On a finite family of variational inclusions with the constraints of generalized mixed equilibrium and fixed point problems

Abstract

In this paper, we introduce two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.

MSC:49J30, 47H09, 47J20, 49M05.

1 Introduction

Let H be a real Hilbert space with inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$, C be a nonempty closed convex subset of H and $P_C$ be the metric projection of H onto C. Let $S:C\to H$ be a nonlinear mapping on C. We denote by $Fix(S)$ the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant $\overline{\gamma }\in (0,1]$ such that

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

A mapping $S:C\to H$ is called L-Lipschitz-continuous if there exists a constant $L>0$ such that

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

In particular, if $L=1$ then S is called a nonexpansive mapping; if $L\in (0,1)$ then A is called a contraction.

Let $\phi :C\to \mathbf{R}$ be a real-valued function, $A:H\to H$ be a nonlinear mapping and $\Theta :C\times C\to \mathbf{R}$ be a bifunction. We consider the generalized mixed equilibrium problem (GMEP) [1] of finding $x\in C$ such that

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

(1.1)

We denote the set of solutions of GMEP (1.1) by $GMEP(\Theta ,\phi ,A)$. The GMEP (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied in, e.g., [2–8].

Throughout this paper, it is assumed as in [1] that $\Theta :C\times C\to \mathbf{R}$ is a bifunction satisfying conditions (H1)-(H4) and $\phi :C\to \mathbf{R}$ is a lower semicontinuous and convex function with restriction (H5), where

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

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

• (H3) Θ is upper-hemicontinuous, i.e., for each $x,y,z\in C$,

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

• (H4) $\Theta (x,\cdot )$ is convex and lower semicontinuous for each $x\in C$;

• (H5) for each $x\in H$ and $r>0$, there exist a bounded subset $D_x\subset C$ and $y_x\in C$ such that for any $z\in C\setminus D_x$,

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

Let ${\Theta }_1,{\Theta }_2:C\times C\to \mathbf{R}$ be two bifunctions, and $B_1,B_2:C\to H$ be two nonlinear mappings. Consider the system of generalized equilibrium problems (SGEP): find $(x^{\ast },y^{\ast })\in C\times C$ such that

$$\{\begin{array}{l}{\Theta }_1(x^{\ast },x)+〈B_1{y}^{\ast },x-x^{\ast }〉+\frac{1}{{\mu }_1}〈x^{\ast }-y^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\\ {\Theta }_2(y^{\ast },y)+〈B_2{x}^{\ast },y-y^{\ast }〉+\frac{1}{{\mu }_2}〈y^{\ast }-x^{\ast },y-y^{\ast }〉\ge 0,\mathrm{\forall }y\in C,\end{array}$$

(1.2)

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

Let ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be an infinite family of nonexpansive self-mappings on C and ${\{{\lambda }_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonnegative numbers in $[0,1]$. For any $n\ge 1$, define a self-mapping $W_n$ on H as follows:

$$\{\begin{array}{l}{U}_{n,n+1}=I,\\ U_{n,n}={\lambda }_n{T}_n{U}_{n,n+1}+(1-{\lambda }_n)I,\\ U_{n,n-1}={\lambda }_{n-1}{T}_{n-1}{U}_{n,n}+(1-{\lambda }_{n-1})I,\\ ⋮\\ U_{n,k}={\lambda }_k{T}_k{U}_{n,k+1}+(1-{\lambda }_k)I,\\ U_{n,k-1}={\lambda }_{k-1}{T}_{k-1}{U}_{n,k}+(1-{\lambda }_{k-1})I,\\ ⋮\\ U_{n,2}={\lambda }_2{T}_2{U}_{n,3}+(1-{\lambda }_2)I,\\ W_n=U_{n,1}={\lambda }_1{T}_1{U}_{n,2}+(1-{\lambda }_1)I.\end{array}$$

(1.3)

Such a mapping $W_n$ is called the W-mapping generated by $T_n,T_{n-1},\dots ,T_1$ and ${\lambda }_n,{\lambda }_{n-1},\dots ,{\lambda }_1$.

Let $f:H\to H$ be a contraction and V be a strongly positive bounded linear operator on H. Assume that $\phi :H\to \mathbf{R}$ is a lower semicontinuous and convex functional, that $\Theta ,{\Theta }_1,{\Theta }_2:H\times H\to \mathbf{R}$ satisfy conditions (H1)-(H4), and that $A,B_1,B_2:H\to H$ are inverse-strongly monotone. Very recently, motivated by Yao et al. [3], Cai and Bu [4] introduced the following hybrid extragradient-like iterative algorithm:

$$\{\begin{array}{l}{z}_n=S_{r_n}^{(\Theta ,\phi )}(x_n-r_{n}A{x}_n),\\ y_n=T_{{\mu }_1}^{{\Theta }_1}(I-{\mu }_1{B}_1)T_{{\mu }_2}^{{\Theta }_2}(I-{\mu }_2{B}_2)z_n,\\ x_{n+1}={\alpha }_n(u+\gamma f(x_n))+{\beta }_n{x}_n+((1-{\beta }_n)I-{\alpha }_n(I+\mu V))W_n{y}_n,\mathrm{\forall }n\ge 0,\end{array}$$

(1.4)

for finding a common solution of GMEP (1.1), SGEP (1.2), and the fixed point problem of an infinite family of nonexpansive mappings ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ on H, where $\{r_n\}\subset (0,\mathrm{\infty })$, $\{{\alpha }_n\},\{{\beta }_n\}\subset (0,1)$, and $x_0,u\in H$ are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (1.4) to a point $x^{\ast }\in ({\bigcap }_{i=1}^{\mathrm{\infty }}Fix(T_i))\cap GMEP(\Theta ,\phi ,A)\cap SGEP(G)$ under some suitable conditions, where $SGEP(G)$ is the fixed point set of the mapping $G:=T_{{\mu }_1}^{{\Theta }_1}(I-{\mu }_1{B}_1)T_{{\mu }_2}^{{\Theta }_2}(I-{\mu }_2{B}_2)$. This point $x^{\ast }$ also solves the following optimization problem:

$$\underset{x\in ({\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n))\cap GMEP(\Theta ,\phi ,A)\cap SGEP(G)}{min}\frac{\mu }{2}〈Vx,x〉+\frac{1}{2}{\parallel x-u\parallel }^2-h(x),$$

(OP1)

where $h:H\to \mathbf{R}$ is the potential function of γf.

Let B be a single-valued mapping of C into H and R be a set-valued mapping with $D(R)=C$. Consider the following variational inclusion: find a point $x\in C$ such that

$$0\in Bx+Rx.$$

(1.5)

We denote by $\mathrm{I}(B,R)$ the solution set of the variational inclusion (1.5). In particular, if $B=R=0$, then $\mathrm{I}(B,R)=C$. If $B=0$, then problem (1.5) becomes the inclusion problem introduced by Rockafellar [9]. It is known that problem (1.5) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping $R:D(R)\subset H\to 2^H$ be maximal monotone. We define the resolvent operator $J_{R,\lambda }:H\to \overline{D(R)}$ associated with R and λ as follows:

$$J_{R,\lambda }={(I+\lambda R)}^{-1},\mathrm{\forall }x\in H,$$

where λ is a positive number.

In 1998, Huang [10] studied problem (1.5) in the case where R is maximal monotone and B is strongly monotone and Lipschitz-continuous with $D(R)=C=H$. Subsequently, Zeng et al. [11] further studied problem (1.5) in the case which is more general than Huang’s [10]. Moreover, the authors [11] obtained the same strong convergence conclusion as in Huang’s result [10]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [5, 12–17] and the references therein.

In 2011, for the case where $C=H$, Yao et al. [5] introduced and analyzed an iterative algorithms for finding a common element of the set of solutions of the GMEP (1.1), the set of solutions of the variational inclusion (1.5) for maximal monotone and inverse-strongly monotone mappings and the set of fixed points of a countable family of nonexpansive mappings on H.

Recently, Kim and Xu [18] introduced the concept of asymptotically κ-strict pseudocontractive mappings in a Hilbert space.

Definition 1.1 Let C be a nonempty subset of a Hilbert space H. A mapping $S:C\to C$ is said to be an asymptotically κ-strict pseudocontractive mapping with sequence $\{{\gamma }_n\}$ if there exist a constant $\kappa \in [0,1)$ and a sequence $\{{\gamma }_n\}$ in $[0,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$ such that

$${\parallel S^{n}x-S^{n}y\parallel }^2\le (1+{\gamma }_n){\parallel x-y\parallel }^2+\kappa {\parallel x-S^{n}x-(y-S^{n}y)\parallel }^2,\mathrm{\forall }n\ge 1,\mathrm{\forall }x,y\in C.$$

Subsequently, Sahu et al. [19] considered the concept of asymptotically κ-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 1.2 Let C be a nonempty subset of a Hilbert space H. A mapping $S:C\to C$ is said to be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence $\{{\gamma }_n\}$ if there exist a constant $\kappa \in [0,1)$ and a sequence $\{{\gamma }_n\}$ in $[0,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}({\parallel S^{n}x-S^{n}y\parallel }^2-(1+{\gamma }_n){\parallel x-y\parallel }^2-\kappa {\parallel x-S^{n}x-(y-S^{n}y)\parallel }^2)\le 0.$$

(1.6)

Put $c_n:=max\{0,{sup}_{x,y\in C}({\parallel S^{n}x-S^{n}y\parallel }^2-(1+{\gamma }_n){\parallel x-y\parallel }^2-\kappa {\parallel x-S^{n}x-(y-S^{n}y)\parallel }^2)\}$. Then $c_n\ge 0$ ($\mathrm{\forall }n\ge 1$), $c_n\to 0$ ($n\to \mathrm{\infty }$), and (1.6) reduce to the relation

$${\parallel S^{n}x-S^{n}y\parallel }^2\le (1+{\gamma }_n){\parallel x-y\parallel }^2+\kappa {\parallel x-S^{n}x-(y-S^{n}y)\parallel }^2+c_n,\mathrm{\forall }n\ge 1,\mathrm{\forall }x,y\in C.$$

(1.7)

Whenever $c_n=0$ for all $n\ge 1$ in (1.7), then S is an asymptotically κ-strict pseudocontractive mapping with sequence $\{{\gamma }_n\}$. The authors [19] derived the weak and strong convergence of the modified Mann iteration processes for an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence $\{{\gamma }_n\}$. More precisely, they first established one weak convergence theorem for the following iterative scheme:

$$\{\begin{array}{l}{x}_1=x\in C\text{ chosen arbitrarily},\\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{S}^n{x}_n,\mathrm{\forall }n\ge 1,\end{array}$$

where $0<\delta \le {\alpha }_n\le 1-\kappa -\delta$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n{c}_n<\mathrm{\infty }$, and ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n<\mathrm{\infty }$; and then obtained another strong convergence theorem for the following iterative scheme:

$$\{\begin{array}{l}{x}_1=x\in C\text{ chosen arbitrary},\\ y_n=(1-{\alpha }_n)x_n+{\alpha }_n{S}^n{x}_n,\\ C_n=\{z\in C:{\parallel y_n-z\parallel }^2\le {\parallel x_n-z\parallel }^2+{\theta }_n\},\\ Q_n=\{z\in C:〈x_n-z,x-x_n〉\ge 0\},\\ x_{n+1}=P_{C_n\cap Q_n}x,\mathrm{\forall }n\ge 1,\end{array}$$

where $0<\delta \le {\alpha }_n\le 1-\kappa$, ${\theta }_n=c_n+{\gamma }_n{\Delta }_n$, and ${\Delta }_n=sup\{{\parallel x_n-z\parallel }^2:z\in Fix(S)\}<\mathrm{\infty }$.

Inspired by the above facts, we in this paper introduce two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions. The results presented in this paper are the supplement, extension, improvement, and generalization of the previously known results in this area.

2 Preliminaries

Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$, respectively. Let C be a nonempty closed convex subset of H. We write $x_n\rightharpoonup x$ to indicate that the sequence $\{x_n\}$ converges weakly to x and $x_n\to x$ to indicate that the sequence $\{x_n\}$ converges strongly to x. Moreover, we use ${\omega }_w(x_n)$ to denote the weak ω-limit set of the sequence $\{x_n\}$, i.e.,

$${\omega }_w(x_n):=\{x\in H:x_{n_i}\rightharpoonup x\text{ for some subsequence }\{x_{n_i}\}\text{ of }\{x_n\}\}.$$

Definition 2.1 A mapping $A:C\to H$ is called

• (i) monotone if

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

• (ii) η-strongly monotone if there exists a constant $\eta >0$ such that

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

• (iii) ζ-inverse-strongly monotone if there exists a constant $\zeta >0$ such that

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

It is easy to see that the projection $P_C$ is 1-inverse-strongly monotone (in short, 1-ism). Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

Definition 2.2 A differentiable function $K:H\to \mathbf{R}$ is called:

• (i) convex, if

  $$K(y)-K(x)\ge 〈K^{\prime }(x),y-x〉,\mathrm{\forall }x,y\in H,$$

  where $K^{\prime }(x)$ is the Frechet derivative of K at x;

• (ii) strongly convex, if there exists a constant $\sigma >0$ such that

  $$K(y)-K(x)-〈K^{\prime }(x),y-x〉\ge \frac{\sigma }{2}{\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H.$$

It is easy to see that if $K:H\to \mathbf{R}$ is a differentiable strongly convex function with constant $\sigma >0$ then $K^{\prime }:H\to H$ is strongly monotone with constant $\sigma >0$.

The metric (or nearest point) projection from H onto C is the mapping $P_C:H\to C$ which assigns to each point $x\in H$ the unique point $P_{C}x\in C$ satisfying the property

$$\parallel x-P_{C}x\parallel =\underset{y\in C}{inf}\parallel x-y\parallel =:d(x,C).$$

Some important properties of projections are gathered in the following proposition.

Proposition 2.1 For given $x\in H$ and $z\in C$:

• (i) $z=P_{C}x\iff 〈x-z,y-z〉\le 0$, $\mathrm{\forall }y\in C$;

• (ii) $z=P_{C}x\iff {\parallel x-z\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel y-z\parallel }^2$, $\mathrm{\forall }y\in C$;

• (iii) $〈P_{C}x-P_{C}y,x-y〉\ge {\parallel P_{C}x-P_{C}y\parallel }^2$, $\mathrm{\forall }y\in H$. (This implies that $P_C$ is nonexpansive and monotone.)

By using the technique of [20], we can readily obtain the following elementary result.

Proposition 2.2 (see [[6], Lemma 1 and Proposition 1])

Let C be a nonempty closed convex subset of a real Hilbert space H and let $\phi :C\to \mathbf{R}$ be a lower semicontinuous and convex function. Let $\Theta :C\times C\to \mathbf{R}$ be a bifunction satisfying the conditions (H1)-(H4). Assume that

• (i) $K:H\to \mathbf{R}$ is strongly convex with constant $\sigma >0$ and the function $x\mapsto 〈y-x,K^{\prime }(x)〉$ is weakly upper semicontinuous for each $y\in H$;

• (ii) for each $x\in H$ and $r>0$, there exist a bounded subset $D_x\subset C$ and $y_x\in C$ such that for any $z\in C\setminus D_x$,

  $$\Theta (z,y_x)+\phi (y_x)-\phi (z)+\frac{1}{r}〈K^{\prime }(z)-K^{\prime }(x),y_x-z〉<0.$$

Then the following hold:

• (a) for each $x\in H$, $S_r^{(\Theta ,\phi )}(x)\ne \mathrm{\varnothing }$;

• (b) $S_r^{(\Theta ,\phi )}$ is single-valued;

• (c) $S_r^{(\Theta ,\phi )}$ is nonexpansive if $K^{\prime }$ is Lipschitz-continuous with constant $\nu >0$ and

  $$〈K^{\prime }(x_1)-K^{\prime }(x_2),u_1-u_2〉\ge 〈K^{\prime }(u_1)-K^{\prime }(u_2),u_1-u_2〉,\mathrm{\forall }(x_1,x_2)\in H\times H,$$

  where $u_i=S_r^{(\Theta ,\phi )}(x_i)$ for $i=1,2$;

• (d) for all $s,t>0$ and $x\in H$

  $$\begin{array}{r}〈K^{\prime }\left(S_s^{(\Theta ,\phi )}x\right)-K^{\prime }\left(S_t^{(\Theta ,\phi )}x\right),S_s^{(\Theta ,\phi )}x-S_t^{(\Theta ,\phi )}x〉\\ \le \frac{s-t}{s}〈K^{\prime }\left(S_s^{(\Theta ,\phi )}x\right)-K^{\prime }(x),S_s^{(\Theta ,\phi )}x-S_t^{(\Theta ,\phi )}x〉;\end{array}$$

• (e) $Fix(S_r^{(\Theta ,\phi )})=MEP(\Theta ,\phi )$;

• (f) $MEP(\Theta ,\phi )$ is closed and convex.

In particular, whenever $\Theta :C\times C\to \mathbf{R}$ is a bifunction satisfying the conditions (H1)-(H4) and $K(x)=\frac{1}{2}{\parallel x\parallel }^2$, $\mathrm{\forall }x\in H$, then, for any $x,y\in H$,

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

($S_r^{(\Theta ,\phi )}$ is firmly nonexpansive) and

$$\parallel S_s^{(\Theta ,\phi )}x-S_t^{(\Theta ,\phi )}x\parallel \le \frac{|s-t|}{s}\parallel S_s^{(\Theta ,\phi )}x-x\parallel ,\mathrm{\forall }s,t>0,x\in H.$$

In this case, $S_r^{(\Theta ,\phi )}$ is rewritten as $T_r^{(\Theta ,\phi )}$. If, in addition, $\phi \equiv 0$, then $T_r^{(\Theta ,\phi )}$ is rewritten as $T_r^{\Theta }$ (see [[21], Lemma 2.1] for more details).

We need some facts and tools in a real Hilbert space H which are listed as lemmas below.

Lemma 2.1 Let X be a real inner product space. Then we have the following inequality:

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉,\mathrm{\forall }x,y\in X.$$

Lemma 2.2 Let H be a real Hilbert space. Then the following hold:

• (a) ${\parallel x-y\parallel }^2={\parallel x\parallel }^2-{\parallel y\parallel }^2-2〈x-y,y〉$ for all $x,y\in H$;

• (b) ${\parallel \lambda x+\mu y\parallel }^2=\lambda {\parallel x\parallel }^2+\mu {\parallel y\parallel }^2-\lambda \mu {\parallel x-y\parallel }^2$ for all $x,y\in H$ and $\lambda ,\mu \in [0,1]$ with $\lambda +\mu =1$;

• (c) If $\{x_n\}$ is a sequence in H such that $x_n\rightharpoonup x$, it follows that

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel x_n-y\parallel }^2=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel x_n-x\parallel }^2+{\parallel x-y\parallel }^2,\mathrm{\forall }y\in H.$$

Lemma 2.3 ([[19], Lemma 2.5])

Let H be a real Hilbert space. Given a nonempty closed convex subset of H and points $x,y,z\in H$ and given also a real number $a\in \mathbf{R}$, the set

$$\{v\in C:{\parallel y-v\parallel }^2\le {\parallel x-v\parallel }^2+〈z,v〉+a\}$$

is convex (and closed).

Lemma 2.4 ([[19], Lemma 2.6])

Let C be a nonempty subset of a Hilbert space H and $S:C\to C$ be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence $\{{\gamma }_n\}$. Then

$$\parallel S^{n}x-S^{n}y\parallel \le \frac{1}{1-\kappa }(\kappa \parallel x-y\parallel +\sqrt{(1+(1-\kappa ){\gamma }_n){\parallel x-y\parallel }^2+(1-\kappa )c_n})$$

for all $x,y\in C$ and $n\ge 1$.

Lemma 2.5 ([[19], Lemma 2.7])

Let C be a nonempty subset of a Hilbert space H and $S:C\to C$ be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence $\{{\gamma }_n\}$. Let $\{x_n\}$ be a sequence in C such that $\parallel x_n-x_{n+1}\parallel \to 0$ and $\parallel x_n-S^n{x}_n\parallel \to 0$ as $n\to \mathrm{\infty }$. Then $\parallel x_n-S{x}_n\parallel \to 0$ as $n\to \mathrm{\infty }$.

Lemma 2.6 (Demiclosedness principle [[19], Proposition 3.1])

Let C be a nonempty closed convex subset of a Hilbert space H and $S:C\to C$ be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence $\{{\gamma }_n\}$. Then $I-S$ is demiclosed at zero in the sense that if $\{x_n\}$ is a sequence in C such that $x_n\rightharpoonup x\in C$ and ${lim\hspace{0.17em}sup}_{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\parallel x_n-S^m{x}_n\parallel =0$, then $(I-S)x=0$.

Lemma 2.7 ([[19], Proposition 3.2])

Let C be a nonempty closed convex subset of a Hilbert space H and $S:C\to C$ be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence $\{{\gamma }_n\}$ such that $Fix(S)\ne \mathrm{\varnothing }$. Then $Fix(S)$ is closed and convex.

Remark 2.1 Lemmas 2.6 and 2.7 give some basic properties of an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence $\{{\gamma }_n\}$. Moreover, Lemma 2.6 extends the demiclosedness principles studied for certain classes of nonlinear mappings; see [19] for more details.

Lemma 2.8 ([[22], p.80])

Let ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$, ${\{b_n\}}_{n=1}^{\mathrm{\infty }}$, and ${\{{\delta }_n\}}_{n=1}^{\mathrm{\infty }}$ be sequences of nonnegative real numbers satisfying the inequality

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

If ${\sum }_{n=1}^{\mathrm{\infty }}{\delta }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists. If, in addition, ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ has a subsequence which converges to zero, then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Recall that a Banach space X is said to satisfy the Opial condition [23] if, for any given sequence $\{x_n\}\subset X$ which converges weakly to an element $x\in X$, we have the inequality

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

It is well known in [23] that every Hilbert space H satisfies the Opial condition.

Lemma 2.9 (see [[24], Proposition 3.1])

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

$${\parallel x_{n+1}-p\parallel }^2\le (1+{\lambda }_n){\parallel x_n-p\parallel }^2+{\delta }_n,\mathrm{\forall }p\in C,n\ge 1,$$

where $\{{\lambda }_n\}$ and $\{{\delta }_n\}$ are sequences of nonnegative real numbers such that ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\delta }_n<\mathrm{\infty }$. Then $\{P_C{x}_n\}$ converges strongly in C.

Lemma 2.10 (see [25])

Let C be a closed convex subset of a real Hilbert space H. Let $\{x_n\}$ be a sequence in H and $u\in H$. Let $q=P_{C}u$. If $\{x_n\}$ is such that ${\omega }_w(x_n)\subset C$ and satisfies the condition

$$\parallel x_n-u\parallel \le \parallel u-q\parallel ,\mathit{\text{for all }}n,$$

then $x_n\to q$ as $n\to \mathrm{\infty }$.

Lemma 2.11 (see [[26], Lemma 3.2])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonexpansive self-mappings on C such that ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)\ne \mathrm{\varnothing }$ and let $\{{\lambda }_n\}$ be a sequence in $(0,b]$ for some $b\in (0,1)$. Then, for every $x\in C$ and $k\ge 1$ the limit ${lim}_{n\to \mathrm{\infty }}{U}_{n,k}x$ exists.

Remark 2.2 (see [[27], Remark 3.1])

It can be known from Lemma 2.11 that if D is a nonempty bounded subset of C, then for $ϵ>0$ there exists $n_0\ge k$ such that for all $n>n_0$

$$\underset{x\in D}{sup}\parallel U_{n,k}x-U_{k}x\parallel \le ϵ.$$

Remark 2.3 (see [[27], Remark 3.2])

Utilizing Lemma 2.11, we define a mapping $W:C\to C$ as follows:

$$Wx=\underset{n\to \mathrm{\infty }}{lim}{W}_{n}x=\underset{n\to \mathrm{\infty }}{lim}{U}_{n,1}x,\mathrm{\forall }x\in C.$$

Such a W is called the W-mapping generated by $T_1,T_2,\dots$ and ${\lambda }_1,{\lambda }_2,\dots$ . Since $W_n$ is nonexpansive, $W:C\to C$ is also nonexpansive. Indeed, observe that for each $x,y\in C$

$$\parallel Wx-Wy\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel W_{n}x-W_{n}y\parallel \le \parallel x-y\parallel .$$

If $\{x_n\}$ is a bounded sequence in C, then we put $D=\{x_n:n\ge 1\}$. Hence, it is clear from Remark 2.2 that for an arbitrary $ϵ>0$ there exists $N_0\ge 1$ such that for all $n>N_0$

$$\parallel W_n{x}_n-W{x}_n\parallel =\parallel U_{n,1}{x}_n-U_1{x}_n\parallel \le \underset{x\in D}{sup}\parallel U_{n,1}x-U_{1}x\parallel \le ϵ.$$

This implies that

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

Lemma 2.12 (see [[26], Lemma 3.3])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonexpansive self-mappings on C such that ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)\ne \mathrm{\varnothing }$, and let $\{{\lambda }_n\}$ be a sequence in $(0,b]$ for some $b\in (0,1)$. Then $Fix(W)={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)$.

Lemma 2.13 (see [[28], Theorem 10.4 (Demiclosedness Principle)])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be nonexpansive. Then $I-T$ is demiclosed on C. That is, whenever $\{x_n\}$ is a sequence in C weakly converging to some $x\in C$ and the sequence $\{(I-T)x_n\}$ strongly converges to some y, it follows that $(I-T)x=y$. Here I is the identity operator of H.

Recall that a set-valued mapping $R:D(R)\subset H\to 2^H$ is called monotone if, for all $x,y\in D(R)$, $f\in R(x)$, and $g\in R(y)$ imply

$$〈f-g,x-y〉\ge 0.$$

A set-valued mapping R is called maximal monotone if R is monotone and $(I+\lambda R)D(R)=H$ for each $\lambda >0$, where I is the identity mapping of H. We denote by $G(R)$ the graph of R. It is known that a monotone mapping R is maximal if and only if, for $(x,f)\in H\times H$, $〈f-g,x-y〉\ge 0$ for every $(y,g)\in G(R)$, we have $f\in R(x)$. We illustrate the concept of maximal monotone mapping with the following example.

Let $A:C\to H$ be a monotone, k-Lipschitz-continuous mapping and let $N_{C}v$ be the normal cone to C at $v\in C$, i.e.,

$$N_{C}v=\{w\in H:〈v-u,w〉\ge 0,\mathrm{\forall }u\in C\}.$$

Define

$$Tv=\{\begin{array}{ll}Av+N_{C}v& \text{if }v\in C,\\ \mathrm{\varnothing }& \text{if }v\notin C.\end{array}$$

Then T is maximal monotone and $0\in Tv$ if and only if $〈Av,y-v〉\ge 0$ for all $y\in C$ (see [9]).

Assume that $R:D(R)\subset H\to 2^H$ is a maximal monotone mapping. Let $\lambda >0$. In terms of Huang [10] (see also [11]), we have the following property for the resolvent operator $J_{R,\lambda }:H\to \overline{D(R)}$.

Lemma 2.14 $J_{R,\lambda }$ is single-valued and firmly nonexpansive, i.e.,

$$〈J_{R,\lambda }x-J_{R,\lambda }y,x-y〉\ge {\parallel J_{R,\lambda }x-J_{R,\lambda }y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Consequently, $J_{R,\lambda }$ is nonexpansive and monotone.

Lemma 2.15 (see [14])

Let R be a maximal monotone mapping with $D(R)=C$. Then for any given $\lambda >0$, $u\in C$ is a solution of problem (1.6) if and only if $u\in C$ satisfies

$$u=J_{R,\lambda }(u-\lambda Bu).$$

Lemma 2.16 (see [11])

Let R be a maximal monotone mapping with $D(R)=C$ and let $B:C\to H$ be a strongly monotone, continuous, and single-valued mapping. Then for each $z\in H$, the equation $z\in (B+\lambda R)x$ has a unique solution $x_{\lambda }$ for $\lambda >0$.

Lemma 2.17 (see [14])

Let R be a maximal monotone mapping with $D(R)=C$ and $B:C\to H$ be a monotone, continuous and single-valued mapping. Then $(I+\lambda (R+B))C=H$ for each $\lambda >0$. In this case, $R+B$ is maximal monotone.

Lemma 2.18 (see [29])

Let C be a nonempty closed convex subset of a real Hilbert space H, and $g:C\to \mathbf{R}\cup +\mathrm{\infty }$ be a proper lower semicontinuous differentiable convex function. If $x^{\ast }$ is a solution the minimization problem

$$g\left(x^{\ast }\right)=\underset{x\in C}{inf}g(x),$$

then

$$〈g^{\prime }(x),x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C.$$

In particular, if $x^{\ast }$ solves (OP1), then

$$〈u+(\gamma f-(I+\mu V))x^{\ast },x-x^{\ast }〉\le 0.$$

3 Strong convergence theorems

In this section, we introduce and analyze an iterative algorithm for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from $C\times C$ to R satisfying (H1)-(H4) and $\phi :C\to \mathbf{R}$ be a lower semicontinuous and convex functional. Let $R_i:C\to 2^H$ be a maximal monotone mapping and let $A:H\to H$ and $B_i:C\to H$ be ζ-inverse-strongly monotone and ${\eta }_i$-inverse-strongly monotone, respectively, where $i\in \{1,2,\dots ,N\}$. Let $S:C\to C$ be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some $0\le \kappa <1$ with sequence $\{{\gamma }_n\}\subset [0,\mathrm{\infty })$ such that ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$ and $\{c_n\}\subset [0,\mathrm{\infty })$ such that ${lim}_{n\to \mathrm{\infty }}{c}_n=0$. Let ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonexpansive self-mappings on C and $\{{\lambda }_n\}$ be a sequence in $(0,b]$ for some $b\in (0,1)$. Let V be a $\overline{\gamma }$-strongly positive bounded linear operator and $f:H\to H$ be an l-Lipschitzian mapping with $\gamma l<(1+\mu )\overline{\gamma }$. Assume that $\Omega :=({\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n))\cap GMEP(\Theta ,\phi ,A)\cap ({\bigcap }_{i=1}^N\mathrm{I}(B_i,R_i))\cap Fix(S)$ is nonempty and bounded. Let $W_n$ be the W-mapping defined by (1.4) and $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\delta }_n\}$ be three sequences in $(0,1)$ such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and $\kappa \le {\delta }_n\le d<1$. Assume that:

• (i) $K:H\to \mathbf{R}$ is strongly convex with constant $\sigma >0$ and its derivative $K^{\prime }$ is Lipschitz-continuous with constant $\nu >0$ such that the function $x\mapsto 〈y-x,K^{\prime }(x)〉$ is weakly upper semicontinuous for each $y\in H$;

• (ii) for each $x\in H$, there exist a bounded subset $D_x\subset C$ and $z_x\in C$ such that for any $y\notin D_x$,

  $$\Theta (y,z_x)+\phi (z_x)-\phi (y)+\frac{1}{r}〈K^{\prime }(y)-K^{\prime }(x),z_x-y〉<0;$$

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$;

• (iv) $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$, $\mathrm{\forall }i\in \{1,2,\dots ,N\}$, and $\{r_n\}\subset [0,2\zeta ]$ satisfies

  $$0<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{r}_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{r}_n<2\zeta .$$

Pick any $x_0\in H$ and set $C_1=C$, $x_1=P_{C_1}{x}_0$. Let $\{x_n\}$ be a sequence generated by the following algorithm:

$$\{\begin{array}{l}{u}_n=S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)x_n,\\ z_n=J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N)J_{R_{N-1},{\lambda }_{N-1,n}}(I-{\lambda }_{N-1,n}{B}_{N-1})\cdots J_{R_1,{\lambda }_{1,n}}(I-{\lambda }_{1,n}{B}_1)u_n,\\ k_n={\delta }_n{z}_n+(1-{\delta }_n)S^n{z}_n,\\ y_n={\alpha }_n(u+\gamma f(x_n))+{\beta }_n{k}_n+((1-{\beta }_n)I-{\alpha }_n(I+\mu V))W_n{z}_n,\\ C_{n+1}=\{z\in C_n:{\parallel y_n-z\parallel }^2\le {\parallel x_n-z\parallel }^2+{\theta }_n\},\\ x_{n+1}=P_{C_{n+1}}{x}_0,\mathrm{\forall }n\ge 0,\end{array}$$

(3.1)

where ${\theta }_n=({\alpha }_n+{\gamma }_n){\Delta }_n\varrho +c_n\varrho$, ${\Delta }_n=sup\{{\parallel x_n-p\parallel }^2+{\parallel u+(\gamma f-I-\mu V)p\parallel }^2:p\in \Omega \}<\mathrm{\infty }$, and $\varrho =\frac{1}{1-{sup}_{n\ge 1}{\alpha }_n}<\mathrm{\infty }$. If $S_r^{(\Theta ,\phi )}$ is firmly nonexpansive, then the following statements hold:

• (I) $\{x_n\}$ converges strongly to $P_{\Omega }{x}_0$;

• (II) $\{x_n\}$ converges strongly to $P_{\Omega }{x}_0$, which solves the optimization problem

  $$\underset{x\in \Omega }{min}\frac{\mu }{2}〈Vx,x〉+\frac{1}{2}{\parallel x-u\parallel }^2-h(x),$$

  (OP2)

  provided ${\gamma }_n+c_n+\parallel x_n-y_n\parallel =o({\alpha }_n)$ additionally, where $h:H\to \mathbf{R}$ is the potential function of γf.

Proof Since ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$, we may assume, without loss of generality, that ${\alpha }_n\le (1-{\beta }_n){(1+\mu \parallel V\parallel )}^{-1}$. Since V is a $\overline{\gamma }$-strongly positive bounded linear operator on H, we know that

$$\parallel V\parallel =sup\{〈Vu,u〉:u\in H,\parallel u\parallel =1\}.$$

Observe that

$$\begin{array}{rl}〈((1-{\beta }_n)I-{\alpha }_n(I+\mu V))u,u〉& =1-{\beta }_n-{\alpha }_n-{\alpha }_n\mu 〈Vu,u〉\\ \ge 1-{\beta }_n-{\alpha }_n-{\alpha }_n\mu \parallel V\parallel \\ \ge 0,\end{array}$$

that is, $(1-{\beta }_n)I-{\alpha }_n(I+\mu V)$ is positive. It follows that

$$\begin{array}{rl}\parallel (1-{\beta }_n)I-{\alpha }_n(I+\mu V)\parallel & =sup\{〈((1-{\beta }_n)I-{\alpha }_n(I+\mu V))u,u〉:u\in H,\parallel u\parallel =1\}\\ =sup\{1-{\beta }_n-{\alpha }_n-{\alpha }_n\mu 〈Vu,u〉:u\in H,\parallel u\parallel =1\}\\ \le 1-{\beta }_n-{\alpha }_n-{\alpha }_n\mu \overline{\gamma }.\end{array}$$

Put

$${\Lambda }_n^i=J_{R_i,{\lambda }_{i,n}}(I-{\lambda }_{i,n}{B}_i)J_{R_{i-1},{\lambda }_{i-1,n}}(I-{\lambda }_{i-1,n}{B}_{i-1})\cdots J_{R_1,{\lambda }_{1,n}}(I-{\lambda }_{1,n}{B}_1)$$

for all $i\in \{1,2,\dots ,N\}$ and $n\ge 1$, and ${\Lambda }_n^0=I$, where I is the identity mapping on H. Then we have that $z_n={\Lambda }_n^N{u}_n$. We divide the rest of the proof into several steps.

Step 1. We show that $\{x_n\}$ is well defined. It is obvious that $C_n$ is closed and convex. As the defining inequality in $C_n$ is equivalent to the inequality

$$〈2(x_n-z_n),z〉\le {\parallel x_n\parallel }^2-{\parallel z_n\parallel }^2+{\theta }_n,$$

by Lemma 2.3 we know that $C_n$ is convex and closed for every $n\ge 1$.

First of all, we show that $\Omega \subset C_n$ for all $n\ge 1$. Suppose that $\Omega \subset C_n$ for some $n\ge 1$. Take $p\in \Omega$ arbitrarily. Since $p=S_{r_n}^{(\Theta ,\phi )}(p-r_{n}Ap)$, A is ζ-inverse strongly monotone and $0\le r_n\le 2\zeta$, we have

$$\begin{array}{rl}{\parallel u_n-p\parallel }^2& ={\parallel S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)x_n-S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)p\parallel }^2\\ \le {\parallel (I-r_{n}A)x_n-(I-r_{n}A)p\parallel }^2\\ ={\parallel (x_n-p)-r_n(A{x}_n-Ap)\parallel }^2\\ ={\parallel x_n-p\parallel }^2-2r_n〈x_n-p,A{x}_n-Ap〉+r_n^2{\parallel A{x}_n-Ap\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-2r_n\zeta {\parallel A{x}_n-Ap\parallel }^2+r_n^2{\parallel A{x}_n-Ap\parallel }^2\\ ={\parallel x_n-p\parallel }^2+r_n(r_n-2\zeta ){\parallel A{x}_n-Ap\parallel }^2\\ \le {\parallel x_n-p\parallel }^2.\end{array}$$

(3.2)

Since $p=J_{R_i,{\lambda }_{i,n}}(I-{\lambda }_{i,n}{B}_i)p$, ${\Lambda }_n^{i}p=p$, and $B_i$ is ${\eta }_i$-inverse-strongly monotone, where ${\eta }_i\in (0,2{\eta }_i)$, $i\in \{1,2,\dots ,N\}$, by Lemma 2.14 we deduce that

$$\begin{array}{rl}{\parallel z_n-p\parallel }^2& ={\parallel J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}p\parallel }^2\\ \le {\parallel (I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n-(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}p\parallel }^2\\ ={\parallel ({\Lambda }_n^{N-1}{u}_n-{\Lambda }_n^{N-1}p)-{\lambda }_{N,n}(B_N{\Lambda }_n^{N-1}{u}_n-B_N{\Lambda }_n^{N-1}p)\parallel }^2\\ \le {\parallel {\Lambda }_n^{N-1}{u}_n-{\Lambda }_n^{N-1}p\parallel }^2+{\lambda }_{N,n}({\lambda }_{N,n}-2{\eta }_N){\parallel B_N{\Lambda }_n^{N-1}{u}_n-B_N{\Lambda }_n^{N-1}p\parallel }^2\\ \le {\parallel {\Lambda }_n^{N-1}{u}_n-{\Lambda }_n^{N-1}p\parallel }^2\\ ⋮\\ \le {\parallel {\Lambda }_n^0{u}_n-{\Lambda }_n^{0}p\parallel }^2\\ ={\parallel u_n-p\parallel }^2.\end{array}$$

(3.3)

Combining (3.2) and (3.3), we have

$$\parallel z_n-p\parallel \le \parallel x_n-p\parallel .$$

(3.4)

By Lemma 2.2(b), we deduce from (3.1) and (3.4) that

$$\begin{array}{rl}{\parallel k_n-p\parallel }^2=& {\parallel {\delta }_n(z_n-p)+(1-{\delta }_n)(S^n{z}_n-p)\parallel }^2\\ =& {\delta }_n{\parallel z_n-p\parallel }^2+(1-{\delta }_n){\parallel S^n{z}_n-p\parallel }^2-{\delta }_n(1-{\delta }_n){\parallel z_n-S^n{z}_n\parallel }^2\\ \le & {\delta }_n{\parallel z_n-p\parallel }^2+(1-{\delta }_n)[(1+{\gamma }_n){\parallel z_n-p\parallel }^2\\ +\kappa {\parallel z_n-S^n{z}_n\parallel }^2+c_n]-{\delta }_n(1-{\delta }_n){\parallel z_n-S^n{z}_n\parallel }^2\\ =& [1+{\gamma }_n(1-{\delta }_n)]{\parallel z_n-p\parallel }^2+(1-{\delta }_n)(\kappa -{\delta }_n){\parallel z_n-S^n{z}_n\parallel }^2+(1-{\delta }_n)c_n\\ \le & (1+{\gamma }_n){\parallel z_n-p\parallel }^2+(1-{\delta }_n)(\kappa -{\delta }_n){\parallel z_n-S^n{z}_n\parallel }^2+c_n\\ \le & (1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n.\end{array}$$

(3.5)

Set $\overline{V}=I+\mu V$. Then, for $\gamma l\le (1+\mu )\overline{\gamma }$, by Lemma 2.1 we obtain from (3.1), (3.4), and (3.5)

$$\begin{array}{r}{\parallel y_n-p\parallel }^2\\ ={\parallel {\alpha }_n(u+\gamma f(x_n))+{\beta }_n{k}_n+((1-{\beta }_n)I-{\alpha }_n\overline{V})W_n{z}_n-p\parallel }^2\\ ={\parallel {\alpha }_n(u+\gamma f(x_n)-\overline{V}p)+{\beta }_n(k_n-p)+((1-{\beta }_n)I-{\alpha }_n\overline{V})(W_n{z}_n-p)\parallel }^2\\ =\parallel {\alpha }_n(u+\gamma f(p)-\overline{V}p)+{\alpha }_n\gamma (f(x_n)-f(p))\\ {+{\beta }_n(k_n-p)+((1-{\beta }_n)I-{\alpha }_n\overline{V})(W_n{z}_n-p)\parallel }^2\\ \le {\parallel {\alpha }_n\gamma (f(x_n)-f(p))+{\beta }_n(k_n-p)+((1-{\beta }_n)I-{\alpha }_n\overline{V})(W_n{z}_n-p)\parallel }^2\\ +2{\alpha }_n〈(u+\gamma f(p)-\overline{V}p),y_n-p〉\\ \le {[{\alpha }_n\gamma \parallel f(x_n)-f(p)\parallel +{\beta }_n\parallel k_n-p\parallel +\parallel ((1-{\beta }_n)I-{\alpha }_n\overline{V})(W_n{z}_n-p)\parallel ]}^2\\ +2{\alpha }_n〈(u+\gamma f(p)-\overline{V}p),y_n-p〉\\ \le {[{\alpha }_n\gamma l\parallel x_n-p\parallel +{\beta }_n\parallel k_n-p\parallel +(1-{\beta }_n-{\alpha }_n-{\alpha }_n\mu \overline{\gamma })\parallel W_n{z}_n-p\parallel ]}^2\\ +2{\alpha }_n\parallel u+\gamma f(p)-\overline{V}p\parallel \parallel y_n-p\parallel \\ \le {[{\alpha }_n(1+\mu )\overline{\gamma }\parallel x_n-p\parallel +{\beta }_n\parallel k_n-p\parallel +(1-{\beta }_n-{\alpha }_n(1+\mu )\overline{\gamma })\parallel z_n-p\parallel ]}^2\\ +{\alpha }_n({\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+{\parallel y_n-p\parallel }^2)\\ \le {\alpha }_n(1+\mu )\overline{\gamma }{\parallel x_n-p\parallel }^2+{\beta }_n{\parallel k_n-p\parallel }^2+(1-{\beta }_n-{\alpha }_n(1+\mu )\overline{\gamma }){\parallel z_n-p\parallel }^2\\ +{\alpha }_n({\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+{\parallel y_n-p\parallel }^2)\\ \le {\alpha }_n(1+\mu )\overline{\gamma }{\parallel x_n-p\parallel }^2+{\beta }_n((1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n)\\ +(1-{\beta }_n-{\alpha }_n(1+\mu )\overline{\gamma }){\parallel z_n-p\parallel }^2+{\alpha }_n({\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+{\parallel y_n-p\parallel }^2)\\ \le {\alpha }_n(1+\mu )\overline{\gamma }{\parallel x_n-p\parallel }^2+{\beta }_n((1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n)\\ +(1-{\beta }_n-{\alpha }_n(1+\mu )\overline{\gamma })((1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n)\\ +{\alpha }_n({\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+{\parallel y_n-p\parallel }^2)\\ ={\alpha }_n(1+\mu )\overline{\gamma }{\parallel x_n-p\parallel }^2+(1-{\alpha }_n(1+\mu )\overline{\gamma })((1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n)\\ +{\alpha }_n({\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+{\parallel y_n-p\parallel }^2)\\ \le {\alpha }_n(1+\mu )\overline{\gamma }((1+{\gamma }_n){\parallel x_n-p\parallel }^2+c_n)\\ +(1-{\alpha }_n(1+\mu )\overline{\gamma })((1+{\gamma }_n){\parallel x_n-p\parallel }^2+c_n)\\ +{\alpha }_n({\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+{\parallel y_n-p\parallel }^2)\\ =(1+{\gamma }_n){\parallel x_n-p\parallel }^2+c_n+{\alpha }_n({\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+{\parallel y_n-p\parallel }^2),\end{array}$$

which hence yields

$$\begin{array}{rl}{\parallel y_n-p\parallel }^2& \le \frac{1+{\gamma }_n}{1-{\alpha }_n}{\parallel x_n-p\parallel }^2+\frac{{\alpha }_n}{1-{\alpha }_n}{\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+\frac{1}{1-{\alpha }_n}{c}_n\\ =(1+\frac{{\alpha }_n+{\gamma }_n}{1-{\alpha }_n}){\parallel x_n-p\parallel }^2+\frac{{\alpha }_n}{1-{\alpha }_n}{\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+\frac{1}{1-{\alpha }_n}{c}_n\\ \le (1+\frac{{\alpha }_n+{\gamma }_n}{1-{\alpha }_n}){\parallel x_n-p\parallel }^2+\frac{{\alpha }_n+{\gamma }_n}{1-{\alpha }_n}{\parallel u+\gamma f(p)-\overline{V}p\parallel }^2+\frac{1}{1-{\alpha }_n}{c}_n\\ ={\parallel x_n-p\parallel }^2+\frac{{\alpha }_n+{\gamma }_n}{1-{\alpha }_n}({\parallel x_n-p\parallel }^2+{\parallel u+\gamma f(p)-\overline{V}p\parallel }^2)+\frac{1}{1-{\alpha }_n}{c}_n\\ \le {\parallel x_n-p\parallel }^2+({\alpha }_n+{\gamma }_n)\varrho ({\parallel x_n-p\parallel }^2+{\parallel u+\gamma f(p)-\overline{V}p\parallel }^2)+\varrho c_n\\ \le {\parallel x_n-p\parallel }^2+({\alpha }_n+{\gamma }_n){\Delta }_n\varrho +c_n\varrho \\ ={\parallel x_n-p\parallel }^2+{\theta }_n,\end{array}$$

(3.6)

where ${\theta }_n=({\alpha }_n+{\gamma }_n){\Delta }_n\varrho +c_n\varrho$, ${\Delta }_n=sup\{{\parallel x_n-p\parallel }^2+{\parallel u+\gamma f(p)-\overline{V}p\parallel }^2:p\in \Omega \}<\mathrm{\infty }$, and $\varrho =\frac{1}{1-{sup}_{n\ge 1}{\alpha }_n}<\mathrm{\infty }$ (due to $\{{\alpha }_n\}\subset (0,1)$ and ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$). Hence $p\in C_{n+1}$. This implies that $\Omega \subset C_n$ for all $n\ge 1$. Therefore, $\{x_n\}$ is well defined.

Step 2. We prove that $\parallel x_n-k_n\parallel \to 0$ as $n\to \mathrm{\infty }$.

Indeed, let $v=P_{\Omega }{x}_0$. From $x_n=P_{C_n}{x}_0$ and $v\in \Omega \subset C_n$, we obtain

$$\parallel x_n-x_0\parallel \le \parallel v-x_0\parallel .$$

(3.7)

This implies that $\{x_n\}$ is bounded and hence $\{u_n\}$, $\{z_n\}$, $\{k_n\}$, and $\{y_n\}$ are also bounded. Since $x_{n+1}\in C_{n+1}\subset C_n$ and $x_n=P_{C_n}{x}_0$, we have

$$\parallel x_n-x_0\parallel \le \parallel x_{n+1}-x_0\parallel ,\mathrm{\forall }n\ge 1.$$

Therefore ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists. From $x_n=P_{C_n}{x}_0$, $x_{n+1}\in C_{n+1}\subset C_n$, by Proposition 2.1(ii) we obtain

$${\parallel x_{n+1}-x_n\parallel }^2\le {\parallel x_0-x_{n+1}\parallel }^2-{\parallel x_0-x_n\parallel }^2,$$

which implies

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

(3.8)

It follows from $x_{n+1}\in C_{n+1}$ that ${\parallel y_n-x_{n+1}\parallel }^2\le {\parallel x_n-x_{n+1}\parallel }^2+{\theta }_n$ and hence

$$\begin{array}{rl}{\parallel x_n-y_n\parallel }^2& \le 2({\parallel x_n-x_{n+1}\parallel }^2+{\parallel x_{n+1}-y_n\parallel }^2)\\ \le 2({\parallel x_n-x_{n+1}\parallel }^2+{\parallel x_n-x_{n+1}\parallel }^2+{\theta }_n)\\ =2(2{\parallel x_n-x_{n+1}\parallel }^2+{\theta }_n).\end{array}$$

From (3.8) and ${lim}_{n\to \mathrm{\infty }}{\theta }_n=0$, we have

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

(3.9)

Also, utilizing Lemmas 2.1 and 2.2(b) we obtain from (3.1), (3.4), and (3.5)

$$\begin{array}{r}{\parallel y_n-p\parallel }^2\\ ={\parallel {\alpha }_n(u+\gamma f(x_n)-\overline{V}{W}_n{z}_n)+{\beta }_n(k_n-p)+(1-{\beta }_n)(W_n{z}_n-p)\parallel }^2\\ \le {\parallel {\beta }_n(k_n-p)+(1-{\beta }_n)(W_n{z}_n-p)\parallel }^2+2{\alpha }_n〈u+\gamma f(x_n)-\overline{V}{W}_n{z}_n,y_n-p〉\\ ={\beta }_n{\parallel k_n-p\parallel }^2+(1-{\beta }_n){\parallel W_n{z}_n-p\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel k_n-W_n{z}_n\parallel }^2\\ +2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel \\ \le {\beta }_n{\parallel k_n-p\parallel }^2+(1-{\beta }_n){\parallel z_n-p\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel k_n-W_n{z}_n\parallel }^2\\ +2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel \\ \le {\beta }_n((1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n)+(1-{\beta }_n){\parallel z_n-p\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel k_n-W_n{z}_n\parallel }^2\\ +2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel \\ \le {\beta }_n((1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n)+(1-{\beta }_n)((1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n)\\ -{\beta }_n(1-{\beta }_n){\parallel k_n-W_n{z}_n\parallel }^2+2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel \\ =(1+{\gamma }_n){\parallel z_n-p\parallel }^2+c_n-{\beta }_n(1-{\beta }_n){\parallel k_n-W_n{z}_n\parallel }^2\\ +2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel \\ \le (1+{\gamma }_n){\parallel x_n-p\parallel }^2+c_n-{\beta }_n(1-{\beta }_n){\parallel k_n-W_n{z}_n\parallel }^2\\ +2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel ,\end{array}$$

which leads to

$$\begin{array}{r}{\beta }_n(1-{\beta }_n){\parallel k_n-W_n{z}_n\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ +2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel \\ \le \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel )+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ +2{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel \parallel y_n-p\parallel .\end{array}$$

Since ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$, and ${lim}_{n\to \mathrm{\infty }}{c}_n=0$, it follows from (3.9) and condition (iii) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel k_n-W_n{z}_n\parallel =0.$$

(3.10)

Note that

$$y_n-k_n={\alpha }_n(u+\gamma f(x_n)-\overline{V}{W}_n{z}_n)+(1-{\beta }_n)(W_n{z}_n-k_n),$$

which yields

$$\begin{array}{rl}\parallel x_n-k_n\parallel & \le \parallel x_n-y_n\parallel +\parallel y_n-k_n\parallel \\ \le \parallel x_n-y_n\parallel +\parallel {\alpha }_n(u+\gamma f(x_n)-\overline{V}{W}_n{z}_n)+(1-{\beta }_n)(W_n{z}_n-k_n)\parallel \\ \le \parallel x_n-y_n\parallel +{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel +(1-{\beta }_n)\parallel W_n{z}_n-k_n\parallel \\ \le \parallel x_n-y_n\parallel +{\alpha }_n\parallel u+\gamma f(x_n)-\overline{V}{W}_n{z}_n\parallel +\parallel W_n{z}_n-k_n\parallel .\end{array}$$

So, from (3.9), (3.10), and ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, we get

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

(3.11)

Step 3. We prove that $\parallel x_n-u_n\parallel \to 0$, $\parallel u_n-z_n\parallel \to 0$, $\parallel z_n-W{z}_n\parallel \to 0$, and $\parallel z_n-S^n{z}_n\parallel \to 0$ as $n\to \mathrm{\infty }$.

Indeed, taking into consideration that $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{r}_n<2\zeta$, we may assume, without loss of generality, that $\{r_n\}\subset [c,d]\subset (0,2\zeta )$. From (3.4) and (3.5) it follows that

$$\begin{array}{rl}{\parallel k_n-p\parallel }^2& \le [1+{\gamma }_n(1-{\delta }_n)]{\parallel z_n-p\parallel }^2+(1-{\delta }_n)(k-{\delta }_n){\parallel z_n-S^n{z}_n\parallel }^2+(1-{\delta }_n)c_n\\ \le {\parallel z_n-p\parallel }^2+{\gamma }_n{\parallel z_n-p\parallel }^2+c_n\\ \le {\parallel z_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n.\end{array}$$

(3.12)

Next we prove that

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

(3.13)

For $p\in \Omega$, we find that

$$\begin{array}{rl}{\parallel u_n-p\parallel }^2& ={\parallel S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)x_n-S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)p\parallel }^2\\ \le {\parallel (I-r_{n}A)x_n-(I-r_{n}A)p\parallel }^2\\ ={\parallel x_n-p-r_n(A{x}_n-Ap)\parallel }^2\\ \le {\parallel x_n-p\parallel }^2+r_n(r_n-2\zeta ){\parallel A{x}_n-Ap\parallel }^2.\end{array}$$

(3.14)

By (3.3), (3.12), and (3.14), we obtain

$$\begin{array}{rl}{\parallel k_n-p\parallel }^2& \le {\parallel z_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le {\parallel u_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le {\parallel x_n-p\parallel }^2+r_n(r_n-2\zeta ){\parallel A{x}_n-Ap\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n,\end{array}$$

which implies that

$$\begin{array}{rl}c(2\zeta -d){\parallel A{x}_n-Ap\parallel }^2& \le r_n(2\zeta -r_n){\parallel A{x}_n-Ap\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-{\parallel k_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le \parallel x_n-k_n\parallel (\parallel x_n-p\parallel +\parallel k_n-p\parallel )+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n.\end{array}$$

From ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$, ${lim}_{n\to \mathrm{\infty }}{c}_n=0$, and (3.11), we have

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

(3.15)

By the firm nonexpansivity of $S_{r_n}^{(\mathrm{\Theta },\phi )}$ and Lemma 2.2(a), we have

$$\begin{array}{r}{\parallel u_n-p\parallel }^2\\ ={\parallel S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)x_n-S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)p\parallel }^2\\ \le 〈(I-r_{n}A)x_n-(I-r_{n}A)p,u_n-p〉\\ =\frac{1}{2}[{\parallel (I-r_{n}A)x_n-(I-r_{n}A)p\parallel }^2+{\parallel u_n-p\parallel }^2\\ -{\parallel (I-r_{n}A)x_n-(I-r_{n}A)p-(u_n-p)\parallel }^2]\\ \le \frac{1}{2}[{\parallel x_n-p\parallel }^2+{\parallel u_n-p\parallel }^2-{\parallel x_n-u_n-r_n(A{x}_n-Ap)\parallel }^2]\\ =\frac{1}{2}[{\parallel x_n-p\parallel }^2+{\parallel u_n-p\parallel }^2-{\parallel x_n-u_n\parallel }^2+2r_n〈A{x}_n-Ap,x_n-u_n〉\\ -r_n^2{\parallel A{x}_n-Ap\parallel }^2],\end{array}$$

which implies that

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

(3.16)

Combining (3.12) and (3.16), we have

$$\begin{array}{rl}{\parallel k_n-p\parallel }^2& \le {\parallel z_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le {\parallel u_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le {\parallel x_n-p\parallel }^2-{\parallel x_n-u_n\parallel }^2+2r_n\parallel A{x}_n-Ap\parallel \parallel x_n-u_n\parallel +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n,\end{array}$$

which implies

$$\begin{array}{r}{\parallel x_n-u_n\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-{\parallel k_n-p\parallel }^2+2r_n\parallel A{x}_n-Ap\parallel \parallel x_n-u_n\parallel +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le \parallel x_n-k_n\parallel (\parallel x_n-p\parallel +\parallel k_n-p\parallel )+2r_n\parallel A{x}_n-Ap\parallel \parallel x_n-u_n\parallel \\ +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n.\end{array}$$

From ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$, ${lim}_{n\to \mathrm{\infty }}{c}_n=0$, (3.11), and (3.15), we know that (3.13) holds.

Next we show that ${lim}_{n\to \mathrm{\infty }}\parallel B_i{\Lambda }_n^i{u}_n-B_{i}p\parallel =0$, $i=1,2,\dots ,N$. It follows from Lemma 2.14 that

$$\begin{array}{rl}{\parallel {\Lambda }_n^i{u}_n-p\parallel }^2& ={\parallel J_{R_i,{\lambda }_{i,n}}(I-{\lambda }_{i,n}{B}_i){\Lambda }_n^{i-1}{u}_n-J_{R_i,{\lambda }_{i,n}}(I-{\lambda }_{i,n}{B}_i)p\parallel }^2\\ \le {\parallel (I-{\lambda }_{i,n}{B}_i){\Lambda }_n^{i-1}{u}_n-(I-{\lambda }_{i,n}{B}_i)p\parallel }^2\\ \le {\parallel {\Lambda }_n^{i-1}{u}_n-p\parallel }^2+{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2\\ \le {\parallel u_n-p\parallel }^2+{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2\\ \le {\parallel x_n-p\parallel }^2+{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2.\end{array}$$

(3.17)

Combining (3.12) and (3.17), we have

$$\begin{array}{rl}{\parallel k_n-p\parallel }^2\le & {\parallel z_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le & {\parallel {\Lambda }_n^i{u}_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le & {\parallel x_n-p\parallel }^2+{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2\\ +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n,\end{array}$$

together with $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$, $i\in \{1,2,\dots ,N\}$, implies

$$\begin{array}{rl}{a}_i(2{\eta }_i-b_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2\le & {\lambda }_{i,n}(2{\eta }_i-{\lambda }_{i,n}){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-{\parallel k_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le & \parallel x_n-k_n\parallel (\parallel x_n-p\parallel +\parallel k_n-p\parallel )\\ +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n.\end{array}$$

From ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$, ${lim}_{n\to \mathrm{\infty }}{c}_n=0$, and (3.11), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel =0,i=1,2,\dots ,N.$$

(3.18)

By Lemma 2.14 and Lemma 2.2(a), we obtain

$$\begin{array}{rl}{\parallel {\Lambda }_n^i{u}_n-p\parallel }^2=& {\parallel J_{R_i,{\lambda }_{i,n}}(I-{\lambda }_{i,n}{B}_i){\Lambda }_n^{i-1}{u}_n-J_{R_i,{\lambda }_{i,n}}(I-{\lambda }_{i,n}{B}_i)p\parallel }^2\\ \le & 〈(I-{\lambda }_{i,n}{B}_i){\Lambda }_n^{i-1}{u}_n-(I-{\lambda }_{i,n}{B}_i)p,{\Lambda }_n^i{u}_n-p〉\\ =& \frac{1}{2}({\parallel (I-{\lambda }_{i,n}{B}_i){\Lambda }_n^{i-1}{u}_n-(I-{\lambda }_{i,n}{B}_i)p\parallel }^2+{\parallel {\Lambda }_n^i{u}_n-p\parallel }^2\\ -{\parallel (I-{\lambda }_{i,n}{B}_i){\Lambda }_n^{i-1}{u}_n-(I-{\lambda }_{i,n}{B}_i)p-({\Lambda }_n^i{u}_n-p)\parallel }^2)\\ \le & \frac{1}{2}({\parallel {\Lambda }_n^{i-1}{u}_n-p\parallel }^2+{\parallel {\Lambda }_n^i{u}_n-p\parallel }^2\\ -{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n-{\lambda }_{i,n}(B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p)\parallel }^2)\\ \le & \frac{1}{2}({\parallel u_n-p\parallel }^2+{\parallel {\Lambda }_n^i{u}_n-p\parallel }^2\\ -{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n-{\lambda }_{i,n}(B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p)\parallel }^2)\\ \le & \frac{1}{2}({\parallel x_n-p\parallel }^2+{\parallel {\Lambda }_n^i{u}_n-p\parallel }^2\\ -{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n-{\lambda }_{i,n}(B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p)\parallel }^2),\end{array}$$

which implies

$$\begin{array}{r}{\parallel {\Lambda }_n^i{u}_n-p\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n-{\lambda }_{i,n}(B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p)\parallel }^2\\ ={\parallel x_n-p\parallel }^2-{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel }^2-{\lambda }_{i,n}^2{\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2\\ +2{\lambda }_{i,n}〈{\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n,B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p〉\\ \le {\parallel x_n-p\parallel }^2-{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel }^2\\ +2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel .\end{array}$$

(3.19)

Combining (3.12) and (3.19) we get

$$\begin{array}{rl}{\parallel k_n-p\parallel }^2\le & {\parallel z_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le & {\parallel {\Lambda }_n^i{u}_n-p\parallel }^2+{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le & {\parallel x_n-p\parallel }^2-{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel }^2\\ +2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel \\ +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n,\end{array}$$

which implies

$$\begin{array}{r}{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel }^2\\ \le {\parallel x_n-p\parallel }^2-{\parallel k_n-p\parallel }^2+2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel \\ +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n\\ \le \parallel x_n-k_n\parallel (\parallel x_n-p\parallel +\parallel k_n-p\parallel )+2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel \\ +{\gamma }_n{\parallel x_n-p\parallel }^2+c_n.\end{array}$$

From (3.11), (3.18), ${lim}_{n\to \mathrm{\infty }}{\gamma }_n=0$, and ${lim}_{n\to \mathrm{\infty }}{c}_n=0$, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel =0,i=1,2,\dots ,N.$$

(3.20)

From (3.20) we get

$$\begin{array}{rl}\parallel u_n-z_n\parallel & =\parallel {\Lambda }_n^0{u}_n-{\Lambda }_n^N{u}_n\parallel \\ \le \parallel {\Lambda }_n^0{u}_n-{\Lambda }_n^1{u}_n\parallel +\parallel {\Lambda }_n^1{u}_n-{\Lambda }_n^2{u}_n\parallel +\cdots +\parallel {\Lambda }_n^{N-1}{u}_n-{\Lambda }_n^N{u}_n\parallel \\ \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(3.21)

By (3.13) and (3.21), we have

$$\begin{array}{rl}\parallel x_n-z_n\parallel & \le \parallel x_n-u_n\parallel +\parallel u_n-z_n\parallel \\ \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(3.22)

From (3.8) and (3.22), we have

$$\begin{array}{rl}\parallel z_{n+1}-z_n\parallel & \le \parallel z_{n+1}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel +\parallel x_n-z_n\parallel \\ \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(3.23)

By (3.11), (3.13), and (3.21), we get

$$\begin{array}{rl}\parallel k_n-z_n\parallel & \le \parallel k_n-x_n\parallel +\parallel x_n-u_n\parallel +\parallel u_n-z_n\parallel \\ \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(3.24)

We observe that

$$k_n-z_n=(1-{\delta }_n)(S^n{z}_n-z_n).$$

From ${\delta }_n\le d<1$ and (3.24), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S^n{z}_n-z_n\parallel =0.$$

(3.25)

We note that

$$\begin{array}{rl}\parallel S^n{z}_n-S^{n+1}{z}_n\parallel \le & \parallel S^n{z}_n-z_n\parallel +\parallel z_n-z_{n+1}\parallel +\parallel z_{n+1}-S^{n+1}{z}_{n+1}\parallel \\ +\parallel S^{n+1}{z}_{n+1}-S^{n+1}{z}_n\parallel .\end{array}$$

From (3.23), (3.25), and Lemma 2.4, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S^n{z}_n-S^{n+1}{z}_n\parallel =0.$$

(3.26)

On the other hand, we note that

$$\parallel z_n-S{z}_n\parallel \le \parallel z_n-S^n{z}_n\parallel +\parallel S^n{z}_n-S^{n+1}{z}_n\parallel +\parallel S^{n+1}{z}_n-S{z}_n\parallel .$$

From (3.25), (3.26), and the uniform continuity of S, we have

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

(3.27)

In addition, note that

$$\parallel z_n-W{z}_n\parallel \le \parallel z_n-k_n\parallel +\parallel k_n-W_n{z}_n\parallel +\parallel W_n{z}_n-W{z}_n\parallel .$$

So, from (3.10), (3.24), and Remark 2.3 it follows that

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

(3.28)

Step 4. we prove that $x_n\to v=P_{\Omega }{x}_0$ as $n\to \mathrm{\infty }$.

Indeed, since $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}$ which converges weakly to some w. From (3.13) and (3.20)-(3.22), we see that $u_{n_i}\rightharpoonup w$, ${\Lambda }_{n_i}^m{u}_{n_i}\rightharpoonup w$, and $z_{n_i}\rightharpoonup w$, where $m\in \{1,2,\dots ,N\}$. Since S is uniformly continuous, by (3.27) we get ${lim}_{n\to \mathrm{\infty }}\parallel z_n-S^m{z}_n\parallel =0$ for any $m\ge 1$. Hence from Lemma 2.6, we obtain $w\in Fix(S)$. In the meantime, utilizing Lemma 2.13, we deduce from (3.28) and $z_{n_i}\rightharpoonup w$ that $w\in Fix(W)={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)$ (due to Lemma 2.12). Next, we prove that $w\in {\bigcap }_{m=1}^N\mathrm{I}(B_m,R_m)$. As a matter of fact, since $B_m$ is ${\eta }_m$-inverse-strongly monotone, $B_m$ is a monotone and Lipschitz-continuous mapping. It follows from Lemma 2.17 that $R_m+B_m$ is maximal monotone. Let $(v,g)\in G(R_m+B_m)$, i.e., $g-B_{m}v\in R_{m}v$. Again, since ${\Lambda }_n^m{u}_n=J_{R_m,{\lambda }_{m,n}}(I-{\lambda }_{m,n}{B}_m){\Lambda }_n^{m-1}{u}_n$, $n\ge 1$, $m\in \{1,2,\dots ,N\}$, we have

$${\Lambda }_n^{m-1}{u}_n-{\lambda }_{m,n}{B}_m{\Lambda }_n^{m-1}{u}_n\in (I+{\lambda }_{m,n}{R}_m){\Lambda }_n^m{u}_n,$$

that is,

$$\frac{1}{{\lambda }_{m,n}}({\Lambda }_n^{m-1}{u}_n-{\Lambda }_n^m{u}_n-{\lambda }_{m,n}{B}_m{\Lambda }_n^{m-1}{u}_n)\in R_m{\Lambda }_n^m{u}_n.$$

In terms of the monotonicity of $R_m$, we get

$$〈v-{\Lambda }_n^m{u}_n,g-B_{m}v-\frac{1}{{\lambda }_{m,n}}({\Lambda }_n^{m-1}{u}_n-{\Lambda }_n^m{u}_n-{\lambda }_{m,n}{B}_m{\Lambda }_n^{m-1}{u}_n)〉\ge 0$$

and hence

$$\begin{array}{r}〈v-{\Lambda }_n^m{u}_n,g〉\\ \ge 〈v-{\Lambda }_n^m{u}_n,B_{m}v+\frac{1}{{\lambda }_{m,n}}({\Lambda }_n^{m-1}{u}_n-{\Lambda }_n^m{u}_n-{\lambda }_{m,n}{B}_m{\Lambda }_n^{m-1}{u}_n)〉\\ =〈v-{\Lambda }_n^m{u}_n,B_{m}v-B_m{\Lambda }_n^m{u}_n+B_m{\Lambda }_n^m{u}_n-B_m{\Lambda }_n^{m-1}{u}_n+\frac{1}{{\lambda }_{m,n}}({\Lambda }_n^{m-1}{u}_n-{\Lambda }_n^m{u}_n)〉\\ \ge 〈v-{\Lambda }_n^m{u}_n,B_m{\Lambda }_n^m{u}_n-B_m{\mathrm{\Lambda }}_n^{m-1}{u}_n〉+〈v-{\Lambda }_n^m{u}_n,\frac{1}{{\lambda }_{m,n}}({\Lambda }_n^{m-1}{u}_n-{\Lambda }_n^m{u}_n)〉.\end{array}$$

In particular,

$$\begin{array}{rl}〈v-{\Lambda }_{n_i}^m{u}_{n_i},g〉\ge & 〈v-{\Lambda }_{n_i}^m{u}_{n_i},B_m{\Lambda }_{n_i}^m{u}_{n_i}-B_m{\Lambda }_{n_i}^{m-1}{u}_{n_i}〉\\ +〈v-{\Lambda }_{n_i}^m{u}_{n_i},\frac{1}{{\lambda }_{m,n_i}}({\Lambda }_{n_i}^{m-1}{u}_{n_i}-{\Lambda }_{n_i}^m{u}_{n_i})〉.\end{array}$$

Since $\parallel {\Lambda }_n^m{u}_n-{\Lambda }_n^{m-1}{u}_n\parallel \to 0$ (due to (3.20)) and $\parallel B_m{\Lambda }_n^m{u}_n-B_m{\Lambda }_n^{m-1}{u}_n\parallel \to 0$ (due to the Lipschitz-continuity of $B_m$), we conclude from ${\Lambda }_{n_i}^m{u}_{n_i}\rightharpoonup w$ and $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$, $i\in \{1,2,\dots ,N\}$ that

$$\underset{i\to \mathrm{\infty }}{lim}〈v-{\Lambda }_{n_i}^m{u}_{n_i},g〉=〈v-w,g〉\ge 0.$$

It follows from the maximal monotonicity of $B_m+R_m$ that $0\in (R_m+B_m)w$, i.e., $w\in \mathrm{I}(B_m,R_m)$. Therefore, $w\in {\bigcap }_{m=1}^N\mathrm{I}(B_m,R_m)$.

Next, we show that $w\in GMEP(\Theta ,\phi ,A)$. In fact, from $z_n=S_{r_n}^{(\Theta ,\phi )}(I-r_{n}A)x_n$, we know that

$$\Theta (u_n,y)+\phi (y)-\phi (u_n)+〈A{x}_n,y-u_n〉+\frac{1}{r_n}〈K^{\prime }(u_n)-K^{\prime }(x_n),y-u_n〉\ge 0,\mathrm{\forall }y\in C.$$

From (H2) it follows that

$$\phi (y)-\phi (u_n)+〈A{x}_n,y-u_n〉+\frac{1}{r_n}〈K^{\prime }(u_n)-K^{\prime }(x_n),y-u_n〉\ge \Theta (y,u_n),\mathrm{\forall }y\in C.$$

Replacing n by $n_i$, we have

$$\begin{array}{r}\phi (y)-\phi (u_{n_i})+〈A{x}_{n_i},y-u_{n_i}〉+〈\frac{K^{\prime }(u_{n_i})-K^{\prime }(x_{n_i})}{r_{n_i}},y-u_{n_i}〉\ge \Theta (y,u_{n_i}),\\ \mathrm{\forall }y\in C.\end{array}$$

(3.29)

Put $u_t=ty+(1-t)w$ for all $t\in (0,1]$ and $y\in C$. Then, from (3.29), we have

$$\begin{array}{r}〈u_t-u_{n_i},A{u}_t〉\\ \ge 〈u_t-\end{array}$$