Strong convergence theorems for a generalized mixed equilibrium problem and variational inequality problems

Abstract

In this paper, a new iterative scheme based on the extragradient-like method for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a strongly positive linear bounded operator and the set of solutions of a mixed equilibrium problem is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Our results extend recent results announced by many others.

MSC:49J30, 49J40, 47J25, 47H09.

1 Introduction

Let H be a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$. Let C be a nonempty closed convex subset of H. Recall that a mapping $T:C\to C$ is said to be nonexpansive if

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

We denote by $F(T)$ the set of fixed points of T. Let $P_C$ be the projection of H onto the convex subset C. Moreover, we also denote by ℝ the set of all real numbers.

Peng and Yao [1] considered the generalized mixed equilibrium problem of finding $x\in C$ such that

$$\mathrm{\Theta }(x,y)+\phi (y)-\phi (x)+〈Fx,y-x〉\ge 0,\mathrm{\forall }y\in C,$$

(1.1)

where $F:C\to H$ is a nonlinear mapping and $\phi :C\to \mathbb{R}$ is a function and $\mathrm{\Theta }:C\times C\to \mathbb{R}$ is a bifunction. The set of solutions of problem (1.1) is denoted by GMEP.

In the case of $F=0$, problem (1.1) reduces to the mixed equilibrium problem of finding $x\in C$ such that

$$\mathrm{\Theta }(x,y)+\phi (y)-\phi (x)\ge 0,\mathrm{\forall }y\in C,$$

which was considered by Ceng and Yao [2]. GMEP is denoted by MEP.

In the case of $\phi =0$, problem (1.1) reduces to the generalized equilibrium problem of finding $x\in C$ such that

$$\mathrm{\Theta }(x,y)+〈Fx,y-x〉\ge 0,\mathrm{\forall }y\in C,$$

which was studied by Takahashi and Takahashi [3] and many others, for example, [4–10].

In the case of $\phi =0$ and $F=0$, problem (1.1) reduces to the equilibrium problem of finding $x\in C$ such that

$$\mathrm{\Theta }(x,y)\ge 0,\mathrm{\forall }y\in C.$$

(1.2)

The set of solutions of (1.2) is denoted by $EP(\mathrm{\Theta })$.

In the case $\mathrm{\Theta }=0$, $\phi =0$ and $F=A$, problem (1.1) reduces to the classical variational inequality problem of finding $x\in C$ such that

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

(1.3)

The set of solutions of problem (1.3) is denoted by $VI(A,C)$.

Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, for instance, [2, 3, 11]. Peng and Yao [1] considered iterative methods for finding a common element of the set of solutions of problem (1.1), the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping.

Let $G_1,G_2:C\times C\to \mathbb{R}$ be two bifunctions and let $B_1,B_2:C\to H$ be two nonlinear mappings. We consider the generalized equilibrium problem $(\overline{x},\overline{y})\in C\times C$ such that

$$\{\begin{array}{c}{G}_1(\overline{x},x)+〈B_1\overline{y},x-\overline{x}〉+\frac{1}{{\mu }_1}〈\overline{x}-\overline{y},x-\overline{x}〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ G_2(\overline{y},y)+〈B_2\overline{x},y-\overline{y}〉+\frac{1}{{\mu }_2}〈\overline{y}-\overline{x},y-\overline{y}〉\ge 0,\mathrm{\forall }y\in C,\hfill \end{array}$$

(1.4)

where ${\mu }_1>0$ and ${\mu }_2>0$ are two constants.

In the case $G_1=G_2=0$, problem (1.4) reduces to the general system of variational inequalities of finding $(\overline{x},\overline{y})\in C\times C$ such that

$$\{\begin{array}{c}〈{\mu }_1{B}_1\overline{y}+\overline{x}-\overline{y},x-\overline{x}〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ 〈{\mu }_2{B}_2\overline{x}+\overline{y}-\overline{x},y-\overline{y}〉\ge 0,\mathrm{\forall }y\in C,\hfill \end{array}$$

(1.5)

where ${\mu }_1>0$ and ${\mu }_2>0$ are two constants, which was considered by Ceng, Wang and Yao [12]. In particular, if $B_1=B_2=A$, then problem (1.5) reduces to the system of variational inequalities of finding $(\overline{x},\overline{y})\in C\times C$ such that

$$\{\begin{array}{c}〈{\mu }_{1}A\overline{y}+\overline{x}-\overline{y},x-\overline{x}〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ 〈{\mu }_{2}A\overline{x}+\overline{y}-\overline{x},y-\overline{y}〉\ge 0,\mathrm{\forall }y\in C,\hfill \end{array}$$

(1.6)

which was studied by Verma [13].

If $\overline{x}=\overline{y}$ in (1.6), then (1.6) reduces to the classical variational inequality (1.3). Further, problem (1.6) is equivalent to the following projection formulas:

$$\{\begin{array}{c}\overline{x}=P_C(I-{\mu }_{1}A)\overline{y},\hfill \\ \overline{y}=P_C(I-{\mu }_{2}A)\overline{x}.\hfill \end{array}$$

Recently, Ceng et al. [12] introduced and studied a relaxed extragradient method for finding solutions of problem (1.5).

Let $\{T_i\}$ be an infinite family of nonexpansive mappings of C into itself and $\{{\lambda }_{n1}\},\{{\lambda }_{n2}\},\dots ,\{{\lambda }_{nN}\}$ be real sequences such that ${\lambda }_{n1},{\lambda }_{n2},\dots ,{\lambda }_{nN}\in (0,1]$ for every $n\in N$. For any $n\ge 1$, we define a mapping $W_n$ of C into itself as follows:

$$\begin{array}{c}{U}_{n0}=I,\hfill \\ U_{n1}={\lambda }_{n1}{T}_1{U}_{n0}+(1-{\lambda }_{n1})I,\hfill \\ U_{n2}={\lambda }_{n2}{T}_2{U}_{n1}+(1-{\lambda }_{n2})I,\hfill \\ ⋮\hfill \\ U_{n,N-1}={\lambda }_{n,N-1}{T}_{N-1}{U}_{n,N-2}+(1-{\lambda }_{n,N-1})I,\hfill \\ W_n=U_{n,N}={\lambda }_{n,N}{T}_N{U}_{n,N-1}+(1-{\lambda }_{n,N})I.\hfill \end{array}$$

Such a mapping $W_n$ is called the W-mapping generated by $T_1,T_2,\dots ,T_N$ and $\{{\lambda }_{n1}\},\{{\lambda }_{n2}\},\dots ,\{{\lambda }_{nN}\}$. Nonexpansivity of each $T_i$ ensures the nonexpansivity of $W_n$. Moreover, in [1], it is shown that $F(W_n)={\bigcap }_{i=1}^{N}F(T_i)$.

Throughout this article, let us assume that a bifunction $\mathrm{\Theta }:C\times C\to \mathbb{R}$ and a convex function $\phi :C\to \mathbb{R}$ satisfy the following conditions:

• (H1) $\mathrm{\Theta }(x,x)=0$ for all $x\in C$;

• (H2) Θ is monotone, i.e., $\mathrm{\Theta }(x,y)+\mathrm{\Theta }(y,x)\le 0$ for all $x,y\in C$;

• (H3) for each $y\in C$, $x\mapsto \mathrm{\Theta }(x,y)$ is weakly upper semicontinuous;

• (H4) for each $x\in C$, $y\mapsto \mathrm{\Theta }(x,y)$ is convex and lower semicontinuous;

• (A1) for each $x\in H$ and $r>0$, there exists a bounded subset $D_x\subset C$ and $y_x\in C$ such that for any $z\in C\mathrm{\setminus }{D}_x$,

  $$\mathrm{\Theta }(z,y_x)+\phi (y_x)-\phi (z)+\frac{1}{r}〈y_x-z,z-x〉<0;$$

• (A2) C is a bounded set.

Recently, Qin et al. [8] studied the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. More precisely, they proved the following theorem.

Theorem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from $C\times C$ to ℝ which satisfies (H 1)-(H 4). Let $T_1,T_2,\dots ,T_N$ be a finite family of nonexpansive mappings of C into H and let B be a μ-Lipschitz, relaxed $(u,v)$-cocoercive mapping of C into H such that $F={\bigcap }_{i=1}^{N}F(T_i)\cap EP(\mathrm{\Theta })\cap VI(A,C)\ne \varphi$. Let f be a contraction of H into itself with a coefficient α ($0<\alpha <1$) and let A be a strongly positive linear bounded operator with a coefficient $\overline{\gamma }>0$ such that $\parallel A\parallel \le 1$. Assume that $0<\gamma <\frac{\overline{\gamma }}{\alpha }$. Let $\{x_n\}$ and $\{y_n\}$ be sequences generated by $x_1\in H$ and

$$\{\begin{array}{c}\mathrm{\Theta }(y_n,\eta )+\frac{1}{r_n}〈\eta -y_n,y_n-x_n〉\ge 0,\mathrm{\forall }\eta \in C,\hfill \\ x_{n+1}={\alpha }_n\gamma f(W_n{x}_n)+(1-{\alpha }_{n}A)W_n{P}_C(I-s_{n}B)y_n,n\ge 1,\hfill \end{array}$$

where ${\alpha }_n\in (0,1]$ and $\{r_n\},\{s_n\}\subset [0,\mathrm{\infty })$ satisfy

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}|s_{n+1}-s_n|<\mathrm{\infty }$;

• (iii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$;

• (iv) $\{s_n\}\subset [a,b]$ for some a, b with $0\le a\le b\le \frac{2(v-u{\mu }^2)}{{\mu }^2}$, $v\ge u{\mu }^2$;

• (v) ${\sum }_{n=0}^{\mathrm{\infty }}|{\lambda }_{n,i}-{\lambda }_{n-1,i}|<\mathrm{\infty }$ for all $i=1,2,\dots ,N$.

Then, both $\{x_n\}$ and $\{y_n\}$ converge strongly to $q\in F$, where $q=P_F(\gamma f+(I-A))(q)$, which solves the following variational inequality:

$$〈\gamma f(q)-Aq,p-q〉\le 0,\mathrm{\forall }p\in F.$$

In this paper, motivated by Takahashi and Takahashi [3], Ceng, Wang and Yao [12], Peng and Yao [1] and Qin, Shang and Su [8], we introduce the general iterative scheme for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of the generalized mixed equilibrium problem (1.1) and the set of solutions of the generalized equilibrium problem (1.4), which solves the variational inequality

$$〈(A-\gamma f)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \mathfrak{F},$$

where $\mathfrak{F}={\bigcap }_{i=1}^{N}F(T_i)\cap \mathrm{GMEP}\cap \mathrm{\Omega }$ and Ω is the set of solutions of the generalized equilibrium problem (1.4). The results obtained in this paper improve and extend the recent results announced by Qin et al. [8], Chen et al. [14], Combetters and Hirstoaga [15], Iiduka and Takahashi [16], Marino and Xu [17], Takahashi and Takahashi [18], Wittmann [19] and many others.

2 Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H. For every point $x\in H$, there exists a unique nearest point of C, denoted by $P_{C}x$, such that $\parallel x-P_{C}x\parallel \le \parallel x-y\parallel$ for all $y\in C$. Such a $P_C$ is called the metric projection of H onto C. We know that $P_C$ is a firmly nonexpansive mapping of H onto C, i.e.,

$$〈x-y,P_{C}x-P_{C}y〉\ge {\parallel P_{C}x-P_{C}y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Further, for any $x\in H$ and $z\in C$, $z=P_{C}x$ if and only if

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

It is also known that H satisfies Opial’s condition [20] if for each sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ in H which converges weakly to a point $x\in H$, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-x\parallel <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-y\parallel ,\mathrm{\forall }y\in H,y\ne x.$$

Moreover, we assume that A is a bounded strongly positive operator on H with a constant $\overline{\gamma }$, that is, there exists $\overline{\gamma }>0$ such that

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

A mapping $B:C\to H$ is called β-inverse strongly monotone if there exists $\beta >0$ such that

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

It is obvious that any inverse strongly monotone mapping is Lipschitz continuous.

In order to prove our main results in the next section, we need the following lemmas and proposition.

Lemma 2.1 [2]

Let C be a nonempty closed convex subset of H. Let $\mathrm{\Theta }:C\times C\to \mathbb{R}$ be a bifunction satisfying conditions (H1)-(H4) and let $\phi :C\to \mathbb{R}$ be a lower semicontinuous and convex function. For $r>0$ and $x\in H$, define a mapping

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

for all $x\in H$. Assume that either (A1) or (A2) holds. Then the following results hold:

• (i) $T_r^{(\mathrm{\Theta },\phi )}(x)\ne \varphi$ for each $x\in H$ and $T_r^{(\mathrm{\Theta },\phi )}$ is single-valued;

• (ii) $T_r^{(\mathrm{\Theta },\phi )}$ is firmly nonexpansive, i.e., for any $x,y\in H$,

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

• (iii) $F(T_r^{(\mathrm{\Theta },\phi )})=MEP(\mathrm{\Theta },\phi )$;

• (iv) $MEP(\mathrm{\Theta },\phi )$ is closed and convex.

Remark 2.1 If $\phi =0$, then $T_r^{(\mathrm{\Theta },\phi )}$ is rewritten as $T_r^{\mathrm{\Theta }}$.

By a similar argument as that in the proof of Lemma 2.1 in [12], we have the following result.

Lemma 2.2 Let C be a nonempty closed convex subset of H. Let $G_1,G_2:C\times C\to \mathbb{R}$ be two bifunctions satisfying conditions (H1)-(H4) and let the mappings $B_1,B_2:C\to H$ be ${\beta }_1$-inverse strongly monotone and ${\beta }_2$-inverse strongly monotone, respectively. Then, for given $\overline{x},\overline{y}\in C$, $(\overline{x},\overline{y})$ is a solution of (1.4) if and only if $\overline{x}$ is a fixed point of the mapping $\mathrm{\Gamma }:C\to C$ defined by

$$\mathrm{\Gamma }(x)=T_{{\mu }_1}^{G_1}[T_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)-{\mu }_1{B}_1{T}_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)],\mathrm{\forall }x\in C,$$

where $\overline{y}=T_{{\mu }_2}^{G_2}(\overline{x}-{\mu }_2{B}_2\overline{x})$.

The set of fixed points of the mapping Γ is denoted by Ω.

Proposition 2.1 [3]

Let C, H, Θ, φ and $T_r^{(\mathrm{\Theta },\phi )}$ be as in Lemma  2.1. Then the following holds:

$${\parallel T_s^{(\mathrm{\Theta },\phi )}x-T_t^{(\mathrm{\Theta },\phi )}x\parallel }^2\le \frac{s-t}{s}〈T_s^{(\mathrm{\Theta },\phi )}x-T_t^{(\mathrm{\Theta },\rho )}x,T_s^{(\mathrm{\Theta },\phi )}x-x〉$$

for all $s,t>0$ and $x\in H$.

Lemma 2.3 [21]

Assume that T is a nonexpansive self-mapping of a nonempty closed convex subset C of H. If T has a fixed point, then $I-T$ is demi-closed; that is, when $\{x_n\}$ is a sequence in C converging weakly to some $x\in C$ and the sequence $\{(I-T)x_n\}$ converges strongly to some y, it follows that $(I-T)x=y$.

Lemma 2.4 [22]

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

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\delta }_n,\mathrm{\forall }n\ge 1,$$

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

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

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

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

Lemma 2.5 [17]

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

The following lemma is an immediate consequence of an inner product.

Lemma 2.6 In a real Hilbert space H, the following inequality holds:

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉$$

for all $x,y\in H$.

3 Main results

Now we state and prove our main results.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $\mathrm{\Theta },G_1,G_2:C\times C\to \mathbb{R}$ be three bifunctions which satisfy assumptions (H1)-(H4) and $\phi :C\to \mathbb{R}$ be a lower semicontinuous and convex function with assumption (A1) or (A2). Let the mappings $F,B_1,B_2:C\to H$ be ζ-inverse strongly monotone, ${\beta }_1$-inverse strongly monotone and ${\beta }_2$-inverse strongly monotone, respectively. Let $T_1,T_2,\dots ,T_N$ be a finite family of nonexpansive mappings of C into H such that $\mathfrak{F}={\bigcap }_{i=1}^{N}F(T_i)\cap \mathrm{GMEP}\cap \mathrm{\Omega }\ne \varphi$. Let f be a contraction of C into itself with a constant α ($0<\alpha <1$) and let A be a strongly positive linear bounded operator with a coefficient $\overline{\gamma }>0$ such that $\parallel A\parallel \le 1$. Assume that $0<\gamma <\frac{\overline{\gamma }}{\alpha }$. Let $x_1\in C$ and let $\{x_n\}$ be a sequence defined by

$$\{\begin{array}{c}{z}_n=T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n),\hfill \\ y_n=T_{{\mu }_1}^{G_1}[T_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)-{\mu }_1{B}_1{T}_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)],\hfill \\ x_{n+1}={\alpha }_n\gamma f(W_n{x}_n)+(1-{\alpha }_{n}A)W_n{y}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where ${\alpha }_n\in [0,1]$, ${\mu }_1\in (0,2{\beta }_1)$, ${\mu }_2\in (0,2{\beta }_2)$ and $\{{\delta }_n\}\subset [0,2\zeta ]$ satisfy the following conditions:

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

• (ii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n<2\zeta$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$;

• (iii) ${lim}_{n\to \mathrm{\infty }}{\lambda }_{n,i}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n,i}-{\lambda }_{n-1,i}|<\mathrm{\infty }$ for all $i=1,2,\dots ,N$.

Then $\{x_n\}$ converges strongly to $x^{\ast }=P_{\mathfrak{F}}(\gamma f+(I-A))(x^{\ast })$, which solves the following variational inequality:

$$〈(A-\gamma f)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \mathfrak{F},$$

and $(x^{\ast },y^{\ast })$ is a solution of problem (1.4), where $y^{\ast }=T_{{\mu }_2}^{G_2}(x^{\ast }-{\mu }_2{B}_2{x}^{\ast })$.

Proof We divide the proof into several steps.

Step 1. $\{x_n\}$ is bounded.

Indeed, take $p\in \mathfrak{F}={\bigcap }_{i=1}^{N}F(T_i)\cap \mathrm{GMEP}\cap \mathrm{\Omega }\ne \varphi$ arbitrarily. Since $p=T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(p-{\delta }_{n}Fp)$, F is ζ-inverse strongly monotone and $0\le {\delta }_n\le 2\zeta$, we obtain that for any $n\ge 1$,

$$\begin{array}{rcl}{\parallel z_n-p\parallel }^2& =& {\parallel T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(p-{\delta }_{n}Fp)\parallel }^2\\ \le & {\parallel (x_n-p)-{\delta }_n(F{x}_n-Fp)\parallel }^2\\ =& {\parallel x_n-p\parallel }^2-2{\delta }_n〈x_n-p,F{x}_n-Fp〉+{\delta }_n^2{\parallel F{x}_n-Fp\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-2{\delta }_n\zeta {\parallel F{x}_n-Fp\parallel }^2+{\delta }_n^2{\parallel F{x}_n-Fp\parallel }^2\\ =& {\parallel x_n-p\parallel }^2+{\delta }_n({\delta }_n-2\zeta ){\parallel F{x}_n-Fp\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2.\end{array}$$

(3.2)

Putting $u_n=T_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)$ and $u=T_{{\mu }_2}^{G_2}(p-{\mu }_2{B}_{2}p)$, we have

$$\begin{array}{rcl}{\parallel u_n-u\parallel }^2& =& {\parallel T_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)-T_{{\mu }_2}^{G_2}(p-{\mu }_2{B}_{2}p)\parallel }^2\\ \le & {\parallel (z_n-p)-{\mu }_2(B_2{z}_n-B_{2}p)\parallel }^2\\ =& {\parallel z_n-p\parallel }^2-2{\mu }_2〈z_n-p,B_2{z}_n-B_{2}p〉+{\mu }_2^2{\parallel B_2{z}_n-B_{2}p\parallel }^2\\ \le & {\parallel z_n-p\parallel }^2-2{\mu }_2{\beta }_2{\parallel B_2{z}_n-B_{2}p\parallel }^2+{\mu }_2^2{\parallel B_2{z}_n-B_{2}p\parallel }^2\\ =& {\parallel z_n-p\parallel }^2+{\mu }_2({\mu }_2-2{\beta }_2){\parallel B_2{z}_n-B_{2}p\parallel }^2\\ \le & {\parallel z_n-p\parallel }^2.\end{array}$$

(3.3)

And since $p=T_{{\mu }_1}^{G_1}(u-{\mu }_1{B}_{1}u)$, we know that for any $n\ge 1$,

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& =& {\parallel T_{{\mu }_1}^{G_1}(u_n-{\mu }_1{B}_1{u}_n)-T_{{\mu }_1}^{G_1}(u-{\mu }_1{B}_{1}u)\parallel }^2\\ \le & {\parallel (u_n-u)-{\mu }_1(B_1{u}_n-B_{1}u)\parallel }^2\\ \le & {\parallel u_n-u\parallel }^2-2{\mu }_1〈u_n-u,B_1{u}_n-B_{1}u〉+{\mu }_1^2{\parallel B_1{u}_n-B_{1}u\parallel }^2\\ \le & {\parallel u_n-u\parallel }^2-2{\mu }_1{\beta }_1\parallel B_1{u}_n-B_{1}u\parallel +{\mu }_1{\parallel B_1{u}_n-b_{1}u\parallel }^2\\ \le & {\parallel u_n-u\parallel }^2+{\mu }_1({\mu }_1-2{\beta }_1)\parallel B_1{u}_n-B_{1}u\parallel \\ \le & {\parallel u_n-u\parallel }^2\\ \le & {\parallel z_n-p\parallel }^2.\end{array}$$

(3.4)

Furthermore, from (3.1), we have

$$\begin{array}{rcl}\parallel x_{n+1}-p\parallel & =& \parallel {\alpha }_n\gamma f(W_n{x}_n)+(1-{\alpha }_{n}A)W_n{y}_n-p\parallel \\ =& \parallel {\alpha }_n(\gamma f(W_n{x}_n)-Ap)+(1-{\alpha }_{n}A)(W_n{y}_n-p)\parallel \\ \le & {\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel +\parallel I-{\alpha }_{n}A\parallel \parallel W_n{y}_n-p\parallel \\ \le & {\alpha }_n(\gamma \parallel f(W_n{x}_n)-f(p)\parallel +\parallel \gamma f(p)-Ap\parallel )+(1-{\alpha }_n\overline{\gamma })\parallel y_n-p\parallel \\ \le & {\alpha }_n(\gamma \alpha \parallel x_n-p\parallel +\parallel \gamma f(p)-Ap\parallel )+(1-{\alpha }_n\overline{\gamma })\parallel x_n-p\parallel \\ \le & [1-(\overline{\gamma }-\gamma \alpha ){\alpha }_n]\parallel x_n-p\parallel +{\alpha }_n\parallel \gamma f(p)-Ap\parallel .\end{array}$$

By induction, we obtain that for all $n\ge 1$,

$$\parallel x_n-p\parallel \le max\{\parallel x_1-p\parallel ,\frac{1}{\overline{\gamma }-\gamma \alpha }\parallel \gamma f(p)-Ap\parallel \}.$$

Hence $\{x_n\}$ is bounded. Consequently, we deduce immediately that $\{z_n\}$, $\{y_n\}$, $\{f(W_n{x}_n)\}$ and $\{W_n(y_n)\}$ are bounded.

Step 2. ${lim}_{n\to \mathrm{\infty }}\parallel W_{n+1}{y}_n-W_n{y}_n\parallel =0$.

It follows from the definition of $W_n$ that

$$\begin{array}{c}\parallel W_{n+1}{y}_n-W_n{y}_n\parallel \hfill \\ =\parallel {\lambda }_{n+1,N}{T}_N{U}_{n+1,N-1}{y}_n-(1-{\lambda }_{n+1,N})y_n-{\lambda }_{n,N}{T}_N{U}_{n,N-1}{y}_n-(1-{\lambda }_{n,N})y_n\parallel \hfill \\ \le |{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel y_n\parallel +\parallel {\lambda }_{n+1,N}{T}_N{U}_{n+1,N-1}{y}_n-{\lambda }_{n,N}{T}_N{U}_{n,N-1}{y}_n\parallel \hfill \\ \le |{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel y_n\parallel +\parallel {\lambda }_{n+1,N}(T_N{U}_{n+1,N-1}{y}_n-T_N{U}_{n,N-1}{y}_n)\parallel \hfill \\ +|{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel T_N{U}_{n,N-1}{y}_n\parallel \hfill \\ \le |{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel y_n\parallel +{\lambda }_{n+1,N}\parallel U_{n+1,N-1}{y}_n-U_{n,N-1}{y}_n\parallel \hfill \\ +|{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel T_N{U}_{n,N-1}{y}_n\parallel .\hfill \end{array}$$

Since $\{y_n\}$ is bounded and $T_k$, $U_{n,k}$ are nonexpansive, ${lim}_{n\to \mathrm{\infty }}\parallel W_{n+1}{y}_n-W_n{y}_n\parallel =0$.

Step 3. ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$.

We estimate $\parallel y_{n+1}-y_n\parallel$, $\parallel W_{n+1}{x}_{n+1}-W_n{x}_n\parallel$ and $\parallel W_{n+1}{y}_{n+1}-W_n{y}_n\parallel$. From (3.1) we have

(3.5)

and

$$\begin{array}{c}\parallel z_{n+1}-z_n\parallel \hfill \\ =\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_{n+1}-{\delta }_{n+1}F{x}_{n+1})-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel \hfill \\ \le \parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_{n+1}-{\delta }_{n+1}F{x}_{n+1})-T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel \hfill \\ +\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel \hfill \\ \le \parallel (x_{n+1}-{\delta }_{n+1}F{x}_{n+1})-(x_n-{\delta }_{n}F{x}_n)\parallel \hfill \\ +\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel \hfill \\ \le \parallel x_{n+1}-x_n\parallel +|{\delta }_{n+1}-{\delta }_n|\parallel F{x}_n\parallel \hfill \\ +\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel .\hfill \end{array}$$

(3.6)

It follows from (3.5) and (3.6) that

$$\begin{array}{rcl}\parallel y_{n+1}-y_n\parallel & \le & \parallel z_{n+1}-z_n\parallel \\ \le & \parallel x_{n+1}-x_n\parallel +|{\delta }_{n+1}-{\delta }_n|\parallel F{x}_n\parallel \\ +\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel .\end{array}$$

(3.7)

Without loss of generality, let us assume that there exists a real number a such that ${\delta }_n>a>0$ for all n. Utilizing Proposition 2.1, we have

$$\begin{array}{c}\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel \hfill \\ \le \frac{|{\delta }_{n+1}-{\delta }_n|}{{\delta }_{n+1}}\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(I-{\delta }_{n}F)x_n\parallel \hfill \\ \le \frac{|{\delta }_{n+1}-{\delta }_n|}{a}\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(I-{\delta }_{n}F)x_n\parallel .\hfill \end{array}$$

(3.8)

It follows from the definition of $W_n$ that

$$\begin{array}{c}\parallel W_{n+1}{y}_n-W_n{y}_n\parallel \hfill \\ =\parallel {\lambda }_{n+1,N}{T}_N{U}_{n+1,N-1}{y}_n+(1-{\lambda }_{n+1,N})y_n-{\lambda }_{n,N}{T}_N{U}_{n,N-1}{y}_n-(1-{\lambda }_{n,N})y_n\parallel \hfill \\ \le |{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel y_n\parallel +\parallel {\lambda }_{n+1,N}{T}_N{U}_{n+1,N-1}{y}_n-{\lambda }_{n,N}{T}_N{U}_{n,N-1}{y}_n\parallel \hfill \\ \le |{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel y_n\parallel +\parallel {\lambda }_{n+1,N}(T_N{U}_{n+1,N-1}{y}_n-T_N{U}_{n,N-1}{y}_n)\parallel \hfill \\ +|{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel T_N{U}_{n,N-1}{y}_n\parallel \hfill \\ \le |{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel y_n\parallel +{\lambda }_{n+1,N}\parallel U_{n+1,N-1}{y}_n-U_{n,N-1}{y}_n\parallel \hfill \\ +|{\lambda }_{n+1,N}-{\lambda }_{n,N}|\parallel T_N{U}_{n,N-1}{y}_n\parallel \hfill \\ \le M_1|{\lambda }_{n+1,N}-{\lambda }_{n,N}|+{\lambda }_{n+1,N}\parallel U_{n+1,N-1}{y}_n-U_{n,N-1}{y}_n\parallel ,\hfill \end{array}$$

(3.9)

where $M_1$ is a constant such that $M_1\ge 2max\{{sup}_{n\ge 1}\parallel y_n\parallel ,{sup}_{n\ge 1}\parallel T_N{U}_{n,N-1}{y}_n\parallel \}$. Next, we consider

$$\begin{array}{c}\parallel U_{n+1,N-1}{y}_n-U_{n,N-1}{y}_n\parallel \hfill \\ =\parallel {\lambda }_{n+1,N-1}{T}_{N-1}{U}_{n+1,N-2}{y}_n+(1-{\lambda }_{n+1,N-1})y_n\hfill \\ -{\lambda }_{n,N-1}{T}_{N-1}{U}_{n,N-2}{y}_n-(1-{\lambda }_{n,N-1})y_n\parallel \hfill \\ \le |{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}|\parallel y_n\parallel +\parallel {\lambda }_{n+1,N-1}{T}_{N-1}{U}_{n+1,N-2}{y}_n-{\lambda }_{n,N-1}{T}_{N-1}{U}_{n,N-2}{y}_n\parallel \hfill \\ \le |{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}|\parallel y_n\parallel +{\lambda }_{n+1,N-1}\parallel T_{N-1}{U}_{n+1,N-2}{y}_n-T_{N-1}{U}_{n,N-2}{y}_n\parallel \hfill \\ +|{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}|\parallel T_{N-1}{U}_{n,N-2}{y}_n\parallel \hfill \\ \le M_2|{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}|+\parallel U_{n+1,N-2}{y}_n-U_{n,N-2}{y}_n\parallel ,\hfill \end{array}$$

where $M_2$ is a constant that $M_2\ge 2max\{{sup}_{n\ge 1}\parallel y_n\parallel ,{sup}_{n\ge 1}\parallel T_{N-1}{U}_{n,N-2}{y}_n\parallel \}$. In a similar way, we obtain

$$\parallel U_{n+1,N-1}{y}_n-U_{n,N-1}{y}_n\parallel \le M_3\sum _{i=1}^{N-1}|{\lambda }_{n+1,i}-{\lambda }_{n,i}|,$$

(3.10)

where $M_3$ is an appropriate constant. Substituting (3.10) into (3.9), we have that

$$\begin{array}{rcl}\parallel W_{n+1}{y}_n-W_n{y}_n\parallel & \le & M_1|{\lambda }_{n+1,N}-{\lambda }_{n,N}|+{\lambda }_{n+1,N}{M}_3\sum _{i=1}^{N-1}|{\lambda }_{n+1,i}-{\lambda }_{n,i}|\\ \le & M_4\sum _{i=1}^N|{\lambda }_{n+1,i}-{\lambda }_{n,i}|,\end{array}$$

(3.11)

where $M_4$ is a constant such that $M_4\ge max\{M_1,M_3\}$. Similarly, we have

$$\parallel W_{n+1}{x}_n-W_n{x}_n\parallel \le M_5\sum _{i=1}^N|{\lambda }_{n+1,i}-{\lambda }_{n,i}|,$$

(3.12)

where $M_5$ is an appropriate constant. Hence it follows from (3.1), (3.7), (3.8), (3.11) and (3.12) that

$$\begin{array}{c}\parallel x_{n+2}-x_{n+1}\parallel \hfill \\ =\parallel (I-{\alpha }_{n+1}A)(W_{n+1}{y}_{n+1}-W_n{y}_n)-({\alpha }_{n+1}-{\alpha }_n)A{W}_n{y}_n\hfill \\ +\gamma [{\alpha }_{n+1}(f(W_{n+1}{x}_{n+1})-f(W_n{x}_n))+f(W_n{x}_n)({\alpha }_{n+1}-{\alpha }_n)]\parallel \hfill \\ \le (1-{\alpha }_{n+1}\overline{\gamma })(\parallel W_{n+1}{y}_{n+1}-W_{n+1}{y}_n\parallel +\parallel W_{n+1}{y}_n-W_n{y}_n\parallel )\hfill \\ +|{\alpha }_{n+1}-{\alpha }_n|\parallel A{W}_n{y}_n\parallel \hfill \\ +\gamma [{\alpha }_{n+1}\parallel f(W_{n+1}{x}_{n+1})-f(W_n{x}_n)\parallel +f(W_n{x}_n)({\alpha }_{n+1}-{\alpha }_n)]\hfill \\ \le (1-{\alpha }_{n+1}\overline{\gamma })(\parallel y_{n+1}-y_n\parallel +\parallel W_{n+1}{y}_n-W_n{y}_n\parallel )\hfill \\ +|{\alpha }_{n+1}-{\alpha }_n|\parallel A{W}_n{y}_n\parallel \hfill \\ +\gamma [{\alpha }_{n+1}\alpha (\parallel x_{n+1}-x_n\parallel +\parallel W_{n+1}{x}_n-W_n{x}_n\parallel )+|{\alpha }_{n+1}-{\alpha }_n|\parallel f(W_n{x}_n)\parallel ]\hfill \\ \le (1-{\alpha }_{n+1}\overline{\gamma })(\parallel x_{n+1}-x_n\parallel +|{\delta }_{n+1}-{\delta }_n|\parallel F{x}_n\parallel \hfill \\ +\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)\parallel +M_4\sum _{i=1}^N|{\lambda }_{n+1,i}-{\lambda }_{n,i}|)\hfill \\ +|{\alpha }_{n+1}-{\alpha }_n|\parallel A{W}_n{y}_n\parallel \hfill \\ +\gamma \alpha {\alpha }_{n+1}\parallel x_{n+1}-x_n\parallel +\gamma \alpha {\alpha }_{n+1}\parallel W_{n+1}{x}_n-W_n{x}_n\parallel \hfill \\ +\gamma |{\alpha }_{n+1}-{\alpha }_n|\parallel f(W_n{x}_n)\parallel \hfill \\ \le [1-{\alpha }_{n+1}(\overline{\gamma }-\gamma \alpha )]\parallel x_{n+1}-x_n\parallel +|{\delta }_{n+1}-{\delta }_n|\parallel F{x}_n\parallel \hfill \\ +\frac{|{\delta }_{n+1}-{\delta }_n|}{a}\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(I-{\delta }_{n}F)x_n\parallel +M_4\sum _{i=1}^N|{\lambda }_{n+1,i}-{\lambda }_{n,i}|\hfill \\ +|{\alpha }_{n+1}-{\alpha }_n|\parallel A{W}_n{y}_n\parallel +\gamma \alpha {\alpha }_{n+1}{M}_5\sum _{i=1}^N|{\lambda }_{n+1,i}-{\lambda }_{n,i}|\hfill \\ +\gamma |{\alpha }_{n+1}-{\alpha }_n|\parallel f(W_n{x}_n)\parallel \hfill \\ \le [1-{\alpha }_{n+1}(\overline{\gamma }-\gamma \alpha )]\parallel x_{n+1}-x_n\parallel \hfill \\ +M_6[2|{\delta }_{n+1}-{\delta }_n|+\sum _{i=1}^N|{\lambda }_{n+1,i}-{\lambda }_{n,i}|+(1+\gamma )|{\alpha }_{n+1}-{\alpha }_n|],\hfill \end{array}$$

where $M_6$ is a constant such that $M_6\ge max\{{sup}_{n\ge 1}\parallel F{x}_n\parallel ,\frac{1}{a}{sup}_{n\ge 1}\parallel T_{{\delta }_{n+1}}^{(\mathrm{\Theta },\phi )}(I-{\delta }_{n}F)x_n\parallel ,M_4+\gamma M_5,{sup}_{n\ge 1}\parallel A{W}_n{y}_n\parallel ,{sup}_{n\ge 1}\parallel f(W_n{x}_n)\parallel \}$. By Lemma 2.4, we get ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$.

Step 4. ${lim}_{n\to \mathrm{\infty }}\parallel F{x}_n-Fp\parallel =0$, ${lim}_{n\to \mathrm{\infty }}\parallel B_1{u}_n-B_{1}u\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel B_2{z}_n-B_{2}p\parallel =0$.

Indeed, from (3.1)-(3.4) we get

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ ={\parallel {\alpha }_n(\gamma f(W_n{x}_n)-Ap)+(I-{\alpha }_{n}A)(W_n{y}_n-p)\parallel }^2\hfill \\ \le {({\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel +(1-{\alpha }_n\overline{\gamma })\parallel y_n-p\parallel )}^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma }){\parallel y_n-p\parallel }^2\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma })[{\parallel u_n-u\parallel }^2+{\mu }_1({\mu }_1-2{\beta }_1){\parallel B_1{u}_n-B_{1}u\parallel }^2]\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma })[{\parallel z_n-p\parallel }^2+{\mu }_2({\mu }_2-2{\beta }_2){\parallel B_2{z}_n-B_{2}p\parallel }^2\hfill \\ +{\mu }_1({\mu }_1-2{\beta }_1){\parallel B_1{u}_n-B_{1}u\parallel }^2]+2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2+{\delta }_n({\delta }_n-2\zeta ){\parallel F_n-Fp\parallel }^2\hfill \\ +{\mu }_2({\mu }_2-2{\beta }_2){\parallel B_2{z}_n-B_{2}p\parallel }^2+{\mu }_1({\mu }_1-2{\beta }_1){\parallel B_1{u}_n-B_{1}u\parallel }^2\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel .\hfill \end{array}$$

Therefore

$$\begin{array}{c}{\delta }_n(2\zeta -{\delta }_n){\parallel F{x}_n-Fp\parallel }^2+{\mu }_2(2{\beta }_2-{\mu }_2){\parallel B_2{z}_n-B_{2}p\parallel }^2+{\mu }_1(2{\beta }_1-{\mu }_1){\parallel B_1{u}_n-B_{1}u\parallel }^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+\parallel x_n-p\parallel -{\parallel x_{n+1}-p\parallel }^2+2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel \hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel .\hfill \end{array}$$

Since ${\alpha }_n\to 0$ and $\parallel x_n-x_{n+1}\parallel \to 0$ as $n\to \mathrm{\infty }$, we have ${lim}_{n\to \mathrm{\infty }}\parallel F{x}_n-Fp\parallel =0$, ${lim}_{n\to \mathrm{\infty }}\parallel B_1{u}_n-B_{1}u\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel B_2{z}_n-B_{2}p\parallel =0$.

Step 5. ${lim}_{n\to \mathrm{\infty }}\parallel x_n-z_n\parallel =0$, ${lim}_{n\to \mathrm{\infty }}\parallel z_n-y_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$.

Indeed, from (3.2), (3.3) and Lemma 2.1, we have

$$\begin{array}{c}{\parallel u_n-u\parallel }^2\hfill \\ ={\parallel T_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)-T_{{\mu }_2}^{G_2}(p-{\mu }_2{B}_{2}p)\parallel }^2\hfill \\ \le 〈(z_n-{\mu }_2{B}_2{z}_n)-(p-{\mu }_2{B}_{2}p),u_n-u〉\hfill \\ =\frac{1}{2}[{\parallel (z_n-{\mu }_2{B}_2{z}_n)-(p-{\mu }_2{B}_{2}p)\parallel }^2+{\parallel u_n-u\parallel }^2\hfill \\ -{\parallel (z_n-{\mu }_2{B}_2{z}_n)-(p-{\mu }_2{B}_{2}p)-(u_n-u)\parallel }^2]\hfill \\ \le \frac{1}{2}[{\parallel z_n-p\parallel }^2+{\parallel u_n-u\parallel }^2-{\parallel (z_n-u_n)-{\mu }_2(B_2{z}_n-B_{2}p)-(p-u)\parallel }^2]\hfill \\ \le \frac{1}{2}[{\parallel x_n-p\parallel }^2+{\parallel u_n-u\parallel }^2-{\parallel (z_n-u_n)-(p-u)\parallel }^2\hfill \\ +2{\mu }_2〈(z_n-u_n)-(p-u),B_2{z}_n-B_{2}p〉-{\mu }_2^2{\parallel B_2{z}_n-B_{2}p\parallel }^2]\hfill \end{array}$$

and

$$\begin{array}{c}{\parallel y_n-p\parallel }^2\hfill \\ ={\parallel T_{{\mu }_1}^{G_1}(u_n-{\mu }_1{B}_1{u}_n)-T_{{\mu }_1}^{G_1}(u-{\mu }_1{B}_{1}u)\parallel }^2\hfill \\ \le 〈(u_n-{\mu }_1{B}_1{u}_n)-(u-{\mu }_1{B}_{1}u),y_n-p〉\hfill \\ =\frac{1}{2}[{\parallel (u_n-{\mu }_1{B}_1{u}_n)-(u-{\mu }_1{B}_{1}u)\parallel }^2+{\parallel y_n-p\parallel }^2\hfill \\ -{\parallel (u_n-{\mu }_1{B}_1{u}_n)-(u-{\mu }_1{B}_{1}u)-(y_n-p)\parallel }^2]\hfill \\ \le \frac{1}{2}[{\parallel u_n-u\parallel }^2+{\parallel y_n-p\parallel }^2-{\parallel (u_n-y_n)+(p-u)\parallel }^2\hfill \\ +2{\mu }_1〈B_1{u}_n-B_{1}u,(u_n-y_n)+(p-u)〉-{\mu }_1^2{\parallel B_1{u}_n-B_{1}u\parallel }^2]\hfill \\ \le \frac{1}{2}[{\parallel x_n-p\parallel }^2+{\parallel y_n-p\parallel }^2-{\parallel (u_n-y_n)+(p-u)\parallel }^2\hfill \\ +2{\mu }_1〈B_1{u}_n-B_{1}u,(u_n-y_n)+(p-u)〉],\hfill \end{array}$$

which imply that

$$\begin{array}{rcl}{\parallel u_n-u\parallel }^2& \le & {\parallel x_n-p\parallel }^2-{\parallel (z_n-u_n)-(p-u)\parallel }^2\\ +2{\mu }_2〈(z_n-u_n)-(p-u),B_2{z}_n-B_{2}p〉-{\mu }_2^2{\parallel B_2{z}_n-B_{2}p\parallel }^2\end{array}$$

(3.13)

and

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\parallel x_n-p\parallel }^2-{\parallel (u_n-y_n)+(p-u)\parallel }^2\\ +2{\mu }_1\parallel B_1{u}_n-B_{1}u\parallel \parallel (u_n-y_n)+(p-u)\parallel .\end{array}$$

(3.14)

It follows from (3.14) that

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ ={\parallel {\alpha }_n(\gamma f(W_n{x}_n)-Ap)+(I-{\alpha }_{n}A)(W_n{y}_n-p)\parallel }^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma }){\parallel y_n-p\parallel }^2\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma })[{\parallel x_n-p\parallel }^2-{\parallel (u_n-y_n)+(p-u)\parallel }^2\hfill \\ +2{\mu }_1\parallel B_1{u}_n-B_{1}u\parallel \parallel (u_n-y_n)+(p-u)\parallel ]+2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel ,\hfill \end{array}$$

which gives that

$$\begin{array}{c}(1-{\alpha }_n\overline{\gamma }){\parallel (u_n-y_n)+(p-u)\parallel }^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\hfill \\ +2{\mu }_1(1-{\alpha }_n\overline{\gamma })\parallel B_1{u}_n-B_{1}u\parallel \parallel (u_n-y_n)+(p-u)\parallel \hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel \hfill \\ +2{\mu }_1(1-{\alpha }_n\overline{\gamma })\parallel B_1{u}_n-B_{1}u\parallel \parallel (u_n-y_n)+(p-u)\parallel \hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel .\hfill \end{array}$$

Since ${\alpha }_n\to 0$, $\parallel x_{n+1}-x_n\parallel \to 0$ and $\parallel B_1{u}_n-B_{1}u\parallel \to 0$ as $n\to \mathrm{\infty }$, we have

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel (u_n-y_n)+(p-u)\parallel }^2=0.$$

(3.15)

Also, from (3.4) and (3.13), we have

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ ={\parallel {\alpha }_n(\gamma f(W_n{x}_n)-Ap)+(I-{\alpha }_{n}A)(W_n{y}_n-p)\parallel }^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma }){\parallel y_n-p\parallel }^2\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma }){\parallel u_n-u\parallel }^2\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma })[{\parallel x_n-p\parallel }^2-{\parallel (z_n-u_n)-(p-u)\parallel }^2\hfill \\ +2{\mu }_2〈(z_n-u_n)-(p-u),B_2{z}_n-B_{2}p〉]\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel .\hfill \end{array}$$

So, we have

$$\begin{array}{c}(1-{\alpha }_n\overline{\gamma }){\parallel (z_n-u_n)-(p-u)\parallel }^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\hfill \\ +2{\mu }_2\parallel (z_n-u_n)-(p-u)\parallel \parallel B_2{z}_n-B_{2}p\parallel \hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap)\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel \hfill \\ +2{\mu }_2\parallel (z_n-u_n)-(p-u)\parallel \parallel B_2{z}_n-B_{2}p\parallel \hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel .\hfill \end{array}$$

Note that $\parallel B_2{z}_n-B_{2}p\parallel \to 0$ as $n\to \mathrm{\infty }$. Then we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel (z_n-u_n)-(p-u)\parallel =0.$$

(3.16)

In addition, from the firm nonexpansivity of $T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}$, we have

$$\begin{array}{c}{\parallel z_n-p\parallel }^2\hfill \\ ={\parallel T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)-T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(p-{\delta }_{n}Fp)\parallel }^2\hfill \\ \le 〈(x_n-{\delta }_{n}F{x}_n)-(p-{\delta }_{n}Fp),z_n-p〉\hfill \\ =\frac{1}{2}[{\parallel (x_n-{\delta }_{n}F{x}_n)-(p-{\delta }_{n}Fp)\parallel }^2+{\parallel z_n-p\parallel }^2\hfill \\ -{\parallel (x_n-{\delta }_{n}F{x}_n)-(p-{\delta }_{n}Fp)-(z_n-p)\parallel }^2]\hfill \\ \le \frac{1}{2}[{\parallel x_n-p\parallel }^2+{\parallel z_n-p\parallel }^2-{\parallel x_n-z_n-{\delta }_n(F{x}_n-Fp)\parallel }^2]\hfill \\ =\frac{1}{2}[{\parallel x_n-p\parallel }^2+{\parallel z_n-p\parallel }^2-{\parallel x_n-z_n\parallel }^2+2{\delta }_n〈F{x}_n-Fp,x_n-z_n〉\hfill \\ -{\delta }_n^2{\parallel F{x}_n-Fp\parallel }^2],\hfill \end{array}$$

which implies that

$${\parallel z_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_n-z_n\parallel }^2+2{\delta }_n\parallel F{x}_n-Fp\parallel \parallel x_n-z_n\parallel .$$

(3.17)

From (3.1), (3.4) and (3.17), we have

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ ={\parallel {\alpha }_n(\gamma f(W_n{x}_n)-Ap)+(I-{\alpha }_{n}A)(W_n{y}_n-p)\parallel }^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma }){\parallel y_n-p\parallel }^2\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma }){\parallel z_n-p\parallel }^2\hfill \\ +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(1-{\alpha }_n\overline{\gamma })[{\parallel x_n-p\parallel }^2-{\parallel x_n-z_n\parallel }^2\hfill \\ +2{\delta }_n\parallel F{x}_n-Fp\parallel \parallel x_n-z_n\parallel ]+2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel .\hfill \end{array}$$

It follows that

$$\begin{array}{c}(1-{\alpha }_n\overline{\gamma }){\parallel x_n-z_n\parallel }^2\hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\hfill \\ +2(1-{\alpha }_n\overline{\gamma }){\delta }_n\parallel F{x}_n-Fp\parallel \parallel x_n-z_n\parallel +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel \hfill \\ \le {\alpha }_n{\parallel \gamma f(W_n{x}_n)-Ap\parallel }^2+(\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel \hfill \\ +2(1-{\alpha }_n\overline{\gamma }){\delta }_n\parallel F{x}_n-Fp\parallel \parallel x_n-z_n\parallel +2{\alpha }_n\parallel \gamma f(W_n{x}_n)-Ap\parallel \parallel y_n-p\parallel .\hfill \end{array}$$

Since $\parallel F{x}_n-Fp\parallel \to 0$ as $n\to \mathrm{\infty }$, we obtain

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

(3.18)

Thus, from (3.15), (3.16) and (3.18), we obtain that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}\parallel z_n-y_n\parallel & =& \underset{n\to \mathrm{\infty }}{lim}\parallel (z_n-u_n)-(p-u)+(u_n-y_n)+(p-u)\parallel \\ \le & \underset{n\to \mathrm{\infty }}{lim}\parallel z_n-u_n-(p-u)\parallel +\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-y_n+(p-u)\parallel \\ =& 0\end{array}$$

and

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-y_n\parallel & \le & \underset{n\to \mathrm{\infty }}{lim}\parallel x_n-z_n\parallel +\underset{n\to \mathrm{\infty }}{lim}\parallel z_n-y_n\parallel \\ =& 0.\end{array}$$

Step 6. ${lim}_{n\to \mathrm{\infty }}\parallel y_n-W_n{y}_n\parallel =0$.

Indeed, observe that

$$\begin{array}{rcl}\parallel x_n-W_n{y}_n\parallel & \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-W_n{y}_n\parallel \\ =& \parallel x_n-x_{n+1}\parallel +\parallel {\alpha }_n\gamma f(W_n{x}_n)+(I-{\alpha }_{n}A)W_n{y}_n-W_n{y}_n\parallel \\ \le & \parallel x_n-x_{n+1}\parallel +{\alpha }_n[\gamma \parallel f(W_n{x}_n)\parallel +\parallel A\parallel \parallel W_n{y}_n\parallel ].\end{array}$$

From Step 3 and ${\alpha }_n\to 0$ as $n\to \mathrm{\infty }$, we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n-W_n{y}_n\parallel =0$. Consequently,

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-W_n{y}_n\parallel & \le & \underset{n\to \mathrm{\infty }}{lim}(\parallel y_n-x_n\parallel +\parallel x_n-W_n{y}_n\parallel )\\ =& 0.\end{array}$$

Step 7. ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}〈\gamma f(x^{\ast })-A{x}^{\ast },x_n-x^{\ast }〉\le 0$, where $x^{\ast }=P_{\mathfrak{F}}(\gamma f+(I-A))(x^{\ast })$.

Indeed, take a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that

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

Correspondingly, there exists a subsequence $\{y_{n_i}\}$ of $\{y_n\}$. Since $\{y_{n_i}\}$ is bounded, there exists a subsequence of $y_{n_i}$ which converges weakly to w. Without loss of generality, we can assume that $y_{n_i}\rightharpoonup w$. Next we show $w\in \mathfrak{F}$. First, we prove that $w\in \mathrm{\Omega }$. Utilizing Lemma 2.1, we have for all $x,y\in C$

$$\begin{array}{c}{\parallel \mathrm{\Gamma }(x)-\mathrm{\Gamma }(y)\parallel }^2\hfill \\ =\parallel T_{{\mu }_1}^{G_1}[T_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)-{\mu }_1{B}_1{T}_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)]\hfill \\ -T_{{\mu }_1}^{G_1}[T_{{\mu }_2}^{G_2}(y-{\mu }_2{B}_{2}y)-{\mu }_1{B}_1{T}_{{\mu }_2}^{G_2}(y-{\mu }_2{B}_{2}y)]{\parallel }^2\hfill \\ \le \parallel T_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)-T_{{\mu }_2}^{G_2}(y-{\mu }_2{B}_{2}y)\hfill \\ -{\mu }_1[B_1{T}_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)-B_1{T}_{{\mu }_2}^{G_2}(y-{\mu }_2{B}_{2}y)]{\parallel }^2\hfill \\ \le {\parallel T_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)-T_{{\mu }_2}^{G_2}(y-{\mu }_2{B}_{2}y)\parallel }^2\hfill \\ +{\mu }_1({\mu }_1-2{\beta }_1){\parallel B_1{T}_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)-B_1{T}_{{\mu }_2}^{G_2}(y-{\mu }_2{B}_{2}y)\parallel }^2\hfill \\ \le {\parallel T_{{\mu }_2}^{G_2}(x-{\mu }_2{B}_{2}x)-T_{{\mu }_2}^{G_2}(y-{\mu }_2{B}_{2}y)\parallel }^2\hfill \\ \le {\parallel x-y-{\mu }_2(B_{2}x-B_{2}y)\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2+{\mu }_2({\mu }_2-2{\beta }_2){\parallel B_{2}x-B_{2}y\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2.\hfill \end{array}$$

This shows that $\mathrm{\Gamma }:C\to C$ is nonexpansive. Note that

$$\begin{array}{rcl}\parallel y_n-\mathrm{\Gamma }(y_n)\parallel & =& \parallel \mathrm{\Gamma }(z_n)-\mathrm{\Gamma }(y_n)\parallel \\ \le & \parallel z_n-y_n\parallel \\ \to & 0\text{as }n\to \mathrm{\infty }.\end{array}$$

According to Lemma 2.2 and Lemma 2.3, we obtain $w\in \mathrm{\Omega }$.

Next, let us show that $w\in \mathrm{GMEP}$. From $z_n=T_{{\delta }_n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)$, we obtain

$$\mathrm{\Theta }(z_n,y)+\phi (y)-\phi (z_n)+\frac{1}{{\delta }_n}〈y-z_n,z_n-(x_n-{\delta }_{n}F{x}_n)〉\ge 0,\mathrm{\forall }y\in C.$$

It follows from (H2) that

$$\phi (y)-\phi (z_n)+〈y-z_n,F{x}_n〉+\frac{1}{{\delta }_n}〈y-z_n,z_n-x_n〉\ge \mathrm{\Theta }(y,z_n),\mathrm{\forall }y\in C.$$

(3.19)

Replacing n by $n_i$, we have

$$\phi (y)-\phi (z_{n_i})+〈y-z_{n_i},F{x}_{n_i}〉+〈y-z_{n_i},\frac{z_{n_i}-x_{n_i}}{{\delta }_{n_i}}〉\ge \mathrm{\Theta }(y,z_{n_i}),\mathrm{\forall }y\in C.$$

Let $z_t=ty+(1-t)w$ for all $t\in [0,1]$ and $y\in C$. Then we have $z_t\in C$. It follows from (3.19) that

$$\begin{array}{rcl}〈z_t-z_{n_i},F{z}_t〉& \ge & 〈z_t-z_{n_i},F{z}_t〉-\phi (z_t)+\phi (z_{n_i})-〈z_t-z_{n_i},F{x}_{n_i}〉\\ -〈z_t-z_{n_i},\frac{z_{n_i}-x_{n_i}}{{\delta }_{n_i}}〉+\mathrm{\Theta }(z_t,z_{n_i})\\ =& 〈z_t-z_{n_i},F{z}_t-F{z}_{n_i}〉+〈z_t-z_{n_i},F{z}_{n_i}-F{x}_{n_i}〉\\ -\phi (z_t)+\phi (z_{n_i})-〈z_t-z_{n_i},\frac{z_{n_i}-x_{n_i}}{{\delta }_{n_i}}〉+\mathrm{\Theta }(z_t,z_{n_i}).\end{array}$$

Since $\parallel z_{n_i}-x_{n_i}\parallel \to 0$, we have $\parallel F{z}_{n_i}-F{x}_{n_i}\parallel \to 0$. From the monotonicity of F, we have

$$〈F{z}_t-F{z}_{n_i},z_t-z_{n_i}〉\ge 0.$$

From (H4), the weakly lower semicontinuity of φ, $\frac{z_{n_i}-x_{n_i}}{{\delta }_{n_i}}\to 0$ and $z_{n_i}\rightharpoonup w$, we have

$$〈z_t-w,F{z}_t〉\ge -\phi (z_t)+\phi (w)+\mathrm{\Theta }(z_t,w)$$

(3.20)

as $i\to \mathrm{\infty }$. By (H1), (H4) and (3.20), we obtain

$$\begin{array}{rcl}0& =& \mathrm{\Theta }(z_t,z_t)+\phi (z_t)-\phi (z_t)\\ =& \mathrm{\Theta }(z_t,ty+(1-t)w)+\phi (ty+(1-t)w)-\phi (z_t)\\ \le & t\mathrm{\Theta }(z_t,y)+(1-t)\mathrm{\Theta }(z_t,w)+t\phi (y)+(1-t)\phi (w)-\phi (z_t)\\ =& t[\mathrm{\Theta }(z_t,y)+\phi (y)-\phi (z_t)]+(1-t)[\mathrm{\Theta }(z_t,w)+\phi (w)-\phi (z_t)]\\ \le & t[\mathrm{\Theta }(z_t,y)+\phi (y)-\phi (z_t)]+(1-t)〈z_t-w,F{z}_t〉\\ =& t[\mathrm{\Theta }(z_t,y)+\phi (y)-\phi (z_t)]+(1-t)t〈y-w,F{z}_t〉.\end{array}$$

Hence we obtain

$$0\le \mathrm{\Theta }(z_t,y)+\phi (y)-\phi (z_t)+(1-t)〈y-w,F{z}_t〉.$$

Putting $t\to 0$, we have

$$0\le \mathrm{\Theta }(w,y)+\phi (y)-\phi (w)+〈y-w,Fw〉,\mathrm{\forall }y\in C.$$

This implies that $w\in \mathrm{GMEP}$.

Since Hilbert spaces satisfy Opial’s condition, it follows from Step 5 that

$$\begin{array}{rcl}\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel y_{n_i}-w\parallel & <& \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel y_{n_i}-W_{n}w\parallel \\ \le & \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}[\parallel y_{n_i}-W_n{y}_{n_i}\parallel +\parallel W_n{y}_{n_i}-W_{n}w\parallel ]\\ \le & \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel y_{n_i}-W_n{y}_{n_i}\parallel +\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel W_n{y}_{n_i}-W_{n}w\parallel \\ =& \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel W_n{y}_{n_i}-W_{n}w\parallel \\ \le & \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel y_{n_i}-w\parallel ,\end{array}$$

which derives a contraction. This implies that $w\in F(W_n)$. It follows from $F(W_n)={\bigcap }_{i=1}^{N}F(T_i)$ that $w\in {\bigcap }_{i=1}^{N}F(T_i)$.

Since $x^{\ast }=P_{\mathfrak{F}}(\gamma f+(I-A))(x^{\ast })$, we have

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_n-x^{\ast }〉& =& \underset{n\to \mathrm{\infty }}{lim}〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n_i}-x^{\ast }〉\\ =& 〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },w-x^{\ast }〉\\ \le & 0.\end{array}$$

Step 8. $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$.

Indeed, from Lemma 2.6 and (3.4), we have

$$\begin{array}{c}{\parallel x_{n+1}-x^{\ast }\parallel }^2\hfill \\ ={\parallel {\alpha }_n\gamma f(W_n{x}_n)+(I-{\alpha }_{n}A)W_n{y}_n-x^{\ast }\parallel }^2\hfill \\ ={\parallel (I-{\alpha }_{n}A)(W_n{y}_n-x^{\ast })+{\alpha }_n(\gamma f(W_n{x}_n)-A{x}^{\ast })\parallel }^2\hfill \\ \le {\parallel (I-{\alpha }_{n}A)(W_n{y}_n-x^{\ast })\parallel }^2+2{\alpha }_n〈\gamma f(W_n{x}_n)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\hfill \\ \le {(1-{\alpha }_n\overline{\gamma })}^2{\parallel y_n-x^{\ast }\parallel }^2+2{\alpha }_n〈\gamma f(W_n{x}_n)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\hfill \\ \le {(1-{\alpha }_n\overline{\gamma })}^2{\parallel x_n-x^{\ast }\parallel }^2+2{\alpha }_n\gamma 〈f(W_n{x}_n)-f\left(x^{\ast }\right),x_{n+1}-x^{\ast }〉\hfill \\ +2{\alpha }_n〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\hfill \\ \le {(1-{\alpha }_n\overline{\gamma })}^2{\parallel x_n-x^{\ast }\parallel }^2+2{\alpha }_n\gamma \alpha \parallel x_n-x^{\ast }\parallel \parallel x_{n+1}-x^{\ast }\parallel \hfill \\ +2{\alpha }_n〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\hfill \\ \le {(1-{\alpha }_n\overline{\gamma })}^2{\parallel x_n-x^{\ast }\parallel }^2+{\alpha }_n\gamma \alpha ({\parallel x_n-x^{\ast }\parallel }^2+{\parallel x_{n+1}-x^{\ast }\parallel }^2)\hfill \\ +2{\alpha }_n〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉,\hfill \end{array}$$

which implies that

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

Put

$${\gamma }_n=\frac{2{\alpha }_n(\overline{\gamma }-\alpha \gamma )}{1-{\alpha }_n\gamma \alpha }$$

and

$${\xi }_n=\frac{2{\alpha }_n(\overline{\gamma }-\alpha \gamma )}{1-{\alpha }_n\gamma \alpha }[\frac{{\alpha }_n{\overline{\gamma }}^2}{2(\overline{\gamma }-\alpha \gamma )}{\parallel x_n-x^{\ast }\parallel }^2+\frac{1}{\overline{\gamma }-\alpha \gamma }〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉].$$

Then we can write the last inequality as

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\xi }_n.$$

It follows from condition (i) and Step 6 that

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

and

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{{\xi }_n}{{\gamma }_n}& =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\{\frac{{\alpha }_n{\overline{\gamma }}^2}{2(\overline{\gamma }-\alpha \gamma )}{\parallel x_n-x^{\ast }\parallel }^2\\ +\frac{1}{\overline{\gamma }-\alpha \gamma }〈\gamma f\left(x^{\ast }\right)-A{x}^{\ast },x_{n+1}-x^{\ast }〉\}\\ \le & 0.\end{array}$$

Hence, applying Lemma 2.4, we immediately obtain that $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$. This completes the proof. □

As corollaries of Theorem 3.1, we have the following results.

Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $\mathrm{\Theta },G_1,G_2:C\times C\to \mathbb{R}$ be three bifunctions which satisfy assumptions (H1)-(H4) and $\phi :C\to \mathbb{R}$ be a lower semicontinuous and convex function satisfying (A1) or (A2). Let the mappings $B_1,B_2:C\to H$ be ${\beta }_1$-inverse strongly monotone and ${\beta }_2$-inverse strongly monotone, respectively. Let $T_1,T_2,\dots ,T_N$ be a finite family of nonexpansive mappings of C into H such that $\mathfrak{F}={\bigcap }_{i=1}^{N}F(T_i)\cap \mathrm{MEP}\cap \mathrm{\Omega }\ne \varphi$. Let f be a contraction of H into itself with a constant α ($0<\alpha <1$) and let A be a strongly positive linear bounded operator with a coefficient $\overline{\gamma }>0$ such that $\parallel A\parallel \le 1$. Assume that $0<\gamma <\frac{\overline{\gamma }}{\alpha }$. Let $x_1\in C$ and let $\{x_n\}$ be a sequence defined by

$$\{\begin{array}{c}\mathrm{\Theta }(z_n,y)+\phi (y)-\phi (z_n)+\frac{1}{{\delta }_n}〈y-z_n,z_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ y_n=T_{{\mu }_1}^{G_1}[T_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)-{\mu }_1{B}_1{T}_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)],\hfill \\ x_{n+1}={\alpha }_n\gamma f(W_n{x}_n)+(1-{\alpha }_{n}A)W_n{y}_n,n\ge 1,\hfill \end{array}$$

where ${\alpha }_n\in [0,1]$, ${\mu }_1\in (0,2{\beta }_1)$, ${\mu }_2\in (0,2{\beta }_2)$ and $\{{\delta }_n\}\subset (0,\mathrm{\infty })$ satisfy the following conditions:

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

• (ii) ${lim}_{n\to \mathrm{\infty }}{\lambda }_{n,i}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n,i}-{\lambda }_{n-1,i}|<\mathrm{\infty }$ for all $i=1,2,\dots ,N$,

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^N|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$.

Then $\{x_n\}$ converges strongly to $x^{\ast }=P_{{\bigcap }_{i=1}^{N}F(T_i)\cap \mathrm{MEP}\cap \mathrm{\Omega }}(\overline{\gamma }f+(I-A))(x^{\ast })$ and $(x^{\ast },y^{\ast })$ is a solution of problem (1.4), where $y^{\ast }=T_{{\mu }_2}^{G_2}(x^{\ast }-{\mu }_2{B}_2{x}^{\ast })$, which solves the following variational inequality:

$$〈(A-\gamma f)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \bigcap _{i=1}^{N}F(T_i)\cap \mathrm{MEP}\cap \mathrm{\Omega }.$$

Proof In Theorem 3.1, for all $n\ge 0$, $z_n=T_{\delta n}^{(\mathrm{\Theta },\phi )}(x_n-{\delta }_{n}F{x}_n)$ is equivalent to

$$\mathrm{\Theta }(z_n,y)+\phi (y)-\phi (z_n)+〈F{x}_n,y-z_n〉+\frac{1}{{\delta }_n}〈y-z_n,z_n-x_n〉\ge 0,\mathrm{\forall }y\in C.$$

(3.21)

Putting $F\equiv 0$, we obtain

$$\mathrm{\Theta }(z_n,y)+\phi (y)-\phi (z_n)+\frac{1}{{\delta }_n}〈y-z_n,z_n-x_n〉\ge 0,\mathrm{\forall }y\in C.$$

 □

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let $G_1,G_2:C\times C\to \mathbb{R}$ be two bifunctions with satisfy assumptions (H1)-(H4). Let the mappings $F,B_1,B_2:C\to H$ be ζ-inverse strongly monotone, ${\beta }_1$-inverse strongly monotone and ${\beta }_2$-inverse strongly monotone, respectively. Let $T_1,T_2,\dots ,T_N$ be a finite family of nonexpansive mappings of C into H such that ${\bigcap }_{i=1}^{N}F(T_i)\cap VI(A,C)\cap \mathrm{\Omega }\ne \varphi$. Let f be a contraction of H into itself with a constant α ($0<\alpha <1$) and let A be a strongly positive linear bounded operator with a coefficient $\overline{\gamma }>0$ such that $\parallel A\parallel \le 1$. Assume that $0<\gamma <\frac{\overline{\gamma }}{\alpha }$. Let $x_1\in C$ and let $\{x_n\}$ be a sequence defined by

$$\{\begin{array}{c}{z}_n=P_C(x_n-\delta F{x}_n),\hfill \\ y_n=T_{{\mu }_1}^{G_1}[T_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)-{\mu }_1{B}_1{T}_{{\mu }_2}^{G_2}(z_n-{\mu }_2{B}_2{z}_n)],\hfill \\ x_{n+1}={\alpha }_n\gamma f(W_n{x}_n)+(1-{\alpha }_{n}A)W_n{y}_n,n\ge 1,\hfill \end{array}$$

where ${\alpha }_n\in [0,1]$, ${\mu }_1\in (0,2{\beta }_1)$, ${\mu }_2\in (0,2{\beta }_2)$ and $\{{\delta }_n\}\subset [0,2\zeta ]$ satisfy the following conditions:

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

• (ii) ${lim}_{n\to \mathrm{\infty }}{\lambda }_{n,i}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n,i}-{\lambda }_{n-1,i}|<\mathrm{\infty }$ for all $i=1,2,\dots ,N$;

• (iii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n>0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$.

Then $\{x_n\}$ converges strongly to $x^{\ast }=P_{{\bigcap }_{i=1}^{N}F(T_i)\cap VI(A,C)\cap \mathrm{\Omega }}(\gamma f+(I-A))(x^{\ast })$ and $(x^{\ast },y^{\ast })$ is a solution of problem (1.4), where $y^{\ast }=T_{{\mu }_2}^{G_2}(x^{\ast }-{\mu }_2{B}_2{x}^{\ast })$, which solves the following variational inequality:

$$〈(A-\gamma f)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \bigcap _{i=1}^{N}F(T_i)\cap VI(A,C)\cap \mathrm{\Omega }.$$

Proof Put $\mathrm{\Theta }=0$ and $\phi =0$ in Theorem 3.1. Then we have from (3.21) that

$$〈F{x}_n,y-z_n〉+\frac{1}{{\delta }_n}〈y-z_n,z_n-x_n〉\ge 0,\mathrm{\forall }y\in C,n\ge 1.$$

That is,

$$〈y-z_n,x_n-{\delta }_{n}F{x}_n-z_n〉\le 0,\mathrm{\forall }y\in C.$$

It follows that $P_C(x_n-{\delta }_{n}F{x}_n)=z_n$ for all $n\ge 1$. We can obtain the desired conclusion easily. □

Remark 3.1 We can see easily that Takahashi and Takahashi [18], Peng and Yao’s [1] results are special cases of Theorem 3.1.