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

Fixed Point Theory and Applications20132013:240

https://doi.org/10.1186/1687-1812-2013-240

• Received: 18 September 2012
• Accepted: 23 July 2013
• Published:

## 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.

## Keywords

• generalized mixed equilibrium problem
• nonexpansive mapping
• variational inequality
• strong convergence theorem

## 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 $\phi :C\to R\cup \left\{+\mathrm{\infty }\right\}$ be a function and F be a bifunction from $C×C$ to R, where R is the set of real numbers. Peng and Yao  considered the following generalized mixed equilibrium problem:
(1.1)

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

If $B=0$, then the generalized mixed equilibrium problem (1.1) becomes the following mixed equilibrium problem:
(1.2)

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

If $\phi =0$, then the generalized mixed equilibrium problem (1.1) becomes the following generalized equilibrium problem:
(1.3)

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

If $\phi =0$ and $B=0$, then the generalized mixed equilibrium problem (1.1) becomes the following equilibrium problem:
(1.4)

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

If $F\left(x,y\right)=0$ for all $x,y\in C$, the generalized mixed equilibrium problem (1.1) becomes the following generalized variational inequality problem:
(1.5)

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

If $\phi =0$ and $F\left(x,y\right)=0$ for all $x,y\in C$, the generalized mixed equilibrium problem (1.1) becomes the following variational inequality problem:
(1.6)

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

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

For solving the variational inequality problem in the finite-dimensional Euclidean spaces, in 1976, Korpelevich  introduced the following so-called extragradient method:
$\left\{\begin{array}{c}{x}_{1}=x\in C,\hfill \\ {y}_{n}={P}_{C}\left({x}_{n}-\lambda B{x}_{n}\right),\hfill \\ {x}_{n+1}={P}_{C}\left({x}_{n}-\lambda B{y}_{n}\right)\hfill \end{array}$
(1.8)

for every $n=0,1,2,\dots$ , $\lambda \in \left(0,\frac{1}{k}\right)$, 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 $\mathit{VI}\left(C,B\right)$ is nonempty, then the sequences $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$, generated by (1.8), converge to the same point $x\in \mathit{VI}\left(C,A\right)$. 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. , Gárciga Otero and Iuzem , Solodov and Svaiter , Solodov . Moreover, Zeng and Yao  and Nadezhkina and Takahashi  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  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  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 , introduced the following iterative scheme:
${x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)S{P}_{C}\left({x}_{n}-{\lambda }_{n}T{x}_{n}\right),$
(1.9)
where $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left(0,1\right)$, and $\left\{{\lambda }_{n}\right\}$ is a sequence in $\left(0,2\alpha \right)$. They proved that if $F\left(S\right)\cap \mathit{VI}\left(A\right)\ne \mathrm{\varnothing }$, then the sequence $\left\{{x}_{n}\right\}$ generated by (1.9) converges weakly to some $z\in F\left(S\right)\cap \mathit{VI}\left(A\right)$. Recently, Zeng and Yao  introduced the following iterative scheme:
$\left\{\begin{array}{c}{x}_{0}=x\in C,\hfill \\ {y}_{n}={P}_{C}\left({x}_{n}-{\lambda }_{n}{x}_{n}\right),\hfill \\ {x}_{n+1}={\alpha }_{n}{x}_{0}+\left(1-{\alpha }_{n}\right)S{P}_{C}\left({x}_{n}-{\lambda }_{n}A{y}_{n}\right),\hfill \end{array}$
(1.10)

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

In 2006, Nadezhkina and Takahashi  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  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 , Huang, Noor and Al-Said  set the projected residual function by
${R}_{\lambda }\left(x\right)=x-{P}_{C}\left(x-\lambda Ax\right),$
(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 , 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:
$\left\{\begin{array}{c}{x}_{1}=x\in C,\hfill \\ F\left({u}_{n},y\right)+〈B{x}_{n},y-{u}_{n}〉+\phi \left(y\right)-\phi \left({u}_{n}\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {y}_{n}={P}_{C}\left({u}_{n}-{\lambda }_{n}A{u}_{n}\right),\hfill \\ {x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)S\left[{\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){P}_{C}\left({y}_{n}-{\lambda }_{n}A{y}_{n}\right)\right],\hfill \end{array}$

where $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$, $\left\{{r}_{n}\right\}$, $\left\{{\lambda }_{n}\right\}$ satisfy some parameters controlling conditions. We will obtain some strong convergence theorems using the error analysis technique as in . 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
${P}_{C}$ 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
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:
$u\in \mathit{VI}\left(A,C\right)\phantom{\rule{1em}{0ex}}⟺\phantom{\rule{1em}{0ex}}u={P}_{C}\left(u-\lambda Au\right),\phantom{\rule{1em}{0ex}}\lambda >0.$
It is also known that H satisfies Opial’s condition; for any sequence $\left\{{x}_{n}\right\}$ with ${x}_{n}⇀x$, the inequality
$\underset{n\to \mathrm{\infty }}{lim inf}\parallel {x}_{n}-x\parallel <\underset{n\to \mathrm{\infty }}{lim inf}\parallel {x}_{n}-y\parallel$

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:
1. (A1)

$F\left(x,x\right)=0$ for all $x\in C$;

2. (A2)

F is monotone, i.e., $F\left(x,y\right)+F\left(y,x\right)\le 0$ for any $x,y\in C$;

3. (A3)

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

4. (A4)

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

5. (A5)

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

6. (B1)
for each $x\in H$ and $r>0$, there exist a bounded subset $D\left(x\right)\subset C$ and ${y}_{x}\in C\cap dom\left(\phi \right)$ such that for any $z\in C-{D}_{x}$,
$F\left(z,{y}_{x}\right)+\phi \left({y}_{x}\right)+〈Bz,{y}_{x}-z〉+\frac{1}{r}〈{y}_{x}-z,z-x〉\le \phi \left(z\right);$

7. (B2)

C is a bounded set.

Lemma 2.1 

Let C be a nonempty closed convex subset of a Hilbert space H. Let F be a bifunction from $C×C$ to R satisfying (A1)-(A5) and let $\phi :C\to R\cup \left\{+\mathrm{\infty }\right\}$ 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:
${T}_{r}=\left\{z\in C:F\left(z,y\right)+\phi \left(y\right)+〈Bz,y-z〉+\frac{1}{r}〈y-z,z-x〉\le \phi \left(z\right),\mathrm{\forall }y\in C\right\}$
for all $x\in H$. Then the following conclusions hold:
1. (1)

For each $x\in H$, ${T}_{r}\left(x\right)\ne \mathrm{\varnothing }$;

2. (2)

${T}_{r}$ is single-valued;

3. (3)
${T}_{r}$ is firmly nonexpansive, i.e., for any $x,y\in H$,
${\parallel {T}_{r}\left(x\right)-{T}_{r}\left(y\right)\parallel }^{2}\le 〈{T}_{r}\left(x\right)-{T}_{r}\left(y\right),x-y〉;$

4. (4)

$Fix\left({T}_{r}\left(I-rB\right)\right)=\mathit{GMEP}\left(F,\phi ,B\right)$;

5. (5)

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

Lemma 2.2 

For any ${x}^{\ast }\in \mathit{VI}\left(C,A\right)$, if $A:C\to H$ is α-inverse strongly monotone, then ${R}_{\lambda }\left(x\right)$ is $\left(1-\frac{\lambda }{4\alpha }\right)$-inverse strongly monotone for any $\lambda \in \left[0,4\alpha \right]$ and
$〈x-{x}^{\ast },{R}_{\lambda }\left(x\right)〉\ge \left(1-\frac{\lambda }{4\alpha }\right){\parallel {R}_{\lambda }\left(x\right)\parallel }^{2},$

where ${R}_{\lambda }\left(x\right)=x-{P}_{C}\left(x-\lambda Ax\right)$.

Lemma 2.3 

For all $x\in H$ and ${\lambda }^{\prime }\ge \lambda >0$, it holds that
$\parallel {R}_{{\lambda }^{\prime }}\left(x\right)\parallel \ge \parallel {R}_{\lambda }\left(x\right)\parallel ,$

where ${R}_{\lambda }\left(x\right)=x-{P}_{C}\left(x-\lambda Ax\right)$.

Lemma 2.4 

Let $\left\{{a}_{n}\right\}$ and $\left\{{b}_{n}\right\}$ 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 $\left\{{a}_{n}\right\}$ has a subsequence $\left\{{a}_{{n}_{k}}\right\}$ converging to 0, then ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$.

Lemma 2.5 

Let H be a real Hilbert space, and let C be a nonempty, closed and convex subset of H. Let $\left\{{x}_{n}\right\}$ be a sequence in H. Suppose that, for any ${x}^{\ast }\in C$,
$\parallel {x}_{n+1}-{x}^{\ast }\parallel \le \parallel {x}_{n}-{x}^{\ast }\parallel \phantom{\rule{1em}{0ex}}\left(n\in N\right).$

Then ${lim}_{n\to \mathrm{\infty }}{P}_{C}\left({x}_{n}\right)=z$ 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 $C×C\to R$ satisfying (A1)-(A5) and $\phi :C\to R\cup \left\{+\mathrm{\infty }\right\}$ 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 $\mathrm{\Omega }=Fix\left(S\right)\cap \mathit{VI}\left(C,A\right)\cap \mathit{GMEP}\left(F,\phi ,B\right)\ne \mathrm{\varnothing }$. Assume that either (B1) or (B2) holds. Let $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$ and $\left\{{u}_{n}\right\}$ be sequences generated by
$\left\{\begin{array}{c}{x}_{1}=x\in C,\hfill \\ F\left({u}_{n},y\right)+〈B{x}_{n},y-{u}_{n}〉+\phi \left(y\right)-\phi \left({u}_{n}\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {y}_{n}={P}_{C}\left({u}_{n}-{\lambda }_{n}A{u}_{n}\right),\hfill \\ {x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)S\left[{\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){P}_{C}\left({y}_{n}-{\lambda }_{n}A{y}_{n}\right)\right]\hfill \end{array}$
(3.1)

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

Proof We divide the proof into five steps.

Step 1. We claim that $\left\{{x}_{n}\right\}$ is bounded and ${lim}_{n\to \mathrm{\infty }}{R}_{a}\left({u}_{n}\right)={lim}_{n\to \mathrm{\infty }}{R}_{{\lambda }_{n}}\left({u}_{n}\right)=0$.

Put
$\begin{array}{c}{v}_{n}={P}_{C}\left({y}_{n}-{\lambda }_{n}A{y}_{n}\right),\phantom{\rule{2em}{0ex}}{w}_{n}={\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){v}_{n},\hfill \\ {R}_{{\lambda }_{n}}\left({u}_{n}\right)={u}_{n}-{P}_{C}\left({u}_{n}-{\lambda }_{n}A{u}_{n}\right),\phantom{\rule{2em}{0ex}}{R}_{{\lambda }_{n}}\left({y}_{n}\right)={y}_{n}-{P}_{C}\left({y}_{n}-{\lambda }_{n}A{y}_{n}\right)\hfill \end{array}$
for every $n=1,2,\dots$ . Take any $p\in \mathrm{\Omega }$ and let $\left\{{T}_{{r}_{n}}\right\}$ be a sequence of mappings defined as in Lemma 2.1, then $p={P}_{C}\left(p-{\lambda }_{n}Ap\right)={T}_{{r}_{n}}\left(p-{r}_{n}Bp\right)$. From ${u}_{n}={T}_{{r}_{n}}\left({x}_{n}-{r}_{n}B{x}_{n}\right)\in C$, the β-inverse strongly monotonicity of B and $0<{r}_{n}<2\beta$, we have
$\begin{array}{rl}{\parallel {u}_{n}-p\parallel }^{2}& ={\parallel {T}_{{r}_{n}}\left({x}_{n}-{r}_{n}B{x}_{n}\right)-{T}_{{r}_{n}}\left(p-{r}_{n}Bp\right)\parallel }^{2}\\ \le {\parallel {x}_{n}-{r}_{n}B{x}_{n}-\left(p-{r}_{n}Bp\right)\parallel }^{2}\\ \le {\parallel {x}_{n}-p\parallel }^{2}-2{r}_{n}〈{x}_{n}-p,B{x}_{n}-Bp〉+{r}_{n}^{2}{\parallel B{x}_{n}-Bp\parallel }^{2}\\ \le {\parallel {x}_{n}-p\parallel }^{2}-2{r}_{n}\beta {\parallel B{x}_{n}-Bp\parallel }^{2}+{r}_{n}^{2}{\parallel B{x}_{n}-Bp\parallel }^{2}\\ ={\parallel {x}_{n}-p\parallel }^{2}+{r}_{n}\left({r}_{n}-2\beta \right){\parallel B{x}_{n}-Bp\parallel }^{2}\\ \le {\parallel {x}_{n}-p\parallel }^{2},\end{array}$
(3.2)
and from Lemma 2.2, we have
$\begin{array}{rl}{\parallel {y}_{n}-p\parallel }^{2}& ={\parallel {u}_{n}-{R}_{{\lambda }_{n}}\left({u}_{n}\right)-p\parallel }^{2}\\ ={\parallel {u}_{n}-p\parallel }^{2}-2〈{u}_{n}-p,{R}_{{\lambda }_{n}}\left({u}_{n}\right)〉+{\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\\ \le {\parallel {u}_{n}-p\parallel }^{2}-2\left(1-\frac{{\lambda }_{n}}{4\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}+{\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\\ ={\parallel {u}_{n}-p\parallel }^{2}-\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2},\end{array}$
(3.3)
which implies from (3.2) that
${\parallel {y}_{n}-p\parallel }^{2}\le {\parallel {x}_{n}-p\parallel }^{2}-\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}.$
(3.4)
By the same process as in (3.3), we also have from (3.4) that
$\begin{array}{rl}{\parallel {v}_{n}-p\parallel }^{2}& \le {\parallel {y}_{n}-p\parallel }^{2}-\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({y}_{n}\right)\parallel }^{2}\\ \le {\parallel {y}_{n}-p\parallel }^{2}-\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({y}_{n}\right)\parallel }^{2}\\ \le {\parallel {x}_{n}-p\parallel }^{2}-\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}-\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({y}_{n}\right)\parallel }^{2}.\end{array}$
(3.5)
Further, from (3.1) and (3.5), we get
$\begin{array}{rcl}{\parallel {w}_{n}-p\parallel }^{2}& =& {\beta }_{n}^{2}{\parallel {x}_{n}-p\parallel }^{2}+2{\beta }_{n}\left(1-{\beta }_{n}\right)〈{x}_{n}-p,{v}_{n}-p〉+{\left(1-{\beta }_{n}\right)}^{2}{\parallel {v}_{n}-p\parallel }^{2}\\ \le & {\beta }_{n}^{2}{\parallel {x}_{n}-p\parallel }^{2}+2{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel {x}_{n}-p\parallel }^{2}+{\left(1-{\beta }_{n}\right)}^{2}{\parallel {x}_{n}-p\parallel }^{2}\\ -{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\\ \le & {\parallel {x}_{n}-p\parallel }^{2}-{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\\ -{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({y}_{n}\right)\parallel }^{2}.\end{array}$
(3.6)
Hence, from (3.6), the nonexpansive property of the mapping S and $0<{\lambda }_{n}<2\alpha$, we have
$\begin{array}{rcl}{\parallel {x}_{n+1}-p\parallel }^{2}& =& {\alpha }_{n}^{2}{\parallel {x}_{n}-p\parallel }^{2}+{\left(1-{\alpha }_{n}\right)}^{2}{\parallel S{w}_{n}-p\parallel }^{2}+2{\alpha }_{n}\left(1-{\alpha }_{n}\right)〈S{w}_{n}-Sp,{x}_{n}-p〉\\ \le & {\alpha }_{n}^{2}{\parallel {x}_{n}-p\parallel }^{2}+{\left(1-{\alpha }_{n}\right)}^{2}{\parallel {w}_{n}-p\parallel }^{2}+2{\alpha }_{n}\left(1-{\alpha }_{n}\right){\parallel {x}_{n}-p\parallel }^{2}\\ \le & {\alpha }_{n}^{2}{\parallel {x}_{n}-p\parallel }^{2}+{\left(1-{\alpha }_{n}\right)}^{2}{\parallel {x}_{n}-p\parallel }^{2}+2{\alpha }_{n}\left(1-{\alpha }_{n}\right){\parallel {x}_{n}-p\parallel }^{2}\\ -{\left(1-{\alpha }_{n}\right)}^{2}{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\\ =& {\parallel {x}_{n}-p\parallel }^{2}-{\left(1-{\alpha }_{n}\right)}^{2}{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\\ -{\left(1-{\alpha }_{n}\right)}^{2}{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({y}_{n}\right)\parallel }^{2}\\ \le & {\parallel {x}_{n}-p\parallel }^{2}.\end{array}$
(3.7)
Since the sequence $\left\{\parallel {x}_{n}-p\parallel \right\}$ is a bounded and nonincreasing sequence, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-p\parallel$ exists. Hence $\left\{{x}_{n}\right\}$ is bounded. Consequently, the sets $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$, $\left\{{w}_{n}\right\}$, $\left\{{y}_{n}\right\}$ are also bounded. By (3.7), we have
${\left(1-{\alpha }_{n}\right)}^{2}{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\le {\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n+1}-p\parallel }^{2}.$
From the conditions (i) and (ii), there must exist a constant ${M}_{1}>0$ such that
${M}_{1}{\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\le {\left(1-{\alpha }_{n}\right)}^{2}{\left(1-{\beta }_{n}\right)}^{2}\left(1-\frac{{\lambda }_{n}}{2\alpha }\right){\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\le {\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n+1}-p\parallel }^{2},$
from which it follows that
${M}_{1}\sum _{n=1}^{\mathrm{\infty }}{\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel }^{2}\le \sum _{n=1}^{\mathrm{\infty }}\left[{\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n+1}-p\parallel }^{2}\right]={\parallel {x}_{1}-p\parallel }^{2}<\mathrm{\infty }.$
Hence, ${lim}_{n\to \mathrm{\infty }}{R}_{{\lambda }_{n}}\left({u}_{n}\right)={lim}_{n\to \mathrm{\infty }}\parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel =0$. Since ${R}_{{\lambda }_{n}}\left({u}_{n}\right)={u}_{n}-{P}_{C}\left({u}_{n}-{\lambda }_{n}A{u}_{n}\right)={u}_{n}-{y}_{n}$, ${lim}_{n\to \mathrm{\infty }}\parallel {u}_{n}-{y}_{n}\parallel =0$. Notice that ${\lambda }_{n}\ge a$, then by Lemma 2.3, $\parallel {R}_{a}\left({u}_{n}\right)\parallel \le \parallel {R}_{{\lambda }_{n}}\left({u}_{n}\right)\parallel$. Therefore,
$\underset{n\to \mathrm{\infty }}{lim}{R}_{a}\left({u}_{n}\right)=\underset{n\to \mathrm{\infty }}{lim}{R}_{{\lambda }_{n}}\left({u}_{n}\right)=0.$
(3.8)
By the same way, we also get that
$\underset{n\to \mathrm{\infty }}{lim}\parallel {R}_{{\lambda }_{n}}\left({y}_{n}\right)\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {y}_{n}-{v}_{n}\parallel =0,$
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
$\begin{array}{rcl}{\parallel {w}_{n}-p\parallel }^{2}& =& {\beta }_{n}{\parallel {x}_{n}-p\parallel }^{2}+\left(1-{\beta }_{n}\right){\parallel {v}_{n}-p\parallel }^{2}-{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel {x}_{n}-{v}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-p\parallel }^{2}-{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel {x}_{n}-{v}_{n}\parallel }^{2},\end{array}$
which implies that
$\begin{array}{rl}{\parallel {x}_{n+1}-p\parallel }^{2}& ={\alpha }_{n}{\parallel {x}_{n}-p\parallel }^{2}+\left(1-{\alpha }_{n}\right){\parallel {w}_{n}-p\parallel }^{2}-{\alpha }_{n}\left(1-{\alpha }_{n}\right){\parallel S{w}_{n}-{x}_{n}\parallel }^{2}\\ \le {\parallel {x}_{n}-p\parallel }^{2}-{\alpha }_{n}\left(1-{\alpha }_{n}\right){\parallel S{w}_{n}-{x}_{n}\parallel }^{2}-{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel {x}_{n}-{v}_{n}\parallel }^{2}.\end{array}$
(3.10)
Thus, it follows from (3.10) that
${\alpha }_{n}\left(1-{\alpha }_{n}\right){\parallel S{w}_{n}-{x}_{n}\parallel }^{2}\le {\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n+1}-p\parallel }^{2}.$
From the condition (ii), there exists a constant ${M}_{2}>0$ such that
${M}_{2}{\parallel S{w}_{n}-{x}_{n}\parallel }^{2}\le {\alpha }_{n}\left(1-{\alpha }_{n}\right){\parallel S{w}_{n}-{x}_{n}\parallel }^{2}\le {\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n+1}-p\parallel }^{2},$
from which it follows that
${M}_{2}\sum _{n=1}^{\mathrm{\infty }}{\parallel S{w}_{n}-{x}_{n}\parallel }^{2}\le \sum _{n=1}^{\mathrm{\infty }}\left[{\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n+1}-p\parallel }^{2}\right]={\parallel {x}_{1}-p\parallel }^{2}<\mathrm{\infty }.$
Hence
$\underset{n\to \mathrm{\infty }}{lim}\parallel S{w}_{n}-{x}_{n}\parallel =0.$
(3.11)
From (3.10), we also get that
${\beta }_{n}\left(1-{\beta }_{n}\right){\parallel {x}_{n}-{v}_{n}\parallel }^{2}\le {\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n+1}-p\parallel }^{2}.$
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
$\parallel {x}_{n+1}-{x}_{n}\parallel =\left(1-{\alpha }_{n}\right)\parallel S{w}_{n}-{x}_{n}\parallel \le \left(1-c\right)\parallel S{w}_{n}-{x}_{n}\parallel \to 0\phantom{\rule{1em}{0ex}}\left(n\to \mathrm{\infty }\right).$
(3.15)

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

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

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 ${R}_{a}\left(x\right)=x-{P}_{C}\left[x-aAx\right]$ is also continuous. Notice that ${\rho }_{n}\ge a$, then by Lemma 2.3, $\parallel {R}_{x}\left({x}_{n}\right)\parallel \le \parallel {R}_{{\rho }_{n}}\left({x}_{n}\right)\parallel$. Then from Step 1,
$\underset{k\to \mathrm{\infty }}{lim}\parallel {R}_{x}\left({x}_{{n}_{k}}\right)\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {R}_{{\rho }_{n}}\left({x}_{{n}_{k}}\right)\parallel =0.$
Therefore from the continuity of ${R}_{a}\left(x\right)$,
${R}_{a}\left({p}^{\ast }\right)=\underset{n\to \mathrm{\infty }}{lim}{R}_{a}\left({x}_{{n}_{k}}\right)=0.$

This shows that ${p}^{\ast }$ is a solution of the variational inequality (1.6), that is ${p}^{\ast }\in \mathit{VI}\left(C,A\right)$. 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 ${p}^{\ast }\in Fix\left(S\right)$. Finally, by the same argument as in the proof of [, Theorem 3.1], we prove that ${p}^{\ast }\in \mathit{GMEP}\left(F,\phi ,B\right)$. Thus ${p}^{\ast }\in \mathrm{\Omega }=Fix\left(S\right)\cap \mathit{VI}\left(C,A\right)\cap \mathit{GMEP}\left(F,\phi ,B\right)$.

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
$\begin{array}{rcl}{\parallel {x}_{n+1}-{p}^{\ast }\parallel }^{2}& =& 〈{\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)S{w}_{n}-{p}^{\ast },{x}_{n+1}-{p}^{\ast }〉\\ =& {\alpha }_{n}〈{x}_{n}-{p}^{\ast },{x}_{n+1}-{p}^{\ast }〉+\left(1-{\alpha }_{n}\right)〈S{w}_{n}-{p}^{\ast },{x}_{n+1}-{p}^{\ast }〉\\ \le & {\alpha }_{n}{\parallel {x}_{n}-{p}^{\ast }\parallel }^{2}+\left(1-{\alpha }_{n}\right)〈S{w}_{n}-{p}^{\ast },{x}_{n+1}-{p}^{\ast }〉\\ \le & {\alpha }_{n}{\parallel {x}_{n}-{p}^{\ast }\parallel }^{2}+\left(1-{\alpha }_{n}\right)〈S{w}_{n}-{p}^{\ast },{x}_{n+1}-{x}_{n}〉\\ +\left(1-{\alpha }_{n}\right)〈S{w}_{n}-{p}^{\ast },{x}_{n}-{p}^{\ast }〉\\ \le & {\alpha }_{n}{\parallel {x}_{n}-{p}^{\ast }\parallel }^{2}+\left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{p}^{\ast }\parallel }^{2}+\left(1-{\alpha }_{n}\right)〈S{w}_{n}-{p}^{\ast },{x}_{n+1}-{x}_{n}〉\\ =& {\parallel {x}_{n}-{p}^{\ast }\parallel }^{2}+\left(1-{\alpha }_{n}\right)〈S{w}_{n}-{p}^{\ast },{x}_{n+1}-{x}_{n}〉,\end{array}$
which implies
$\begin{array}{rl}{\parallel {x}_{n+1}-{p}^{\ast }\parallel }^{2}-{\parallel {x}_{n}-{p}^{\ast }\parallel }^{2}& \le \left(1-{\alpha }_{n}\right)〈S{w}_{n}-{p}^{\ast },{x}_{n+1}-{x}_{n}〉\\ \le \left(1-c\right)〈S{w}_{n}-{p}^{\ast },{x}_{n+1}-{x}_{n}〉.\end{array}$
(3.16)
From $S{w}_{{n}_{k}}⇀{p}^{\ast }$ and ${x}_{{n}_{k+1}}-{x}_{{n}_{k}}\to 0$ as $k\to \mathrm{\infty }$, it follows from (3.16) that
$\parallel {x}_{{n}_{k+1}}-{p}^{\ast }\parallel \to \parallel {x}_{{n}_{k}}-{p}^{\ast }\parallel \phantom{\rule{1em}{0ex}}\left(k\to \mathrm{\infty }\right).$

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 $\left\{{x}_{n}\right\}$ generated by Algorithm (3.1) converges strongly to ${p}^{\ast }\in \mathrm{\Omega }=Fix\left(S\right)\cap \mathit{VI}\left(C,A\right)\cap \mathit{GMEP}\left(F,\phi ,B\right)$.

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 $\mathrm{\Omega }=Fix\left(S\right)\cap \mathit{VI}\left(C,A\right)\ne \mathrm{\varnothing }$, let $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$ be sequences generated by
$\left\{\begin{array}{c}{x}_{1}=x\in C,\hfill \\ {y}_{n}={P}_{C}\left({x}_{n}-{\lambda }_{n}A{x}_{n}\right),\hfill \\ {x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)S\left[{\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){P}_{C}\left({y}_{n}-{\lambda }_{n}A{y}_{n}\right)\right]\hfill \end{array}$

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

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  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 .

Theorem 3.3 Let C, H, F, A, B, S be as in Theorem  3.1. Assume $\mathrm{\Omega }=Fix\left(S\right)\cap \mathit{VI}\left(C,A\right)\cap \mathit{GEP}\left(F,B\right)\ne \mathrm{\varnothing }$; let $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$ and $\left\{{u}_{n}\right\}$ be sequences generated by
$\left\{\begin{array}{c}{x}_{1}=x\in C,\hfill \\ F\left({u}_{n},y\right)+〈B{x}_{n},y-{u}_{n}〉+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {y}_{n}={P}_{C}\left({u}_{n}-{\lambda }_{n}A{u}_{n}\right),\hfill \\ {x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)S\left[{\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){P}_{C}\left({y}_{n}-{\lambda }_{n}A{y}_{n}\right)\right]\hfill \end{array}$

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

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  because of removing the iterative step ${C}_{n}$ in the algorithm of Theorem 3.1 of .

Theorem 3.4 Let C, H, F, A, S be as in Theorem  3.1. Assume that $\mathrm{\Omega }=Fix\left(S\right)\cap \mathit{VI}\left(C,A\right)\cap \mathit{EP}\left(F\right)\ne \mathrm{\varnothing }$; let $\left\{{x}_{n}\right\}$ and $\left\{{u}_{n}\right\}$ be sequences generated by
$\left\{\begin{array}{c}{x}_{1}=x\in C,\hfill \\ F\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {y}_{n}={P}_{C}\left({u}_{n}-{\lambda }_{n}A{u}_{n}\right),\hfill \\ {x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)S{P}_{C}\left({y}_{n}-{\lambda }_{n}A{y}_{n}\right)\hfill \end{array}$

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

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 .

## Declarations

### Acknowledgements

The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article. The first author was supported by the Natural Science Foundational Committee of Qinhuangdao city (201101A453) and Hebei Normal University of Science and Technology (ZDJS 2009 and CXTD2010-05). The fifth author was supported by the Natural Science Foundational Committee of Qinhuangdao city (2012025A034).

## Authors’ Affiliations

(1)
College of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao, 066004, China
(2)
Institute of Mathematics and Systems Science, Hebei Normal University of Science and Technology, Qinhuangdao, 066004, China
(3)
Hebei Business and Trade School, Shijiazhuang, 050000, China

## References 