consequently, $\parallel x_{n+1}-p^{\ast }\parallel \le \parallel x_n-p^{\ast }\parallel$. Meanwhile, ${lim}_{k\to \mathrm{\infty }}\parallel x_{n_k}-p^{\ast }\parallel =0$ from Step 3. Then from Lemma 2.4, we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p^{\ast }\parallel =0$. Therefore, ${lim}_{n\to \mathrm{\infty }}{x}_n=p^{\ast }$.

Step 5. We claim that $p^{\ast }={lim}_{n\to \mathrm{\infty }}{P}_{\mathrm{\Omega }}{x}_n$.

From (2.1), we have

$$〈x_n-P_{\mathrm{\Omega }}{x}_n,p^{\ast }-P_{\mathrm{\Omega }}{x}_n〉\le 0.$$

(3.17)

By (3.7) and Lemma 2.5, ${lim}_{n\to \mathrm{\infty }}{P}_{\mathrm{\Omega }}{x}_n=q^{\ast }$ for some $q^{\ast }\in \mathrm{\Omega }$. Then in (3.13), let $n\to \mathrm{\infty }$, since ${lim}_{n\to \mathrm{\infty }}{x}_n=p^{\ast }$ by Step 4, we have

$$〈p^{\ast }-q^{\ast },p^{\ast }-q^{\ast }〉\le 0,$$

and, consequently, we have $p^{\ast }=q^{\ast }$. Hence, $p^{\ast }={lim}_{n\to \mathrm{\infty }}{P}_{\mathrm{\Omega }}{x}_n$.

This completes the proof of Theorem 3.1. □

The following theorems can be obtained from Theorem 3.1 immediately.

Theorem 3.2 Let C, H, S be as in Theorem  3.1. Assume that $\mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\ne \mathrm{\varnothing }$, let $\{x_n\}$, $\{y_n\}$ be sequences generated by

$$\{\begin{array}{c}{x}_1=x\in C,\hfill \\ y_n=P_C(x_n-{\lambda }_{n}A{x}_n),\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S[{\beta }_n{x}_n+(1-{\beta }_n)P_C(y_n-{\lambda }_{n}A{y}_n)]\hfill \end{array}$$

for every $n=1,2,\dots$ , where $\{{\lambda }_n\}$, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ satisfy the following conditions: (i) $\{{\lambda }_n\}\subset [a,b]$ for some $a,b\in (0,2\alpha )$ and (ii) $\{{\alpha }_n\}\subset [c,d]$, $\{{\beta }_n\}\subset [e,f]$ for some $c,d,e,f\in (0,1)$, then $\{x_n\}$ converges strongly to $p^{\ast }\in \mathrm{\Omega }$, where $p^{\ast }={lim}_{n\to \mathrm{\infty }}{P}_{\mathrm{\Omega }}(x_n)$.

Proof Putting $B=F=\phi =0$, $r_n=1$ in Theorem 3.1, the conclusion of Theorem 3.2 can be obtained from Theorem 3.1. □

Remark 3.1 The main result of Nadezhkina and Takahashi [14] is a special case of our Theorem 3.2. Indeed, if we take ${\beta }_n=0$ in Theorem 3.2, then we obtain the result of [14].

Theorem 3.3 Let C, H, F, A, B, S be as in Theorem  3.1. Assume $\mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\cap \mathit{GEP}(F,B)\ne \mathrm{\varnothing }$; let $\{x_n\}$, $\{y_n\}$ and $\{u_n\}$ be sequences generated by

$$\{\begin{array}{c}{x}_1=x\in C,\hfill \\ F(u_n,y)+〈B{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ y_n=P_C(u_n-{\lambda }_{n}A{u}_n),\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S[{\beta }_n{x}_n+(1-{\beta }_n)P_C(y_n-{\lambda }_{n}A{y}_n)]\hfill \end{array}$$

for every $n=1,2,\dots$ , where $\{{\lambda }_n\}$, $\{r_n\}$, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ satisfy conditions (i) and (ii) as in Theorem  3.1, then $\{x_n\}$ converges strongly to $p^{\ast }\in \mathrm{\Omega }$, where $p^{\ast }={lim}_{n\to \mathrm{\infty }}{P}_{\mathrm{\Omega }}(x_n)$.

Proof Putting $\phi =0$ in Theorem 3.1, the conclusion of Theorem 3.3 is obtained. □

Remark 3.2 Theorem 3.3 can be viewed as an improvement of Theorem 3.1 of Inchan [25] because of removing the iterative step $C_n$ in the algorithm of Theorem 3.1 of [25].

Theorem 3.4 Let C, H, F, A, S be as in Theorem  3.1. Assume that $\mathrm{\Omega }=Fix(S)\cap \mathit{VI}(C,A)\cap \mathit{EP}(F)\ne \mathrm{\varnothing }$; let $\{x_n\}$ and $\{u_n\}$ be sequences generated by

$$\{\begin{array}{c}{x}_1=x\in C,\hfill \\ F(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ y_n=P_C(u_n-{\lambda }_{n}A{u}_n),\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)S{P}_C(y_n-{\lambda }_{n}A{y}_n)\hfill \end{array}$$

for every $n=1,2,\dots$ , where $\{{\lambda }_n\}$, $\{r_n\}$, $\{{\alpha }_n\}$ satisfy the following conditions: $0<r_n<2\beta$, $\{{\lambda }_n\}\subset [a,b]$ for some $a,b\in (0,2\alpha )$, $\{{\alpha }_n\}\subset [c,d]$ for some $c,d\in (0,1)$, then $\{x_n\}$ converges strongly to $p^{\ast }\in \mathrm{\Omega }$, where $p^{\ast }={lim}_{n\to \mathrm{\infty }}{P}_{\mathrm{\Omega }}(x_n)$.

Proof Taking $B=\phi =0$, ${\beta }_n=0$ in Theorem 3.1, the conclusion of Theorem 3.4 is obtained. □

Remark 3.3 Theorem 3.4 is the strong convergence result of Theorem 3.1 of Jaiboon, Kumam and Humphries [26].