# General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions

## Abstract

In this paper, we modify the general iterative method to approximate a common element of the set of solutions of generalized equilibrium problems and the set of common fixed points of a finite family of k-strictly pseudo-contractive nonself mappings. Strong convergence theorems are established under some suitable conditions in a real Hilbert space, which also solves some variation inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

MSC:47H05, 47H09, 47H10.

## 1 Introduction

Let H be a real Hilbert space with inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$, respectively. Let K be a nonempty closed convex subset of H. Let $A:K\to H$ be a nonlinear mapping and $F:K×K\to \mathbb{R}$ be a bi-function, where $\mathbb{R}$ denotes the set of real numbers. We consider the following generalized equilibrium problem: Find $x\in K$ such that

$F\left(x,y\right)+〈Ax,y-x〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in K.$
(1.1)

We use $\mathit{EP}\left(F,A\right)$ to denote the solution set of the problem (1.1). If $A\equiv 0$, the zero mapping, then the problem (1.1) is reduced to the normal equilibrium problem: Find $x\in K$ such that

$F\left(x,y\right)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in K.$
(1.2)

We use $\mathit{EP}\left(F\right)$ to denote the solution set of the problem (1.2). If $F\equiv 0$, then the problem (1.1) is reduced to the classical variational inequality problem: Find $x\in K$ such that

$〈Ax,y-x〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in K.$

The generalized equilibrium problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, mini-max problems, the Nash equilibrium problem in noncooperative games and others (see, e.g., ).

Recall that a nonself mapping $T:K\to H$ is called a k-strict pseudo-contraction if there exists a constant $k\in \left[0,1\right)$ such that

${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+k{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in K.$
(1.3)

We use $F\left(T\right)$ to denote the fixed point set of T, i.e., $F\left(T\right):=\left\{x\in K:Tx=x\right\}$. As $k=0$, T is said to be nonexpansive, i.e.,

$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in K.$

T is said to be pseudo-contractive if $k=1$, and is also said to be strongly pseudo-contractive if there exists a positive constant $\lambda \in \left(0,1\right)$ such that $T+\lambda I$ is pseudo-contractive. Clearly, the class of k-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions. We remark also that the class of strongly pseudo-contractive mappings is independent of the class of k-strict pseudo-contractions (see, e.g., [4, 5]).

Iterative methods for equilibrium problems and fixed point problems of nonexpansive mappings have been extensively investigated. However, iterative schemes for strict pseudo-contractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn  initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1.3) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudo-contraction. On the other hand, strict pseudo-contractions have more powerful applications than nonexpansive mappings do in solving inverse problems; see, e.g., [618, 2027] and the references therein. Therefore it is interesting to develop the effective iterative methods for equilibrium problems and fixed point problems of strict pseudo-contractions.

In 2006, Marino and Xu  introduced a general iterative method and proved that for a given ${x}_{0}\in H$, the sequence $\left\{{x}_{n}\right\}$ generated by

${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}B\right)T{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in N,$

where T is a self-nonexpansive mapping on H, f is a contraction of H into itself and $\left\{{\alpha }_{n}\right\}\subseteq \left(0,1\right)$ satisfies certain conditions, B is a strongly positive bounded linear operator on H, converges strongly to ${x}^{\ast }\in F\left(T\right)$, which is the unique solution of the following variational inequality:

$〈\left(B-\gamma f\right){x}^{\ast },{x}^{\ast }-x〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in F\left(T\right),$

and is also the optimality condition for some minimization problem.

Recently, Takahashi and Takahashi  considered the equilibrium problem and nonexpansive mapping by viscosity approximation methods. To be more precise, they proved the following theorem.

Theorem of TT Let K be a nonempty closed convex subset of H. Let F be a bi-function from $K×K$ to $\mathbb{R}$ satisfying (A 1)-(A 4) and let $T:K\to H$ be a nonexpansive mapping such that $F\left(T\right)\cap \mathit{EP}\left(F\right)\ne \varphi$. Let $f:H\to H$ be a contraction and let $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ be sequences generated by ${x}_{1}\in H$ and

$\left\{\begin{array}{c}F\left({y}_{n},z\right)+\frac{1}{{r}_{n}}〈z-{y}_{n},{y}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }z\in K,\hfill \\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)T{y}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$

where $\left\{{\alpha }_{n}\right\}\subset \left[0,1\right]$ and $\left\{{r}_{n}\right\}$ satisfy Then $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ converge strongly to $q\in F\left(T\right)\cap \mathit{EP}\left(F\right)$, where $q={P}_{F\left(T\right)\cap \mathit{EP}\left(F\right)}f\left(q\right)$.

In 2009, Ceng et al.  further studied the equilibrium problem and fixed point problems of strict pseudo-contraction mappings T by an iterative scheme for finding an element of $\mathit{EP}\left(F\right)\cap F\left(T\right)$. Very recently, by using the general iterative method Liu  proposed the implicit and explicit iterative processes for finding an element of $\mathit{EP}\left(F\right)\cap F\left(T\right)$ and then obtained some strong convergence theorems, respectively. On the other hand, Takahashi and Takahashi  considered the generalized equilibrium problem and nonexpansive mapping in a Hilbert space. Moreover, they constructed an iterative scheme for finding an element of $\mathit{EP}\left(F,A\right)\cap F\left(T\right)$ and then proved a strong convergence of the iterative sequence under some suitable conditions.

In this paper, inspired and motivated by research going on in this area, we introduce a general iterative method for generalized equilibrium problems and strict pseudo-contractive nonself mappings, which is defined in the following way:

$\left\{\begin{array}{c}F\left({u}_{n},y\right)+〈A{x}_{n},y-{u}_{n}〉+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in K,\hfill \\ {y}_{n}={\beta }_{n}{u}_{n}+\left(1-{\beta }_{n}\right){\sum }_{i=1}^{N}{\eta }_{i}^{\left(n\right)}{T}_{i}{u}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}B\right){y}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$
(1.4)

where constant $\gamma >0$, f is a contraction and A, B are two operators, ${\left\{{T}_{i}\right\}}_{i=i}^{N}:K\to H$ is a finite family of ${k}_{i}$-strict pseudo-contractions, ${\left\{{\eta }_{i}^{\left(n\right)}\right\}}_{i=1}^{N}$ is a finite sequence of positive numbers, $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$ and $\left\{{r}_{n}\right\}$ are some sequences with certain conditions.

Our purpose is not only to modify the general iterative method to the case of a finite family of ${k}_{i}$-strictly pseudo-contractive nonself mappings, but also to establish strong convergence theorems for a generalized equilibrium problem and ${k}_{i}$-strict pseudo-contractions in a real Hilbert space, which also solves some variation inequality problems. Our theorems presented in this paper improve and extend the corresponding results of [12, 15, 16, 18, 20, 21, 25].

## 2 Preliminaries

Let K be a nonempty closed convex subset of a real Hilbert H space with inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$, respectively. Recall that a mapping $f:K\to K$ is a contraction, if there exists a constant $\rho \in \left(0,1\right)$ such that

$\parallel f\left(x\right)-f\left(y\right)\parallel \le \rho \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in K.$

We use ${\mathrm{\Pi }}_{K}$ to denote the collection of all contractions on K. The operator $A:K\to H$ is said to be monotone if

$〈Ax-Ay,x-y〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in K.$

$A:K\to H$ is said to be r-strongly monotone if there exists a constant $r>0$ such that

$〈Ax-Ay,x-y〉\ge r{\parallel x-y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in K.$

$A:K\to H$ is said to be α-inverse strongly monotone if there exists a constant $\alpha >0$ such that

$〈Ax-Ay,x-y〉\ge \alpha {\parallel Ax-Ay\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in K.$

Recall that an operator B is strongly positive if there exists a constant $\overline{\gamma }>0$ with the property

$〈Bx,x〉\ge \overline{\gamma }{\parallel x\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in H.$

To study the generalized equilibrium problem (1.1), we may assume that the bi-function $F:K×K\to \mathbb{R}$ satisfies the following conditions:

(A1) $F\left(x,x\right)=0$ for all $x\in K$;

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

(A3) for each $x,y,z\in K$, ${lim}_{t\to 0}F\left(tz+\left(1-t\right)x,y\right)\le F\left(x,y\right)$;

(A4) for each $x\in K$, $y↦F\left(x,y\right)$ is convex and lower semi-continuous.

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

Lemma 2.1 [1, 3]

Let $F:K×K\to \mathbb{R}$ be a bi-function satisfying (A 1)-(A 4). Then, for any $r>0$ and $x\in H$, there exists $z\in K$ such that

$F\left(z,y\right)+\frac{1}{r}〈y-z,z-x〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in K.$

Further, if ${T}_{r}x=\left\{z\in K:F\left(z,y\right)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in K\right\}$, then the following hold:

1. (1)

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

2. (2)

${T}_{r}$ is firmly nonexpansive, i.e, ${\parallel {T}_{r}x-{T}_{r}y\parallel }^{2}\le 〈{T}_{r}x-{T}_{r}y,x-y〉$ for all $x,y\in H$;

3. (3)

$F\left({T}_{r}\right)=\mathit{EP}\left(F\right)$;

4. (4)

$\mathit{EP}\left(F\right)$ is closed and convex.

Lemma 2.2 

In the Hilbert space H, there hold the following identities:

1. (i)

${\parallel x+y\parallel }^{2}={\parallel x\parallel }^{2}+2〈x,y〉+{\parallel y\parallel }^{2}\le {\parallel x\parallel }^{2}+2〈y,\left(x+y\right)〉$, $\mathrm{\forall }x,y\in H$;

2. (ii)

${\parallel tx+\left(1-t\right)y\parallel }^{2}=t{\parallel x\parallel }^{2}+\left(1-t\right){\parallel y\parallel }^{2}-t\left(1-t\right){\parallel x-y\parallel }^{2}$, $\mathrm{\forall }t\in \left[0,1\right]$, $\mathrm{\forall }x,y\in H$.

Lemma 2.3 

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

Lemma 2.4 

If $T:K\to H$ is a k-strict pseudo-contraction, then the fixed point set $F\left(T\right)$ is closed convex so that the projection ${P}_{F\left(T\right)}$ is well defined.

Lemma 2.5 [2, 10]

Let $T:K\to H$ be a k-strict pseudo-contraction. For $\lambda \in \left[k,1\right)$, define $S:K\to H$ by $Sx=\lambda x+\left(1-\lambda \right)Tx$ for each $x\in K$. Then S is a nonexpansive mapping such that $F\left(S\right)=F\left(T\right)$.

Lemma 2.6 

Assume $\left\{{a}_{n}\right\}$ is a sequence of nonnegative real numbers such that

${a}_{n+1}\le \left(1-{\gamma }_{n}\right){a}_{n}+{\gamma }_{n}{\delta }_{n},\phantom{\rule{1em}{0ex}}n\ge 0,$

where $\left\{{\gamma }_{n}\right\}$ is a sequence in $\left(0,1\right)$ and $\left\{{\delta }_{n}\right\}$ is a real sequence such that

1. (i)

${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_{n}=\mathrm{\infty }$;

2. (ii)

${lim sup}_{n\to \mathrm{\infty }}{\delta }_{n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\gamma }_{n}{\delta }_{n}|<\mathrm{\infty }$.

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

Proposition 2.1 (See, e.g., Acedo and Xu )

Let K be a nonempty closed convex subset of the Hilbert space H. Given an integer $N\ge 1$, assume that ${\left\{{T}_{i}\right\}}_{i=1}^{N}:K\to H$ is a finite family of ${k}_{i}$-strict pseudo-contractions. Suppose that ${\left\{{\lambda }_{i}\right\}}_{i=1}^{N}$ is a positive sequence such that ${\sum }_{i=1}^{N}{\lambda }_{i}=1$. Then ${\sum }_{i=1}^{N}{\lambda }_{i}{T}_{i}$ is a k-strict pseudo-contraction with $k=max\left\{{k}_{i}:1\le i\le N\right\}$.

Proposition 2.2 (See, e.g., Acedo and Xu )

Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ and ${\left\{{\lambda }_{i}\right\}}_{i=1}^{N}$ be given as in Proposition 2.1 above. Then $F\left({\sum }_{i=1}^{N}{\lambda }_{i}{T}_{i}\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$.

## 3 Main results

Theorem 3.1 Let K be a nonempty closed convex subset of the Hilbert space H and $F:K×K\to \mathbb{R}$ be a bi-function satisfying (A 1)-(A 4). Let A be an α-inverse strongly monotone mapping and B be a strongly positive bounded linear operator on H with $\overline{\gamma }>0$. Assume that ${\left\{{T}_{i}\right\}}_{i=1}^{N}:K\to H$ be a finite family of ${k}_{i}$-strict pseudo-contractions such that $\mathcal{F}={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap \mathit{EP}\left(F,A\right)\ne \varphi$. Suppose $f\in {\mathrm{\Pi }}_{K}$ with a coefficient $\rho \in \left(0,1\right)$ and ${\left\{{\eta }_{i}^{\left(n\right)}\right\}}_{i=1}^{N}$ are finite sequences of positive numbers such that ${\sum }_{i=1}^{N}{\eta }_{i}^{\left(n\right)}=1$ for all $n\ge 0$, for a given point ${x}_{0}\in K$, ${\alpha }_{n},{\beta }_{n}\in \left(0,1\right)$, ${r}_{n}\in \left(0,2\alpha \right)$ and $0<\gamma <\frac{\overline{\gamma }}{\rho }$, the following control conditions are satisfied:

1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n}-{\alpha }_{n-1}|<\mathrm{\infty }$;

2. (ii)

${k}_{i}\le {\beta }_{n}\le \lambda <1$, ${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=\lambda$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n}-{\beta }_{n-1}|<\mathrm{\infty }$;

3. (iii)

${\sum }_{n=1}^{\mathrm{\infty }}{\sum }_{i=1}^{N}|{\eta }_{i}^{\left(n\right)}-{\eta }_{i}^{\left(n-1\right)}|<\mathrm{\infty }$;

4. (iv)

${lim inf}_{n\to \mathrm{\infty }}{r}_{n}>0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{r}_{n}-{r}_{n-1}|<\mathrm{\infty }$.

Then the sequence $\left\{{x}_{n}\right\}$ generated by (1.4) converges strongly to $q\in \mathcal{F}$, which solves the variational inequality

$〈\left(B-\gamma f\right)q,q-p〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in \mathcal{F}.$

Proof Putting ${W}_{n}={\sum }_{i=1}^{N}{\eta }_{i}^{\left(n\right)}{T}_{i}$, we have ${W}_{n}:K\to H$ is a k-strict pseudo-contraction and $F\left({W}_{n}\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$ by Proposition 2.1 and 2.2, where $k=max\left\{{k}_{i}:1\le i\le N\right\}$.

First, we show that the mapping $I-{r}_{n}A$ is nonexpansive. Indeed, for each $x,y\in K$, we have

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

It follows from the condition ${r}_{n}\in \left(0,2\alpha \right)$ that the mapping $I-{r}_{n}A$ is nonexpansive. From Lemma 2.1, we see that $\mathit{EP}\left(F,A\right)=F\left({T}_{{r}_{n}}\left(I-{r}_{n}A\right)\right)$. Note that ${u}_{n}$ can be rewritten as ${u}_{n}={T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}$ and $p={T}_{{r}_{n}}\left(I-{r}_{n}A\right)p$ for each $n\ge 1$ as $p\in \mathcal{F}$.

From (1.4), condition (ii) and Lemma 2.2, we have

$\begin{array}{rcl}{\parallel {y}_{n}-p\parallel }^{2}& =& {\parallel {\beta }_{n}\left({u}_{n}-p\right)+\left(1-{\beta }_{n}\right)\left({W}_{n}{u}_{n}-p\right)\parallel }^{2}\\ =& {\beta }_{n}{\parallel {u}_{n}-p\parallel }^{2}+\left(1-{\beta }_{n}\right){\parallel {W}_{n}{u}_{n}-p\parallel }^{2}-{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel {u}_{n}-{W}_{n}{u}_{n}\parallel }^{2}\\ \le & {\beta }_{n}{\parallel {u}_{n}-p\parallel }^{2}+\left(1-{\beta }_{n}\right)\left[{\parallel {u}_{n}-p\parallel }^{2}+k{\parallel {u}_{n}-{W}_{n}{u}_{n}\parallel }^{2}\right]\\ -{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel {u}_{n}-{W}_{n}{u}_{n}\parallel }^{2}\\ =& {\parallel {u}_{n}-p\parallel }^{2}-\left(1-{\beta }_{n}\right)\left({\beta }_{n}-k\right){\parallel {u}_{n}-{W}_{n}{u}_{n}\parallel }^{2}\\ \le & {\parallel {u}_{n}-p\parallel }^{2}.\end{array}$
(3.1)

By ${u}_{n}={T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}$, we obtain

$\parallel {u}_{n}-p\parallel =\parallel {T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}-p\parallel \le \parallel {x}_{n}-p\parallel .$

This together with (3.1), we see that

$\parallel {y}_{n}-p\parallel \le \parallel {u}_{n}-p\parallel \le \parallel {x}_{n}-p\parallel .$
(3.2)

Furthermore, by Lemma 2.3, we have

$\begin{array}{rcl}\parallel {x}_{n+1}-p\parallel & =& \parallel {\alpha }_{n}\left[\gamma f\left({x}_{n}\right)-Bp\right]+\left(I-{\alpha }_{n}B\right)\left({y}_{n}-p\right)\parallel \\ \le & \left(1-{\alpha }_{n}\overline{\gamma }\right)\parallel {y}_{n}-p\parallel +{\alpha }_{n}\parallel \gamma f\left({x}_{n}\right)-Bp\parallel \\ \le & \left(1-{\alpha }_{n}\overline{\gamma }\right)\parallel {y}_{n}-p\parallel +{\alpha }_{n}\left[\parallel \gamma f\left({x}_{n}\right)-\gamma f\left(p\right)\parallel +\parallel \gamma f\left(p\right)-Bp\parallel \right]\\ \le & \left[1-\left(\overline{\gamma }-\gamma \rho \right){\alpha }_{n}\right]\parallel {x}_{n}-p\parallel +{\alpha }_{n}\parallel \gamma f\left(p\right)-Bp\parallel .\end{array}$

It follows from induction that

$\parallel {x}_{n}-p\parallel \le max\left\{\parallel {x}_{0}-p\parallel ,\frac{1}{\overline{\gamma }-\gamma \rho }\parallel \gamma f\left(p\right)-Bp\parallel \right\},\phantom{\rule{1em}{0ex}}n\ge 1,$
(3.3)

which gives that sequence $\left\{{x}_{n}\right\}$ is bounded, and so are $\left\{{u}_{n}\right\}$ and $\left\{{y}_{n}\right\}$.

Define a mapping ${S}_{n}x:={\beta }_{n}x+\left(1-{\beta }_{n}\right){W}_{n}x$ for each $x\in K$. Then ${S}_{n}:K\to H$ is nonexpansive. Indeed, by using (1.3), Lemma 2.2 and condition (ii), we have for all $x,y\in K$ that

$\begin{array}{rcl}{\parallel {S}_{n}x-{S}_{n}y\parallel }^{2}& =& {\parallel {\beta }_{n}\left(x-y\right)+\left(1-{\beta }_{n}\right)\left({W}_{n}x-{W}_{n}y\right)\parallel }^{2}\\ =& {\beta }_{n}{\parallel x-y\parallel }^{2}+\left(1-{\beta }_{n}\right){\parallel {W}_{n}x-{W}_{n}y\parallel }^{2}\\ -{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel x-{W}_{n}x-\left(y-{W}_{n}y\right)\parallel }^{2}\\ \le & {\beta }_{n}{\parallel x-y\parallel }^{2}+\left(1-{\beta }_{n}\right)\left[{\parallel x-y\parallel }^{2}+k{\parallel x-{W}_{n}x-\left(y-{W}_{n}y\right)\parallel }^{2}\right]\\ -{\beta }_{n}\left(1-{\beta }_{n}\right){\parallel x-{W}_{n}x-\left(y-{W}_{n}y\right)\parallel }^{2}\\ =& {\parallel x-y\parallel }^{2}-\left(1-{\beta }_{n}\right)\left({\beta }_{n}-k\right){\parallel x-{W}_{n}x-\left(y-{W}_{n}y\right)\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2},\end{array}$

which shows that ${S}_{n}:K\to H$ is nonexpansive.

Next, we show that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n+1}-{x}_{n}\parallel =0$. From (1.4) and Lemma 2.3, we have

$\begin{array}{rcl}\parallel {x}_{n+1}-{x}_{n}\parallel & =& \parallel {\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}B\right){y}_{n}-\left[{\alpha }_{n-1}\gamma f\left({x}_{n-1}\right)+\left(I-{\alpha }_{n-1}B\right){y}_{n-1}\right]\parallel \\ \le & {\alpha }_{n}\gamma \parallel f\left({x}_{n}\right)-f\left({x}_{n-1}\right)\parallel +|{\alpha }_{n}-{\alpha }_{n-1}|\left[\gamma \parallel f\left({x}_{n-1}\right)\parallel +\parallel B{y}_{n-1}\parallel \right]\\ +\parallel \left(I-{\alpha }_{n}B\right)\left({y}_{n}-{y}_{n-1}\right)\parallel \\ \le & {\alpha }_{n}\gamma \rho \parallel {x}_{n}-{x}_{n-1}\parallel +|{\alpha }_{n}-{\alpha }_{n-1}|{M}_{1}+\left(1-{\alpha }_{n}\overline{\gamma }\right)\parallel {y}_{n}-{y}_{n-1}\parallel ,\end{array}$
(3.4)

where ${M}_{1}={sup}_{n\ge 1}\left\{\gamma \parallel f\left({x}_{n}\right)\parallel +\parallel B{y}_{n}\parallel \right\}<\mathrm{\infty }$. Moreover, we note that ${y}_{n}={S}_{n}{u}_{n}$ and

$\begin{array}{rcl}\parallel {y}_{n}-{y}_{n-1}\parallel & \le & \parallel {S}_{n}{u}_{n}-{S}_{n}{u}_{n-1}\parallel +\parallel {S}_{n}{u}_{n-1}-{S}_{n-1}{u}_{n-1}\parallel \\ \le & \parallel {u}_{n}-{u}_{n-1}\parallel +\parallel {\beta }_{n}{u}_{n-1}+\left(1-{\beta }_{n}\right){W}_{n}{u}_{n-1}\\ -\left[{\beta }_{n-1}{u}_{n-1}+\left(1-{\beta }_{n-1}\right){W}_{n-1}{u}_{n-1}\right]\parallel \\ \le & \parallel {u}_{n}-{u}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel {u}_{n-1}-{W}_{n-1}{u}_{n-1}\parallel \\ +\left(1-{\beta }_{n}\right)\parallel {W}_{n}{u}_{n-1}-{W}_{n-1}{u}_{n-1}\parallel \\ \le & \parallel {u}_{n}-{u}_{n-1}\parallel +|{\beta }_{n}-{\beta }_{n-1}|{M}_{2}+\left(1-{\beta }_{n}\right)\sum _{i=1}^{N}|{\eta }_{i}^{\left(n\right)}-{\eta }_{i}^{\left(n-1\right)}|\parallel {T}_{i}{u}_{n-1}\parallel ,\end{array}$
(3.5)

where ${M}_{2}={sup}_{n\ge 1}\left\{\parallel {u}_{n-1}-{W}_{n-1}{u}_{n-1}\parallel \right\}$. On the other hand, we note that

$\left\{\begin{array}{c}F\left({u}_{n},y\right)+〈A{x}_{n},y-{u}_{n}〉+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\hfill \\ F\left({u}_{n-1},y\right)+〈A{x}_{n-1},y-{u}_{n-1}〉+\frac{1}{{r}_{n-1}}〈y-{u}_{n-1},{u}_{n-1}-{x}_{n-1}〉\ge 0.\hfill \end{array}$
(3.6)

Putting $y={u}_{n-1}$ and $y={u}_{n}$ in (3.6) respectively, we have

$\left\{\begin{array}{c}F\left({u}_{n},{u}_{n-1}\right)+〈A{x}_{n},{u}_{n-1}-{u}_{n}〉+\frac{1}{{r}_{n}}〈{u}_{n-1}-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\hfill \\ F\left({u}_{n-1},{u}_{n}\right)+〈A{x}_{n-1},{u}_{n}-{u}_{n-1}〉+\frac{1}{{r}_{n-1}}〈{u}_{n}-{u}_{n-1},{u}_{n-1}-{x}_{n-1}〉\ge 0.\hfill \end{array}$
(3.7)

It follows from (A2) that

$〈{u}_{n}-{u}_{n-1},\frac{{u}_{n-1}-\left(I-{r}_{n-1}A\right){x}_{n-1}}{{r}_{n-1}}-\frac{{u}_{n}-\left(I-{r}_{n}A\right){x}_{n}}{{r}_{n}}〉\ge 0,$

and hence

$〈{u}_{n}-{u}_{n-1},{u}_{n-1}-{u}_{n}+{u}_{n}-\left(I-{r}_{n-1}A\right){x}_{n-1}-\frac{{r}_{n-1}}{{r}_{n}}\left[{u}_{n}-\left(I-{r}_{n}A\right){x}_{n}\right]〉\ge 0.$

Since ${lim}_{n\to \mathrm{\infty }}{r}_{n}>0$, we assume that there exists a real number μ such that ${r}_{n}>\mu >0$ for all $n\in N$. Consequently, we have

$\begin{array}{rcl}{\parallel {u}_{n}-{u}_{n-1}\parallel }^{2}& \le & 〈{u}_{n}-{u}_{n-1},\left(I-{r}_{n}A\right){x}_{n}-\left(I-{r}_{n-1}A\right){x}_{n-1}\\ +\left(1-\frac{{r}_{n-1}}{{r}_{n}}\right)\left[{u}_{n}-\left(I-{r}_{n}A\right){x}_{n}\right]〉\\ \le & \parallel {u}_{n}-{u}_{n-1}\parallel \left[\parallel {x}_{n}-{x}_{n-1}\parallel +|{r}_{n}-{r}_{n-1}|\parallel A{x}_{n-1}\parallel \\ +\frac{{r}_{n}-{r}_{n-1}}{{r}_{n}}\parallel {u}_{n}-\left(I-{r}_{n}A\right){x}_{n}\parallel \right],\end{array}$

and hence

$\begin{array}{rcl}\parallel {u}_{n}-{u}_{n-1}\parallel & \le & \parallel {x}_{n}-{x}_{n-1}\parallel +|{r}_{n}-{r}_{n-1}|\parallel A{x}_{n-1}\parallel +\frac{{r}_{n}-{r}_{n-1}}{{r}_{n}}\parallel {u}_{n}-\left(I-{r}_{n}A\right){x}_{n}\parallel \\ \le & \parallel {x}_{n}-{x}_{n-1}\parallel +|{r}_{n}-{r}_{n-1}|\left[\parallel A{x}_{n-1}\parallel +\frac{1}{\mu }\parallel {u}_{n}-\left(I-{r}_{n}A\right){x}_{n}\parallel \right]\\ \le & \parallel {x}_{n}-{x}_{n-1}\parallel +|{r}_{n}-{r}_{n-1}|{M}_{3},\end{array}$
(3.8)

where ${M}_{3}=sup\left\{\parallel A{x}_{n-1}\parallel +\frac{1}{\mu }\parallel {u}_{n}-\left(I-{r}_{n}A\right){x}_{n}\parallel ,n\in N\right\}$. Combining (3.4), (3.5) and (3.8), we have

$\begin{array}{rcl}\parallel {x}_{n+1}-{x}_{n}\parallel & \le & {\alpha }_{n}\gamma \rho \parallel {x}_{n}-{x}_{n-1}\parallel +|{\alpha }_{n}-{\alpha }_{n-1}|{M}_{1}+\left(1-{\alpha }_{n}\overline{\gamma }\right)\left[\parallel {x}_{n}-{x}_{n-1}\parallel \\ +|{\beta }_{n}-{\beta }_{n-1}|{M}_{2}+|{r}_{n}-{r}_{n-1}|{M}_{3}+\left(1-{\beta }_{n}\right)\sum _{i=1}^{N}|{\eta }_{i}^{\left(n\right)}-{\eta }_{i}^{\left(n-1\right)}|\parallel {T}_{i}{u}_{n-1}\parallel \right]\\ \le & \left[1-\left(\overline{\gamma }-\gamma \rho \right){\alpha }_{n}\right]\parallel {x}_{n}-{x}_{n-1}\parallel +|{\alpha }_{n}-{\alpha }_{n-1}|{M}_{1}+|{\beta }_{n}-{\beta }_{n-1}|{M}_{2}\\ +|{r}_{n}-{r}_{n-1}|{M}_{3}+\sum _{i=1}^{N}|{\eta }_{i}^{\left(n\right)}-{\eta }_{i}^{\left(n-1\right)}|\parallel {T}_{i}{u}_{n-1}\parallel .\end{array}$

It follows from $0<\gamma <\frac{\overline{\gamma }}{\rho }$ and Lemma 2.6 that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-{x}_{n}\parallel =0.$
(3.9)

Moreover, we observe that

$\parallel {x}_{n}-{y}_{n}\parallel \le \parallel {x}_{n}-{x}_{n+1}\parallel +\parallel {x}_{n+1}-{y}_{n}\parallel \le \parallel {x}_{n}-{x}_{n+1}\parallel +{\alpha }_{n}\parallel \gamma f\left({x}_{n}\right)-B{y}_{n}\parallel .$

It follows from ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and (3.9) that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{y}_{n}\parallel =0.$
(3.10)

For $p\in F\left({S}_{n}\right)\cap \mathit{EP}\left(F,A\right)$, we note that ${u}_{n}={T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}$ and

$\begin{array}{rcl}{\parallel {u}_{n}-p\parallel }^{2}& =& {\parallel {T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}-{T}_{{r}_{n}}\left(I-{r}_{n}A\right)p\parallel }^{2}\le 〈{x}_{n}-p,{u}_{n}-p〉\\ =& \frac{1}{2}\left({\parallel {x}_{n}-p\parallel }^{2}+{\parallel {u}_{n}-p\parallel }^{2}-{\parallel {x}_{n}-{u}_{n}\parallel }^{2}\right),\end{array}$

which implies that

${\parallel {u}_{n}-p\parallel }^{2}\le {\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n}-{u}_{n}\parallel }^{2}.$
(3.11)

From (1.4), (3.2) and (3.11), we have

$\begin{array}{rcl}{\parallel {x}_{n+1}-p\parallel }^{2}& =& {\parallel {\alpha }_{n}\left[\gamma f\left({x}_{n}\right)-Bp\right]+\left(I-{\alpha }_{n}B\right)\left({y}_{n}-p\right)\parallel }^{2}\\ \le & {\left(1-{\alpha }_{n}\overline{\gamma }\right)}^{2}{\parallel {y}_{n}-p\parallel }^{2}+{\alpha }_{n}^{2}{\parallel \gamma f\left({x}_{n}\right)-Bp\parallel }^{2}\\ +2{\alpha }_{n}\left(1-{\alpha }_{n}\overline{\gamma }\right)\parallel \gamma f\left({x}_{n}\right)-Bp\parallel \parallel {y}_{n}-p\parallel \\ \le & {\parallel {u}_{n}-p\parallel }^{2}+{\alpha }_{n}^{2}{\parallel \gamma f\left({x}_{n}\right)-Bp\parallel }^{2}+2{\alpha }_{n}\parallel \gamma f\left({x}_{n}\right)-Bp\parallel \parallel {y}_{n}-p\parallel \\ \le & {\parallel {x}_{n}-p\parallel }^{2}-{\parallel {x}_{n}-{u}_{n}\parallel }^{2}+{\alpha }_{n}^{2}{\parallel \gamma f\left({x}_{n}\right)-Bp\parallel }^{2}\\ +2{\alpha }_{n}\parallel \gamma f\left({x}_{n}\right)-Bp\parallel \parallel {y}_{n}-p\parallel .\end{array}$

Using ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and (3.9) again, we obtain

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{u}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}\parallel =0.$
(3.12)

By the nonexpansion of ${S}_{n}$, we have

$\begin{array}{rcl}\parallel {x}_{n}-{S}_{n}{x}_{n}\parallel & \le & \parallel {x}_{n}-{x}_{n+1}\parallel +\parallel {x}_{n+1}-{S}_{n}{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{x}_{n+1}\parallel +\parallel {\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}B\right){y}_{n}-{S}_{n}{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{x}_{n+1}\parallel +{\alpha }_{n}\left[\parallel \gamma f\left({x}_{n}\right)\parallel +\parallel B{y}_{n}\parallel \right]+\parallel {S}_{n}{u}_{n}-{S}_{n}{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{x}_{n+1}\parallel +{\alpha }_{n}\left[\parallel \gamma f\left({x}_{n}\right)\parallel +\parallel B{y}_{n}\parallel \right]+\parallel {u}_{n}-{x}_{n}\parallel .\end{array}$

This together with (3.9) and (3.12), we obtain

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{S}_{n}{x}_{n}\parallel =0.$
(3.13)

Furthermore, we note that

$\parallel {x}_{n}-{S}_{n}{x}_{n}\parallel =\left(1-{\beta }_{n}\right)\parallel {x}_{n}-{W}_{n}{x}_{n}\parallel .$

It follows from condition (ii) that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{W}_{n}{x}_{n}\parallel =0.$
(3.14)

On the other hand, by condition (iii), we may assume that ${\eta }_{i}^{\left(n\right)}\to {\eta }_{i}$ as $n\to \mathrm{\infty }$ for every $1\le i\le N$. It is easily seen that each ${\eta }_{i}>0$ and ${\sum }_{i=1}^{N}{\eta }_{i}=1$. Define $W={\sum }_{i=1}^{N}{\eta }_{i}{T}_{i}$, then $W:K\to H$ is a k-strict pseudo-contraction such that $F\left(W\right)=F\left({W}_{n}\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$ by Proposition 2.1 and 2.2. Consequently,

$\begin{array}{rcl}\parallel {x}_{n}-W{x}_{n}\parallel & \le & \parallel {x}_{n}-{W}_{n}{x}_{n}\parallel +\parallel {W}_{n}{x}_{n}-W{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{W}_{n}{x}_{n}\parallel +\sum _{i=1}^{N}|{\eta }_{i}^{\left(n\right)}-{\eta }_{i}|\parallel {T}_{i}{x}_{n}\parallel ,\end{array}$

which implies that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-W{x}_{n}\parallel =0.$
(3.15)

Combining (3.14) and (3.15), we obtain

$\underset{n\to \mathrm{\infty }}{lim}\parallel {W}_{n}{x}_{n}-W{x}_{n}\parallel =0.$
(3.16)

Define $S:K\to H$ by $Sx=\lambda x+\left(1-\lambda \right)Wx$. By condition (ii) again, we have ${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=\lambda \in \left[k,1\right)$. Then, S is nonexpansive with $F\left(S\right)=F\left(W\right)$ by Lemma 2.5. Notice that

$\begin{array}{rcl}\parallel {x}_{n}-S{x}_{n}\parallel & \le & \parallel {x}_{n}-{S}_{n}{x}_{n}\parallel +\parallel {S}_{n}{x}_{n}-S{x}_{n}\parallel \\ =& \parallel {x}_{n}-{S}_{n}{x}_{n}\parallel +\parallel {\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){W}_{n}{x}_{n}-\lambda {x}_{n}-\left(1-\lambda \right)W{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{S}_{n}{x}_{n}\parallel +|{\beta }_{n}-\lambda |\parallel {x}_{n}-W{x}_{n}\parallel +\left(1-{\beta }_{n}\right)\parallel {W}_{n}{x}_{n}-W{x}_{n}\parallel .\end{array}$

It follows from (3.13), (3.15) and (3.16) that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-S{x}_{n}\parallel =0.$
(3.17)

Now we claim that ${lim sup}_{n\to \mathrm{\infty }}〈\left(B-\gamma f\right)q,q-{x}_{n}〉\le 0$, where $q={lim}_{t\to 0}{x}_{t}$ with ${x}_{t}$ being the fixed point of the contraction ${\mathrm{\Psi }}_{n}$ on H defined by

${\mathrm{\Psi }}_{n}x=t\gamma f\left(x\right)+\left(I-tB\right){S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right)x,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in H,n\in N,$

where $t\in \left(0,1\right)$. Indeed, by Lemma 2.1 and 2.3, we have

$\begin{array}{rcl}\parallel {\mathrm{\Psi }}_{n}x-{\mathrm{\Psi }}_{n}y\parallel & \le & t\gamma \parallel f\left(x\right)-f\left(y\right)\parallel +\left(1-t\overline{\gamma }\right)\parallel {S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right)x-{S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right)y\parallel \\ \le & t\gamma \rho \parallel x-y\parallel +\left(1-t\overline{\gamma }\right)\parallel {T}_{{r}_{n}}\left(I-{r}_{n}A\right)x-{T}_{{r}_{n}}\left(I-{r}_{n}A\right)y\parallel \\ \le & t\gamma \rho \parallel x-y\parallel +\left(1-t\overline{\gamma }\right)\parallel x-y\parallel \\ =& \left[1-\left(\overline{\gamma }-\gamma \rho \right)t\right]\parallel x-y\parallel ,\end{array}$

for all $x,y\in H$. Since $0<1-\left(\overline{\gamma }-\gamma \rho \right)t<1$, it follows that ${\mathrm{\Psi }}_{n}$ is a contraction. Therefore, by the Banach contraction principle, ${\mathrm{\Psi }}_{n}$ has a unique fixed point ${x}_{t}\in H$ such that

${x}_{t}=t\gamma f\left({x}_{t}\right)+\left(I-tB\right){S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{t}.$

By Lemma 2.2 and (3.10), we have

$\begin{array}{rcl}{\parallel {x}_{t}-{x}_{n}\parallel }^{2}& =& {\parallel \left(I-tB\right)\left[{S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{t}-{x}_{n}\right]+t\left[\gamma f\left({x}_{t}\right)-B{x}_{n}\right]\parallel }^{2}\\ \le & {\left(1-\overline{\gamma }t\right)}^{2}{\parallel {S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{t}-{x}_{n}\parallel }^{2}+2t〈\gamma f\left({x}_{t}\right)-B{x}_{n},{x}_{t}-{x}_{n}〉\\ =& {\left(1-\overline{\gamma }t\right)}^{2}{\parallel {S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{t}-{S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}+{S}_{n}{T}_{{r}_{n}}\left(I-{r}_{n}A\right){x}_{n}-{x}_{n}\parallel }^{2}\\ +2t〈\gamma f\left({x}_{t}\right)-B{x}_{n},{x}_{t}-{x}_{n}〉\\ \le & {\left(1-\overline{\gamma }t\right)}^{2}{\left[\parallel {x}_{t}-{x}_{n}\parallel +\parallel {y}_{n}-{x}_{n}\parallel \right]}^{2}+2t〈\gamma f\left({x}_{t}\right)-B{x}_{n},{x}_{t}-{x}_{n}〉\\ \le & {\left(1-\overline{\gamma }t\right)}^{2}{\parallel {x}_{t}-{x}_{n}\parallel }^{2}+{\psi }_{n}\left(t\right)+2t〈\gamma f\left({x}_{t}\right)-B{x}_{t},{x}_{t}-{x}_{n}〉\\ +2t〈B{x}_{t}-B{x}_{n},{x}_{t}-{x}_{n}〉,\end{array}$
(3.18)

where ${\psi }_{n}\left(t\right)={\left(1-\overline{\gamma }t\right)}^{2}\left(2\parallel {x}_{t}-{x}_{n}\parallel +\parallel {y}_{n}-{x}_{n}\parallel \right)\parallel {y}_{n}-{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty }$. Observe B is strongly positive, we obtain

$〈B{x}_{t}-B{x}_{n},{x}_{t}-{x}_{n}〉=〈B\left({x}_{t}-{x}_{n}\right),{x}_{t}-{x}_{n}〉\ge \overline{\gamma }{\parallel {x}_{t}-{x}_{n}\parallel }^{2}.$
(3.19)

Combining (3.18) and (3.19), we have

$\begin{array}{rcl}2t〈B{x}_{t}-\gamma f\left({x}_{t}\right),{x}_{t}-{x}_{n}〉& \le & \left({\overline{\gamma }}^{2}{t}^{2}-2\overline{\gamma }t\right){\parallel {x}_{t}-{x}_{n}\parallel }^{2}+{\psi }_{n}\left(t\right)+2t〈B{x}_{t}-B{x}_{n},{x}_{t}-{x}_{n}〉\\ \le & \left(\overline{\gamma }{t}^{2}-2t\right)〈B\left({x}_{t}-{x}_{n}\right),{x}_{t}-{x}_{n}〉+{\psi }_{n}\left(t\right)\\ +2t〈B{x}_{t}-B{x}_{n},{x}_{t}-{x}_{n}〉\\ =& \overline{\gamma }{t}^{2}〈B{x}_{t}-B{x}_{n},{x}_{t}-{x}_{n}〉+{\psi }_{n}\left(t\right).\end{array}$

It follows that

$〈B{x}_{t}-\gamma f\left({x}_{t}\right),{x}_{t}-{x}_{n}〉\le \frac{\overline{\gamma }t}{2}〈B{x}_{t}-B{x}_{n},{x}_{t}-{x}_{n}〉+\frac{1}{2t}{\psi }_{n}\left(t\right).$
(3.20)

Let $n\to \mathrm{\infty }$ in (3.20) and note that ${\psi }_{n}\left(t\right)\to 0$ as $n\to \mathrm{\infty }$ yields

$\underset{n\to \mathrm{\infty }}{lim sup}〈B{x}_{t}-\gamma f\left({x}_{t}\right),{x}_{t}-{x}_{n}〉\le \frac{t}{2}{M}_{4},$
(3.21)

where ${M}_{4}$ is an appropriate positive constant such that ${M}_{4}\ge \overline{\gamma }〈B{x}_{t}-B{x}_{n},{x}_{t}-{x}_{n}〉$ for all $t\in \left(0,1\right)$ and $n\ge 1$. Taking $t\to 0$ from (3.21), we have

$\underset{t\to 0}{lim sup}\underset{n\to \mathrm{\infty }}{lim sup}〈B{x}_{t}-\gamma f\left({x}_{t}\right),{x}_{t}-{x}_{n}〉\le 0.$
(3.22)

On the other hand, we have

$\begin{array}{rcl}〈\gamma f\left(q\right)-Bq,{x}_{n}-q〉& =& 〈\gamma f\left(q\right)-Bq,{x}_{n}-q〉-〈\gamma f\left(q\right)-Bq,{x}_{n}-{x}_{t}〉+〈\gamma f\left(q\right)-Bq,{x}_{n}-{x}_{t}〉\\ -〈\gamma f\left(q\right)-B{x}_{t},{x}_{n}-{x}_{t}〉+〈\gamma f\left(q\right)-B{x}_{t},{x}_{n}-{x}_{t}〉\\ -〈\gamma f\left({x}_{t}\right)-B{x}_{t},{x}_{n}-{x}_{t}〉+〈\gamma f\left({x}_{t}\right)-B{x}_{t},{x}_{n}-{x}_{t}〉.\end{array}$

It follows that Therefore, from (3.22) and ${lim}_{t\to 0}{x}_{t}=q$, we have

$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim sup}〈\gamma f\left(q\right)-Bq,{x}_{n}-q〉& =& \underset{t\to 0}{lim sup}\underset{n\to \mathrm{\infty }}{lim sup}〈\gamma f\left(q\right)-Bq,{x}_{n}-q〉\\ \le & \underset{t\to 0}{lim sup}\parallel \gamma f\left(q\right)-Bq\parallel \parallel {x}_{t}-q\parallel \\ +\underset{t\to 0}{lim sup}\parallel B\parallel \parallel {x}_{t}-q\parallel \underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{x}_{t}\parallel \\ +\underset{t\to 0}{lim sup}\gamma \rho \parallel {x}_{t}-q\parallel \underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{x}_{t}\parallel \\ +\underset{t\to 0}{lim sup}\underset{n\to \mathrm{\infty }}{lim sup}〈\gamma f\left({x}_{t}\right)-B{x}_{t},{x}_{n}-{x}_{t}〉\\ \le & 0.\end{array}$
(3.23)

Finally, we prove that ${x}_{n}\to q$ as $n\to \mathrm{\infty }$. From (1.4) and (3.2) again, we have

$\begin{array}{rcl}{\parallel {x}_{n+1}-q\parallel }^{2}& =& 〈{\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}B\right){y}_{n}-q,{x}_{n+1}-q〉\\ =& {\alpha }_{n}〈\gamma f\left({x}_{n}\right)-Bq,{x}_{n+1}-q〉+〈\left(I-{\alpha }_{n}B\right)\left({y}_{n}-q\right),{x}_{n+1}-q〉\\ \le & {\alpha }_{n}\gamma 〈f\left({x}_{n}\right)-f\left(q\right),{x}_{n+1}-q〉+{\alpha }_{n}〈\gamma f\left(q\right)-Bq,{x}_{n+1}-q〉\\ +\left(1-{\alpha }_{n}\overline{\gamma }\right)\parallel {y}_{n}-q\parallel \parallel {x}_{n+1}-q\parallel \\ \le & {\alpha }_{n}\gamma \rho \parallel {x}_{n}-q\parallel \parallel {x}_{n+1}-q\parallel +{\alpha }_{n}〈\gamma f\left(q\right)-Bq,{x}_{n+1}-q〉\\ +\left(1-{\alpha }_{n}\overline{\gamma }\right)\parallel {x}_{n}-q\parallel \parallel {x}_{n+1}-q\parallel \\ =& \left[1-\left(\overline{\gamma }-\gamma \rho \right){\alpha }_{n}\right]\parallel {x}_{n}-q\parallel \parallel {x}_{n+1}-q\parallel +{\alpha }_{n}〈\gamma f\left(q\right)-Bq,{x}_{n+1}-q〉\\ \le & \frac{1-\left(\overline{\gamma }-\gamma \rho \right){\alpha }_{n}}{2}\left({\parallel {x}_{n}-q\parallel }^{2}+{\parallel {x}_{n+1}-q\parallel }^{2}\right)+{\alpha }_{n}〈\gamma f\left(q\right)-Bq,{x}_{n+1}-q〉\\ \le & \frac{1-\left(\overline{\gamma }-\gamma \rho \right){\alpha }_{n}}{2}{\parallel {x}_{n}-q\parallel }^{2}+\frac{1}{2}{\parallel {x}_{n+1}-q\parallel }^{2}+{\alpha }_{n}〈\gamma f\left(q\right)-Bq,{x}_{n+1}-q〉.\end{array}$

It follows that

${\parallel {x}_{n+1}-q\parallel }^{2}\le \left[1-\left(\overline{\gamma }-\gamma \rho \right){\alpha }_{n}\right]{\parallel {x}_{n}-q\parallel }^{2}+2{\alpha }_{n}〈\gamma f\left(q\right)-Bq,{x}_{n+1}-q〉.$
(3.24)

From $0<\gamma <\frac{\overline{\gamma }}{\rho }$, condition (i) and (3.23), we can arrive at the desired conclusion ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-q\parallel =0$ by Lemma 2.6. This completes the proof. □

Theorem 3.2 Let K be a nonempty closed convex subset of the Hilbert space H and $F:K×K\to \mathbb{R}$ be a bi-function satisfying (A 1)-(A 4). Let A be an α-inverse strongly monotone mapping, $f\in {\mathrm{\Pi }}_{K}$ with a coefficient $\rho \in \left(0,1\right)$ and B be a strongly positive bounded linear operator on H with $\overline{\gamma }>0$ and $0<\gamma <\frac{\overline{\gamma }}{\rho }$. Let $T:K\to H$ be a k-strict pseudo-contraction such that $\mathcal{F}=F\left(T\right)\cap \mathit{EP}\left(F,A\right)\ne \varphi$. Let $\left\{{x}_{n}\right\}$ be a sequence generated by ${x}_{0}\in K$ in the following manner:

$\left\{\begin{array}{c}F\left({u}_{n},y\right)+〈A{x}_{n},y-{u}_{n}〉+\frac{1}{r}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in K,\hfill \\ {y}_{n}={\beta }_{n}{u}_{n}+\left(1-{\beta }_{n}\right)T{u}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}B\right){y}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$

where $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ are two sequences in $\left(0,1\right)$, constant $r\in \left(0,2\alpha \right)$. If the following control conditions are satisfied:

1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n}-{\alpha }_{n-1}|<\mathrm{\infty }$;

2. (ii)

$k\le {\beta }_{n}\le \lambda <1$, ${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=\lambda$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n}-{\beta }_{n-1}|<\mathrm{\infty }$.

Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to $q\in \mathcal{F}$, which solves the variational inequality

$〈\left(B-\gamma f\right)q,q-p〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in \mathcal{F}.$

Proof Putting ${r}_{n}=r$ and $N=1$, i.e., ${W}_{n}=T$, the desired conclusion follows immediately from Theorem 3.1. This completes the proof. □

Theorem 3.3 Let K be a nonempty closed convex subset of the Hilbert space H and $F:K×K\to \mathbb{R}$ be a bi-function satisfying (A 1)-(A 4). Let $f\in {\mathrm{\Pi }}_{K}$ with a coefficient $\rho \in \left(0,1\right)$ and B be a strongly positive bounded linear operator on H with $\overline{\gamma }>0$ and $0<\gamma <\frac{\overline{\gamma }}{\rho }$. Let $T:K\to H$ be a k-strict pseudo-contraction such that $\mathcal{F}=F\left(T\right)\cap \mathit{EP}\left(F\right)\ne \varphi$. Let $\left\{{x}_{n}\right\}$ be a sequence generated by ${x}_{0}\in K$ in the following manner:

$\left\{\begin{array}{c}F\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in K,\hfill \\ {y}_{n}={\beta }_{n}{u}_{n}+\left(1-{\beta }_{n}\right)T{u}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}B\right){y}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$

where $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ are two sequences in $\left(0,1\right)$, sequence $\left\{{r}_{n}\right\}\subset \left(0,2\alpha \right)$. If the following control conditions are satisfied:

1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n}-{\alpha }_{n-1}|<\mathrm{\infty }$;

2. (ii)

$k\le {\beta }_{n}\le \lambda <1$, ${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=\lambda$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n}-{\beta }_{n-1}|<\mathrm{\infty }$;

3. (iii)

${lim inf}_{n\to \mathrm{\infty }}{r}_{n}>0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{r}_{n}-{r}_{n-1}|<\mathrm{\infty }$.

Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to $q\in \mathcal{F}$, which solves the variational inequality

$〈\left(B-\gamma f\right)q,q-p〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in \mathcal{F}.$

Proof Putting $N=1$ and $A=0$, i.e., the generalized equilibrium problem (1.1) reduces to the normal equilibrium problem (1.2), the desired conclusion follows immediately from Theorem 3.1. This completes the proof. □

Remark 3.1 Theorem 3.1 and 3.2 improve and extend the main results of Takahashi and Takahashi  and Qin et al.  in different directions.

Remark 3.2 Theorem 3.3 is mainly due to Liu , which improves and extends the main results of Takahashi and Takahashi .

Remark 3.3 If $F=A=0$ and ${u}_{n}={x}_{n}$, then the algorithm (1.4) reduces to approximate the fixed point of k-strict pseudo-contractions, which includes the general iterative method of Marino and Xu  and the parallel algorithm of Acedo and Xu  as special cases.

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## Acknowledgements

Supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Changging (CSTC 2012jjA00039) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 110701).

## Author information

Authors

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Correspondence to Dao-Jun Wen.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Wen, DJ carried out the primary studies for the generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions, participated in the design of iterative methods and drafted the manuscript. Chen YA participated in the convergence analysis and coordination. All authors read and approved the final manuscript.

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Wen, DJ., Chen, YA. General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions. Fixed Point Theory Appl 2012, 125 (2012). https://doi.org/10.1186/1687-1812-2012-125

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• DOI: https://doi.org/10.1186/1687-1812-2012-125

### Keywords

• generalized equilibrium problem
• k-strict pseudo-contractions
• general iterative method
• α-inverse strongly monotone
• common fixed point
• strong convergence 