Hybrid extragradient method for hierarchical variational inequalities

Abstract

In this paper, we consider a hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, a general system of variational inequalities, and the fixed point problem of a strictly pseudocontractive mapping. By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method and Mann’s iteration method, we introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of our HVIP. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a solution of a general system of variational inequalities defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to a unique solution of our HVIP. The results obtained in this paper improve and extend the corresponding results announced by many others.

MSC:49J30, 47H09, 47J20.

1 Introduction and formulations

1.1 Variational inequalities and equilibrium problems

Let C be a nonempty closed convex subset of a real Hilbert space H and $A:C\to H$ be a nonlinear mapping. A variational inequality problem (VIP) is to find a point $x\in C$ such that

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

(1.1)

The solution set of the VIP (1.1) defined by C and A is denoted by $VI(C,A)$. The theory of variational inequalities is a well established subject in nonlinear analysis and optimization. For different aspects of variational inequalities and their generalizations, we refer to [1–3] and the references therein. Several solution methods for solving different kinds of variational inequality have appeared in literature. Korpelevich’s extragradient method [4] is one of them. It is further studied in [5].

Let $\Theta :C\times C\to \mathbf{R}$ be a bifunction. The equilibrium problem (EP) is to find $x\in C$ such that

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

(1.2)

The set of solutions of EP is denoted by $EP(\Theta )$. It is a unified model of several problems, namely, variational inequalities, Nash equilibrium problems, optimization problems, saddle point problems, etc. For further details of EP, we refer to [6, 7] and the references therein.

Let $\phi :C\to \mathbf{R}$ be a real-valued function and $A:H\to H$ be a nonlinear mapping. The generalized mixed equilibrium problem (GMEP) [8] is to find $x\in C$ such that

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

(1.3)

We denote the set of solutions of GMEP (1.3) by $GMEP(\Theta ,\phi ,A)$. The GMEP (1.3) 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 [9, 10] and the references therein.

When $A\equiv 0$, then GMEP (1.3) reduces to the following mixed equilibrium problem (MEP): find $x\in C$ such that

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

The set of solutions of MEP is denoted by $MEP(\Theta ,\phi )$.

The common assumptions on the bifunction $\Theta :C\times C\to \mathbf{R}$ to study GMEP (1.3) or EP (1.2) are the following:

(A1) $\Theta (x,x)=0$ for all $x\in C$;

(A2) Θ is monotone, i.e., $\Theta (x,y)+\Theta (y,x)\le 0$ for any $x,y\in C$;

(A3) Θ is upper-hemicontinuous, i.e., for each $x,y,z\in C$,

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

(A4) $\Theta (x,\cdot )$ is convex and lower semicontinuous for each $x\in C$;

We use the assumption that the function $\phi :C\to \mathbf{R}$ is a lower semicontinuous and convex function with restriction (B1) or (B2), where

(B1) 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 $z\in C\setminus D_x$,

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

(B2) C is a bounded set.

Given a positive number $r>0$. Let $T_r^{(\Theta ,\phi )}:H\to C$ be the solution set of the auxiliary mixed equilibrium problem, that is, for each $x\in H$,

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

Some elementary conclusions related to MEP are given in the following result.

Proposition 1.1 [11]Let $\Theta :C\times C\to \mathbf{R}$ satisfy conditions (A1)-(A4) and $\phi :C\to \mathbf{R}$ be a proper lower semicontinuous and convex function such that either (B1) or (B2) holds. For $r>0$ and $x\in H$, define a mapping $T_r^{(\Theta ,\phi )}:H\to C$ by

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

Then the following conclusions hold:

• (i) For each $x\in H$, $T_r^{(\Theta ,\phi )}(x)$ is nonempty and single-valued;

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

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

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

• (iv) $MEP(\Theta ,\phi )$ is closed and convex;

• (v) ${\parallel T_s^{(\Theta ,\phi )}x-T_t^{(\Theta ,\phi )}x\parallel }^2\le \frac{s-t}{s}〈T_s^{(\Theta ,\phi )}x-T_t^{(\Theta ,\phi )}x,T_s^{(\Theta ,\phi )}x-x〉$, $\mathrm{\forall }s,t>0$ and $x\in H$.

Recently, Cai and Bu [12] considered a problem of finding a common element of the set of solutions of finitely many generalized mixed equilibrium problems, the set of solutions of finitely many variational inequalities mappings and the set of fixed points of an asymptotically k-strict pseudocontractive mapping in the intermediate sense [13] in a real Hilbert space. They proposed and analyzed an algorithm for finding such a solution. The weak convergence result for the proposed algorithm is also presented.

1.2 General system of variational inequalities

Let $F_1,F_2:C\to H$ be two mappings. We consider the general system of variational inequalities (GSVI) of finding $(x^{\ast },y^{\ast })\in C\times C$ such that

$$\{\begin{array}{l}〈{\nu }_1{F}_1{y}^{\ast }+x^{\ast }-y^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\\ 〈{\nu }_2{F}_2{x}^{\ast }+y^{\ast }-x^{\ast },x-y^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\end{array}$$

(1.4)

where ${\nu }_1>0$ and ${\nu }_2>0$ are two constants. It was considered and studied in [5, 14, 15]. In particular, if $F_1=F_2=A$, then the GSVI (1.4) reduces to the problem of finding $(x^{\ast },y^{\ast })\in C\times C$ such that

$$\{\begin{array}{l}〈{\nu }_{1}A{y}^{\ast }+x^{\ast }-y^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,\\ 〈{\nu }_{2}A{x}^{\ast }+y^{\ast }-x^{\ast },x-y^{\ast }〉\ge 0,\mathrm{\forall }x\in C.\end{array}$$

(1.5)

It is called a new system of variational inequalities (NSVI) [9]. It is worth to mention that the above system of two variational inequalities could be used to solve Nash equilibrium problem. For applications of system of variational inequalities to Nash equilibrium problems, we refer to [16–19] and the references therein. Further, if $x^{\ast }=y^{\ast }$ additionally, then the NSVI reduces to the classical VIP (1.1). Putting $G:=P_C(I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)$ and $y^{\ast }=P_C(I-{\nu }_2{F}_2)x^{\ast }$, Ceng et al. [15] transformed the GSVI (1.4) into the following fixed point equation:

$$G{x}^{\ast }=x^{\ast }.$$

(1.6)

1.3 Hierarchical variational inequalities

A variational inequality problem defined over the set of fixed points of a nonexpansive mapping, is called a hierarchical variational inequality problem. Let $S,T:H\to H$ be nonexpansive mappings. We denote by $Fix(T)$ and $Fix(T)$ the set of fixed points of T and S, respectively. If we replace C by $Fix(T)$ in the formulation of VIP (1.1), then VIP (1.1) is defined by $Fix(T)$ and A and it is called a hierarchical variational inequality problem.

Yao et al. [20] considered the hierarchical variational inequality problem (HVIP) in which the mapping A is replaced by the monotone mapping $I-S$. They considered the following form of HVIP: find $\tilde{x}\in Fix(T)$ such that

$$〈(I-S)\tilde{x},p-\tilde{x}〉\ge 0,\mathrm{\forall }p\in Fix(T).$$

(1.7)

The solution set of HVIP (1.7) is denoted by Λ. It is not hard to check that solving HVIP (1.7) is equivalent to the fixed point problem of the composite mapping $P_{Fix(T)}\circ S$, that is, find $\tilde{x}\in C$ such that $\tilde{x}=P_{Fix(T)}S\tilde{x}$. They proposed and analyzed an iterative method for solving this kind of HTVI. For further details and a comprehensive survey on HVIP, we refer to [21] and the references therein.

1.4 Variational inclusions

Let $B:C\to H$ be a single-valued mapping and $R:C\to 2^H$ be a set-valued mapping with $D(R)=C$. We consider the variational inclusion problem of finding a point $x\in C$ such that

$$0\in Bx+Rx.$$

(1.8)

We denote by $\mathrm{I}(B,R)$ the solution set of the variational inclusion (1.8). In particular, if $B\equiv R\equiv 0$, then $\mathrm{I}(B,R)=C$. If $B=0$, then problem (1.8) becomes the inclusion problem introduced by Rockafellar [22]. It is well known that problem (1.8) 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.

1.5 Problem to be considered

In this paper, we introduce and study the following hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many GMEPs, finitely many variational inclusions, a general system of variational inequalities, and a fixed point of a strictly pseudocontractive mapping. Throughout the paper, we denote by M and N set of the positive integers.

Problem 1.1 Assume that

• (i) for $j=1,2$, $F_j:C\to H$ and $F:H\to H$ are mappings;

• (ii) for each $k\in \{1,2,\dots ,M\}$, ${\Theta }_k:C\times C\to \mathbf{R}$ is a bifunction satisfying conditions (A1)-(A4) and ${\phi }_k:C\to \mathbf{R}\cup \{+\mathrm{\infty }\}$ is a proper lower semicontinuous and convex function with restriction (B1) or (B2);

• (iii) for each $k\in \{1,2,\dots ,M\}$ and $i\in \{1,2,\dots ,N\}$, $R_i:C\to 2^H$ is a maximal monotone mapping, and $A_k:H\to H$ and $B_i:C\to H$ are mappings;

• (iv) $T:C\to C$ is a mapping and $S:H\to H$ is a nonexpansive mapping;

• (v) $\Omega :={\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,A_k)\cap {\bigcap }_{i=1}^N\mathrm{I}(B_i,R_i)\cap GSVI(G)\cap Fix(T)\ne \mathrm{\varnothing }$.

Then the objective is to find $x^{\ast }\in \Omega$ such that

$$〈(\mu F-\gamma S)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \Omega .$$

(1.9)

By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [10], and Mann’s iteration method, we introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of Problem 1.1. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a solution of GSVI (1.4) defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to a unique solution of Problem 1.1. The results obtained in this paper improve and extend the corresponding results announced by many others.

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

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

• (ii) 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. Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields. It is obvious that if A is ζ-inverse-strongly monotone, then A is monotone and $\frac{1}{\zeta }$-Lipschitz continuous. Moreover, we also have, for all $u,v\in C$ and $\lambda >0$,

$$\begin{array}{rcl}{\parallel (I-\lambda A)u-(I-\lambda A)v\parallel }^2& =& {\parallel (u-v)-\lambda (Au-Av)\parallel }^2\\ =& {\parallel u-v\parallel }^2-2\lambda 〈Au-Av,u-v〉+{\lambda }^2{\parallel Au-Av\parallel }^2\\ \le & {\parallel u-v\parallel }^2+\lambda (\lambda -2\zeta ){\parallel Au-Av\parallel }^2.\end{array}$$

(2.1)

So, if $\lambda \le 2\zeta$, then $I-\lambda A$ is a nonexpansive mapping from C to H.

Definition 2.2 A mapping $T:H\to H$ is said to be firmly nonexpansive if $2T-I$ is nonexpansive, or equivalently, if T is 1-inverse-strongly monotone (1-ism),

$$〈x-y,Tx-Ty〉\ge {\parallel Tx-Ty\parallel }^2,\mathrm{\forall }x,y\in H;$$

alternatively, T is firmly nonexpansive if and only if T can be expressed as

$$T=\frac{1}{2}(I+S),$$

where $S:H\to H$ is nonexpansive; projections are firmly nonexpansive.

It can easily be seen that if T is nonexpansive, then $I-T$ is monotone.

It is clear that, in a real Hilbert space H, $T:C\to C$ is ξ-strictly pseudocontractive if and only if the following inequality holds:

$$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2-\frac{1-\xi }{2}{\parallel (I-T)x-(I-T)y\parallel }^2,\mathrm{\forall }x,y\in C.$$

This immediately implies that if T is a ξ-strictly pseudocontractive mapping, then $I-T$ is $\frac{1-\xi }{2}$-inverse-strongly monotone; for further details, we refer to [23] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions.

Lemma 2.1 [[23], Proposition 2.1]

Let C be a nonempty closed convex subset of a real Hilbert space H and $T:C\to C$ be a mapping.

• (i) If T is a ξ-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition

  $$\parallel Tx-Ty\parallel \le \frac{1+\xi }{1-\xi }\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

• (ii) If T is a ξ-strictly pseudocontractive mapping, then the mapping $I-T$ is semiclosed at 0, that is, if $\{x_n\}$ is a sequence in C such that $x_n\rightharpoonup \tilde{x}$ and $(I-T)x_n\to 0$, then $(I-T)\tilde{x}=0$.

• (iii) If T is a ξ-(quasi-)strict pseudocontraction, then the fixed-point set $Fix(T)$ of T is closed and convex so that the projection $P_{Fix(T)}$ is well defined.

Lemma 2.2 [24]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a ξ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that $(\gamma +\delta )\xi \le \gamma$. Then

$$\parallel \gamma (x-y)+\delta (Tx-Ty)\parallel \le (\gamma +\delta )\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

Let C be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in $(0,1]$ and let $\mu >0$. Associated with a nonexpansive mapping $T:C\to H$, we define the mapping $T^{\lambda }:C\to H$ by

$$T^{\lambda }x:=Tx-\lambda \mu F(Tx),\mathrm{\forall }x\in C,$$

where $F:H\to H$ is an operator such that, for some positive constants $\kappa ,\eta >0$, F is κ-Lipschitzian and η-strongly monotone on H, that is, F satisfies the conditions

$$\parallel Fx-Fy\parallel \le \kappa \parallel x-y\parallel \text{and}〈Fx-Fy,x-y〉\ge \eta {\parallel x-y\parallel }^2$$

for all $x,y\in H$.

Remark 2.1 Since F is κ-Lipschitzian and η-strongly monotone on H, we get $0<\eta \le \kappa$. Hence, whenever $0<\mu <\frac{2\eta }{{\kappa }^2}$, we have

$$\begin{array}{rl}0& \le {(1-\mu \eta )}^2=1-2\mu \eta +{\mu }^2{\eta }^2\\ \le 1-2\mu \eta +{\mu }^2{\kappa }^2<1-2\mu \eta +\frac{2\eta }{{\kappa }^2}\mu {\kappa }^2=1,\end{array}$$

which implies

$$0<1-\sqrt{1-2\mu \eta +{\mu }^2{\kappa }^2}\le 1.$$

So, $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}\in (0,1]$.

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

$$〈f-g,x-y〉\ge 0.$$

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

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.,

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

Define

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

Then $\tilde{T}$ is maximal monotone (see [22]) such that

$$0\in \tilde{T}v⟺v\in VI(C,A).$$

(2.2)

Let $R:D(R)\subset H\to 2^H$ be a maximal monotone mapping. Let $\lambda ,\mu >0$ be two positive numbers. Associated with R and λ, we define the resolvent operator $J_{R,\lambda }:H\to \overline{D(R)}$ by

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

where λ is a positive number.

The lemma shows that the resolvent operator $J_{R,\lambda }:H\to \overline{D(R)}$ is nonexpansive.

Lemma 2.3 [25]

$J_{R,\lambda }$ 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 2.4 [26]

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 (1.5) if and only if $u\in C$ satisfies

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

Lemma 2.5 [27]

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 2.6 [26]

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 2.7 [28]

We have the resolvent identity

$$J_{R,\lambda }x=J_{R,\mu }(\frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{R,\lambda }x),\mathrm{\forall }x\in H.$$

Remark 2.2 For $\lambda ,\mu >0$, we have the following relation:

$$\parallel J_{R,\lambda }x-J_{R,\mu }y\parallel \le \parallel x-y\parallel +|\lambda -\mu |(\frac{1}{\lambda }\parallel J_{R,\lambda }x-y\parallel +\frac{1}{\mu }\parallel x-J_{R,\mu }y\parallel ),\mathrm{\forall }x,y\in H.$$

(2.3)

Indeed, whenever $\lambda \ge \mu$, utilizing Lemma 2.7 we deduce that

$$\begin{array}{rl}\parallel J_{R,\lambda }x-J_{R,\mu }y\parallel & =\parallel J_{R,\mu }(\frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{R,\lambda }x)-J_{R,\mu }y\parallel \\ \le \parallel \frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{R,\lambda }x-y\parallel \\ \le \frac{\mu }{\lambda }\parallel x-y\parallel +(1-\frac{\mu }{\lambda })\parallel J_{R,\lambda }x-y\parallel \\ \le \parallel x-y\parallel +\frac{|\lambda -\mu |}{\lambda }\parallel J_{R,\lambda }x-y\parallel .\end{array}$$

Similarly, whenever $\lambda <\mu$, we get

$$\parallel J_{R,\lambda }x-J_{R,\mu }y\parallel \le \parallel x-y\parallel +\frac{|\lambda -\mu |}{\mu }\parallel x-J_{R,\mu }y\parallel .$$

Combining the above two cases we conclude that (2.3) holds.

We need following fact and lemmas to establish the strong convergence of the sequences generated by the proposed algorithm.

Lemma 2.8 Let X be a real inner product space. Then

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

Lemma 2.9 Let H be a real Hilbert space. Then:

• (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 2.10 (Demiclosedness principle [29])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C with $Fix(S)\ne \mathrm{\varnothing }$. Then $I-S$ is demiclosed, that is, whenever $\{x_n\}$ is a sequence in C weakly converging to some $x\in C$ and the sequence $\{(I-S)x_n\}$ strongly converges to some y, it follows that $(I-S)x=y$, where I is the identity operator of H.

Lemma 2.11 [[30], Lemma 3.1]

$T^{\lambda }$ is a contraction provided $0<\mu <\frac{2\eta }{{\kappa }^2}$; that is,

$$\parallel T^{\lambda }x-T^{\lambda }y\parallel \le (1-\lambda \tau )\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,$$

where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}\in (0,1]$.

Remark 2.3 (a) Since F is κ-Lipschitzian and η-strongly monotone on H, we get $0<\eta \le \kappa$. Hence, whenever $0<\mu <\frac{2\eta }{{\kappa }^2}$, we have

$$\begin{array}{rl}0& \le {(1-\mu \eta )}^2=1-2\mu \eta +{\mu }^2{\eta }^2\\ \le 1-2\mu \eta +{\mu }^2{\kappa }^2\\ <1-2\mu \eta +\frac{2\eta }{{\kappa }^2}\mu {\kappa }^2=1,\end{array}$$

which implies

$$0<1-\sqrt{1-2\mu \eta +{\mu }^2{\kappa }^2}\le 1.$$

Therefore, $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}\in (0,1]$.

(b) In Lemma 2.11, put $F=\frac{1}{2}I$ and $\mu =2$. Then we know that $\kappa =\eta =\frac{1}{2}$, $0<\mu =2<\frac{2\eta }{{\kappa }^2}=4$ and

$$\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}=1-\sqrt{1-2(2\times \frac{1}{2}-2\times {\left(\frac{1}{2}\right)}^2)}=1.$$

Lemma 2.12 [30]

Let $\{s_n\}$ be a sequence of nonnegative numbers satisfying the conditions

$$s_{n+1}\le (1-{\alpha }_n)s_n+{\alpha }_n{\beta }_n,\mathrm{\forall }n\ge 1,$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences of real numbers such that

• (a) $\{{\alpha }_n\}\subset [0,1]$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, or equivalently,

  $$\prod _{n=1}^{\mathrm{\infty }}(1-{\alpha }_n):=\underset{n\to \mathrm{\infty }}{lim}\prod _{k=1}^n(1-{\alpha }_k)=0;$$

• (b) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n\le 0$, or ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_n{\beta }_n|<\mathrm{\infty }$.

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

3 Algorithms and convergence results

In this section, we will introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of the HVIP (1.9) (over the fixed point set of a strictly pseudocontractive mapping) with constraints of several problems: GSVI (1.4), finitely many GMEPs, and finitely many variational inclusions in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [10], and Mann’s iteration method. We prove the strong convergence of the proposed algorithm to a unique solution of HVIP (1.9) under suitable conditions.

We propose the following algorithm to compute the approximate solution of Problem 1.1.

Algorithm 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. For each $k\in \{1,2,\dots ,M\}$, let ${\Theta }_k:C\times C\to \mathbf{R}$ be a bifunction satisfying conditions (A1)-(A4) and ${\phi }_k:C\to \mathbf{R}\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function with restriction (B1) or (B2). For each $k\in \{1,2,\dots ,M\}$, $i\in \{1,2,\dots ,N\}$, let $R_i:C\to 2^H$ be a maximal monotone mapping, and $A_k:H\to H$ and $B_i:C\to H$ be ${\mu }_k$-inverse-strongly monotone and ${\eta }_i$-inverse-strongly monotone, respectively. Let $T:C\to C$ be a ξ-strictly pseudocontractive mapping, $S:H\to H$ is a nonexpansive mapping and $V:H\to H$ be a ρ-contraction with coefficient $\rho \in [0,1)$. For each $j=1,2$, let $F_j:C\to H$ be ${\zeta }_j$-inverse-strongly monotone and $F:H\to H$ be κ-Lipschitzian and η-strongly monotone with positive constants $\kappa ,\eta >0$ such that $0\le \gamma <\tau$ and $0<\mu <\frac{2\eta }{{\kappa }^2}$ where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Assume that $\Omega :={\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,A_k)\cap {\bigcap }_{i=1}^N\mathrm{I}(B_i,R_i)\cap GSVI(G)\cap Fix(T)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\},\{{\lambda }_n\}\subset (0,1]$, $\{{\beta }_n\},\{{\gamma }_n\},\{{\delta }_n\}\subset [0,1]$, $\{{\rho }_n\}\subset (0,2\alpha ]$, $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$ and $\{r_{k,n}\}\subset [c_k,d_k]\subset (0,2{\mu }_k)$ where $i\in \{1,2,\dots ,N\}$ and $k\in \{1,2,\dots ,M\}$. For arbitrarily given $x_0\in H$, let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{l}{u}_n=T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M)T_{r_{M-1,n}}^{({\Theta }_{M-1},{\phi }_{M-1})}(I-r_{M-1,n}{A}_{M-1})\cdots T_{r_{1,n}}^{({\Theta }_1,{\phi }_1)}(I-r_{1,n}{A}_1)x_n,\\ v_n=J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N)J_{R_{N-1},{\lambda }_{N-1,n}}(I-{\lambda }_{N-1,n}{B}_{N-1})\cdots J_{R_1,{\lambda }_{1,n}}(I-{\lambda }_{1,n}{B}_1)u_n,\\ y_n={\beta }_n{x}_n+{\gamma }_{n}G{v}_n+{\delta }_{n}TG{v}_n,\\ x_{n+1}={\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)+(I-{\lambda }_n\mu F)y_n,\mathrm{\forall }n\ge 0,\end{array}$$

(3.1)

where $G:=P_C(I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)$ with ${\nu }_j\in (0,2{\zeta }_j)$ for $j=1,2$.

Theorem 3.1 In addition to the assumption in Algorithm  3.1, suppose that

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

• (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\alpha }_n}{{\lambda }_n}<\mathrm{\infty }$, ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\lambda }_n}|\frac{1}{{\alpha }_n}-\frac{1}{{\alpha }_{n-1}}|=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\alpha }_n}|1-\frac{{\lambda }_{n-1}}{{\lambda }_n}|=0$;

• (iii) ${lim}_{n\to \mathrm{\infty }}\frac{|{\beta }_n-{\beta }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{|{\gamma }_n-{\gamma }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$;

• (iv) ${lim}_{n\to \mathrm{\infty }}\frac{|{\lambda }_{i,n}-{\lambda }_{i,n-1}|}{{\lambda }_n{\alpha }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{|r_{k,n}-r_{k,n-1}|}{{\lambda }_n{\alpha }_n}=0$ for $i=1,2,\dots ,N$ and $k=1,2,\dots ,M$;

• (v) ${\beta }_n+{\gamma }_n+{\delta }_n=1$ and $({\gamma }_n+{\delta }_n)\xi \le {\gamma }_n$ for all $n\ge 0$;

• (vi) $\{{\beta }_n\}\subset [a,b]\subset (0,1)$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n>0$.

Then

• (a) ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\alpha }_n}=0$;

• (b) ${\omega }_w(x_n)\subset \Omega$;

• (c) $\{x_n\}$ converges strongly to a point $x^{\ast }\in \Omega$, which is a unique solution of HVIP (1.9), that is,

  $$〈(\mu F-\gamma S)x^{\ast },p-x^{\ast }〉\ge 0,\mathrm{\forall }p\in \Omega .$$

Proof First of all, observe that

$$\begin{array}{rl}\mu \eta \ge \tau & ⟺\mu \eta \ge 1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}\\ ⟺\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}\ge 1-\mu \eta \\ ⟺1-2\mu \eta +{\mu }^2{\kappa }^2\ge 1-2\mu \eta +{\mu }^2{\eta }^2\\ ⟺{\kappa }^2\ge {\eta }^2\\ ⟺\kappa \ge \eta \end{array}$$

and

$$\begin{array}{rl}〈(\mu F-\gamma S)x-(\mu F-\gamma S)y,x-y〉& =\mu 〈Fx-Fy,x-y〉-\gamma 〈Sx-Sy,x-y〉\\ \ge \mu \eta {\parallel x-y\parallel }^2-\gamma {\parallel x-y\parallel }^2\\ =(\mu \eta -\gamma ){\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H.\end{array}$$

Since $0\le \gamma <\tau$ and $\kappa \ge \eta$, we know that $\mu \eta \ge \tau >\gamma$ and hence the mapping $\mu F-\gamma S$ is $(\mu \eta -\gamma )$-strongly monotone. Moreover, it is clear that the mapping $\mu F-\gamma S$ is $(\mu \kappa +\gamma )$-Lipschitzian. Thus, there exists a unique solution $x^{\ast }$ in Ω to the VIP

$$〈(\mu F-\gamma S)x^{\ast },p-x^{\ast }〉\ge 0,\mathrm{\forall }p\in \Omega .$$

That is, $\{x^{\ast }\}=VI(\Omega ,\mu F-\gamma S)$. Now, we put

$${\Delta }_n^k=T_{r_{k,n}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n}{A}_k)T_{r_{k-1,n}}^{({\Theta }_{k-1},{\phi }_{k-1})}(I-r_{k-1,n}{A}_{k-1})\cdots T_{r_{1,n}}^{({\Theta }_1,{\phi }_1)}(I-r_{1,n}{A}_1)x_n$$

for all $k\in \{1,2,\dots ,M\}$ and $n\ge 1$,

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

for all $i\in \{1,2,\dots ,N\}$, ${\Delta }_n^0=I$ and ${\Lambda }_n^0=I$, where I is the identity mapping on H. Then we have $u_n={\Delta }_n^M{x}_n$ and $v_n={\Lambda }_n^N{u}_n$.

We divide the rest of the proof into several steps.

Step 1. We prove that $\{x_n\}$ is bounded.

Indeed, take a fixed $p\in \Omega$ arbitrarily. Utilizing (2.1) and Proposition 1.1(ii), we have

$$\begin{array}{rcl}\parallel u_n-p\parallel & =& \parallel T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{B}_M){\Delta }_n^{M-1}{x}_n-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{B}_M){\Delta }_n^{M-1}p\parallel \\ \le & \parallel (I-r_{M,n}{B}_M){\Delta }_n^{M-1}{x}_n-(I-r_{M,n}{B}_M){\Delta }_n^{M-1}p\parallel \\ \le & \parallel {\Delta }_n^{M-1}{x}_n-{\Delta }_n^{M-1}p\parallel \\ \cdots \\ \le & \parallel {\Delta }_n^0{x}_n-{\Delta }_n^{0}p\parallel \\ =& \parallel x_n-p\parallel .\end{array}$$

(3.2)

Utilizing (2.1) and Lemma 2.3, we have

$$\begin{array}{rcl}\parallel v_n-p\parallel & =& \parallel J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}{u}_n-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}p\parallel \\ \le & \parallel (I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}{u}_n-(I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}p\parallel \\ \le & \parallel {\Lambda }_n^{N-1}{u}_n-{\Lambda }_n^{N-1}p\parallel \\ \cdots \\ \le & \parallel {\Lambda }_n^0{u}_n-{\Lambda }_n^{0}p\parallel \\ =& \parallel u_n-p\parallel .\end{array}$$

(3.3)

Combining (3.2) and (3.3), we have

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

(3.4)

Since $p=Gp=P_C(I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)p$, $F_j$ is ${\zeta }_j$-inverse-strongly monotone for $j=1,2$, and $0<{\nu }_j\le 2{\zeta }_j$ for $j=1,2$, we deduce that, for any $n\ge 0$,

$$\begin{array}{c}{\parallel G{v}_n-p\parallel }^2\hfill \\ ={\parallel P_C(I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)v_n-P_C(I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)p\parallel }^2\hfill \\ \le {\parallel (I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)v_n-(I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)p\parallel }^2\hfill \\ ={\parallel [P_C(I-{\nu }_2{F}_2)v_n-P_C(I-{\nu }_2{F}_2)p]-{\nu }_1[F_1{P}_C(I-{\nu }_2{F}_2)v_n-F_1{P}_C(I-{\nu }_2{F}_2)p]\parallel }^2\hfill \\ \le {\parallel P_C(I-{\nu }_2{F}_2)v_n-P_C(I-{\nu }_2{F}_2)p\parallel }^2\hfill \\ +{\nu }_1({\nu }_1-2{\zeta }_1){\parallel F_1{P}_C(I-{\nu }_2{F}_2)v_n-F_1{P}_C(I-{\nu }_2{F}_2)p\parallel }^2\hfill \\ \le {\parallel P_C(I-{\nu }_2{F}_2)v_n-P_C(I-{\nu }_2{F}_2)p\parallel }^2\hfill \\ \le {\parallel (I-{\nu }_2{F}_2)v_n-(I-{\nu }_2{F}_2)p\parallel }^2\hfill \\ ={\parallel (v_n-p)-{\nu }_2(F_2{v}_n-F_{2}p)\parallel }^2\hfill \\ \le {\parallel v_n-p\parallel }^2+{\nu }_2({\nu }_2-2{\zeta }_2){\parallel F_2{v}_n-F_{2}p\parallel }^2\hfill \\ \le {\parallel v_n-p\parallel }^2.\hfill \end{array}$$

(3.5)

(This shows that $G:C\to C$ is a nonexpansive mapping.) Since $({\gamma }_n+{\delta }_n)\xi \le {\gamma }_n$ for all $n\ge 0$ and T is ξ-strictly pseudocontractive, utilizing Lemma 2.2, we obtain from (3.1), (3.4), and (3.5)

$$\begin{array}{rl}\parallel y_n-p\parallel & =\parallel {\beta }_n{x}_n+{\gamma }_{n}G{v}_n+{\delta }_{n}TG{v}_n-p\parallel \\ =\parallel {\beta }_n(x_n-p)+{\gamma }_n(G{v}_n-p)+{\delta }_n(TG{v}_n-p)\parallel \\ \le {\beta }_n\parallel x_n-p\parallel +\parallel {\gamma }_n(G{v}_n-p)+{\delta }_n(TG{v}_n-p)\parallel \\ \le {\beta }_n\parallel x_n-p\parallel +({\gamma }_n+{\delta }_n)\parallel G{v}_n-p\parallel \\ \le {\beta }_n\parallel x_n-p\parallel +({\gamma }_n+{\delta }_n)\parallel v_n-p\parallel \\ \le {\beta }_n\parallel x_n-p\parallel +({\gamma }_n+{\delta }_n)\parallel x_n-p\parallel \\ =\parallel x_n-p\parallel .\end{array}$$

(3.6)

Utilizing Lemma 2.11, we deduce from (3.1), (3.6), and $0\le \gamma <\tau$ that for all $n\ge 0$

$$\begin{array}{c}\parallel x_{n+1}-p\parallel \hfill \\ =\parallel {\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)+(I-{\lambda }_n\mu F)y_n-p\parallel \hfill \\ =\parallel {\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)-{\lambda }_n\mu Fp+(I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel \hfill \\ \le \parallel {\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)-{\lambda }_n\mu Fp\parallel +\parallel (I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel \hfill \\ ={\lambda }_n\parallel {\alpha }_n(\gamma V{x}_n-\mu Fp)+(1-{\alpha }_n)(\gamma S{x}_n-\mu Fp)\parallel +\parallel (I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel \hfill \\ \le {\lambda }_n[{\alpha }_n\parallel \gamma V{x}_n-\mu Fp\parallel +(1-{\alpha }_n)\parallel \gamma S{x}_n-\mu Fp\parallel ]+\parallel (I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel \hfill \\ \le {\lambda }_n[{\alpha }_n(\gamma \parallel V{x}_n-Vp\parallel +\parallel \gamma Vp-\mu Fp\parallel )+(1-{\alpha }_n)(\gamma \parallel S{x}_n-Sp\parallel +\parallel \gamma Sp-\mu Fp\parallel )]\hfill \\ +\parallel (I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel \hfill \\ \le {\lambda }_n[{\alpha }_n(\gamma \rho \parallel x_n-p\parallel +\parallel \gamma Vp-\mu Fp\parallel )+(1-{\alpha }_n)(\gamma \parallel x_n-p\parallel +\parallel \gamma Sp-\mu Fp\parallel )]\hfill \\ +(1-{\lambda }_n\tau )\parallel y_n-p\parallel \hfill \\ \le {\lambda }_n[(1-{\alpha }_n(1-\rho ))\gamma \parallel x_n-p\parallel +max\{\parallel \gamma Vp-\mu Fp\parallel ,\parallel \gamma Sp-\mu Fp\parallel \}]\hfill \\ +(1-{\lambda }_n\tau )\parallel x_n-p\parallel \hfill \\ ={\lambda }_n(1-{\alpha }_n(1-\rho ))\gamma \parallel x_n-p\parallel +{\lambda }_{n}max\{\parallel \gamma Vp-\mu Fp\parallel ,\parallel \gamma Sp-\mu Fp\parallel \}\hfill \\ +(1-{\lambda }_n\tau )\parallel x_n-p\parallel \hfill \\ \le {\lambda }_n\gamma \parallel x_n-p\parallel +{\lambda }_{n}max\{\parallel \gamma Vp-\mu Fp\parallel ,\parallel \gamma Sp-\mu Fp\parallel \}+(1-{\lambda }_n\tau )\parallel x_n-p\parallel \hfill \\ =(1-{\lambda }_n(\tau -\gamma ))\parallel x_n-p\parallel +{\lambda }_{n}max\{\parallel \gamma Vp-\mu Fp\parallel ,\parallel \gamma Sp-\mu Fp\parallel \}\hfill \\ =(1-{\lambda }_n(\tau -\gamma ))\parallel x_n-p\parallel +{\lambda }_n(\tau -\gamma )\frac{max\{\parallel \gamma Vp-\mu Fp\parallel ,\parallel \gamma Sp-\mu Fp\parallel \}}{\tau -\gamma }\hfill \\ \le max\{\parallel x_n-p\parallel ,\frac{\parallel \gamma Vp-\mu Fp\parallel }{\tau -\gamma },\frac{\parallel \gamma Sp-\mu Fp\parallel }{\tau -\gamma }\}.\hfill \end{array}$$

By induction, we get

$$\parallel x_n-p\parallel \le max\{\parallel x_0-p\parallel ,\frac{\parallel \gamma Vp-\mu Fp\parallel }{\tau -\gamma },\frac{\parallel \gamma Sp-\mu Fp\parallel }{\tau -\gamma }\},\mathrm{\forall }n\ge 0.$$

Thus, $\{x_n\}$ is bounded and so are the sequences $\{u_n\}$, $\{v_n\}$, and $\{y_n\}$.

Step 2. We prove that ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\alpha }_n}=0$.

Indeed, utilizing (2.1) and (2.3), we obtain

$$\begin{array}{c}\parallel v_{n+1}-v_n\parallel \hfill \\ =\parallel {\Lambda }_{n+1}^N{u}_{n+1}-{\Lambda }_n^N{u}_n\parallel \hfill \\ =\parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n+1}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \hfill \\ \le \parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n+1}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel \hfill \\ +\parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \hfill \\ \le \parallel (I-{\lambda }_{N,n+1}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel \hfill \\ +\parallel (I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel +|{\lambda }_{N,n+1}-{\lambda }_{N,n}|\hfill \\ \times (\frac{1}{{\lambda }_{N,n+1}}\parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \hfill \\ +\frac{1}{{\lambda }_{N,n}}\parallel (I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel )\hfill \\ \le |{\lambda }_{N,n+1}-{\lambda }_{N,n}|(\parallel B_N{\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel +\tilde{M})+\parallel {\Lambda }_{n+1}^{N-1}{u}_{n+1}-{\Lambda }_n^{N-1}{u}_n\parallel \hfill \\ \le |{\lambda }_{N,n+1}-{\lambda }_{N,n}|(\parallel B_N{\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel +\tilde{M})\hfill \\ +|{\lambda }_{N-1,n+1}-{\lambda }_{N-1,n}|(\parallel B_{N-1}{\Lambda }_{n+1}^{N-2}{u}_{n+1}\parallel +\tilde{M})+\parallel {\Lambda }_{n+1}^{N-2}{u}_{n+1}-{\Lambda }_n^{N-2}{u}_n\parallel \hfill \\ \cdots \hfill \\ \le |{\lambda }_{N,n+1}-{\lambda }_{N,n}|(\parallel B_N{\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel +\tilde{M})+|{\lambda }_{N-1,n+1}-{\lambda }_{N-1,n}|(\parallel B_{N-1}{\Lambda }_{n+1}^{N-2}{u}_{n+1}\parallel +\tilde{M})\hfill \\ +\cdots +|{\lambda }_{1,n+1}-{\lambda }_{1,n}|(\parallel B_1{\Lambda }_{n+1}^0{u}_{n+1}\parallel +\tilde{M})+\parallel {\Lambda }_{n+1}^0{u}_{n+1}-{\Lambda }_n^0{u}_n\parallel \hfill \\ \le {\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+\parallel u_{n+1}-u_n\parallel ,\hfill \end{array}$$

(3.7)

where

$$\begin{array}{c}\underset{n\ge 0}{sup}\{\frac{1}{{\lambda }_{N,n+1}}\parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \hfill \\ +\frac{1}{{\lambda }_{N,n}}\parallel (I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \}\le \tilde{M}\hfill \end{array}$$

for some $\tilde{M}>0$ and ${sup}_{n\ge 0}\{{\sum }_{i=1}^N\parallel B_i{\Lambda }_{n+1}^{i-1}{u}_{n+1}\parallel +\tilde{M}\}\le {\tilde{M}}_0$ for some ${\tilde{M}}_0>0$.

Utilizing Proposition 1.1(ii), (v), we deduce that

$$\begin{array}{c}\parallel u_{n+1}-u_n\parallel \hfill \\ =\parallel {\Delta }_{n+1}^M{x}_{n+1}-{\Delta }_n^M{x}_n\parallel \hfill \\ =\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_n^{M-1}{x}_n\parallel \hfill \\ \le \parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \hfill \\ +\parallel T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_n^{M-1}{x}_n\parallel \hfill \\ \le \parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \hfill \\ +\parallel T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \hfill \\ +\parallel (I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-(I-r_{M,n}{A}_M){\Delta }_n^{M-1}{x}_n\parallel \hfill \\ \le \frac{|r_{M,n+1}-r_{M,n}|}{r_{M,n+1}}\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \hfill \\ +|r_{M,n+1}-r_{M,n}|\parallel A_M{\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel +\parallel {\Delta }_{n+1}^{M-1}{x}_{n+1}-{\Delta }_n^{M-1}{x}_n\parallel \hfill \\ =|r_{M,n+1}-r_{M,n}|[\parallel A_M{\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel +\frac{1}{r_{M,n+1}}\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\hfill \\ -(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel ]+\parallel {\Delta }_{n+1}^{M-1}{x}_{n+1}-{\Delta }_n^{M-1}{x}_n\parallel \hfill \\ \cdots \hfill \\ \le |r_{M,n+1}-r_{M,n}|[\parallel A_M{\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel +\frac{1}{r_{M,n+1}}\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\hfill \\ -(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel ]+\cdots +|r_{1,n+1}-r_{1,n}|[\parallel A_1{\Delta }_{n+1}^0{x}_{n+1}\parallel \hfill \\ +\frac{1}{r_{1,n+1}}\parallel T_{r_{1,n+1}}^{({\Theta }_1,{\phi }_1)}(I-r_{1,n+1}{A}_1){\Delta }_{n+1}^0{x}_{n+1}-(I-r_{1,n+1}{A}_1){\Delta }_{n+1}^0{x}_{n+1}\parallel ]\hfill \\ +\parallel {\Delta }_{n+1}^0{x}_{n+1}-{\Delta }_n^0{x}_n\parallel \hfill \\ \le {\tilde{M}}_1\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|+\parallel x_{n+1}-x_n\parallel ,\hfill \end{array}$$

(3.8)

where ${\tilde{M}}_1>0$ is a constant such that for each $n\ge 0$

$$\begin{array}{c}\sum _{k=1}^M[\parallel A_k{\Delta }_{n+1}^{k-1}{x}_{n+1}\parallel +\frac{1}{r_{k,n+1}}\parallel T_{r_{k,n+1}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n+1}{A}_k){\Delta }_{n+1}^{k-1}{x}_{n+1}-(I-r_{k,n+1}{A}_k){\Delta }_{n+1}^{k-1}{x}_{n+1}\parallel ]\hfill \\ \le {\tilde{M}}_1.\hfill \end{array}$$

Furthermore, we define $y_n={\beta }_n{x}_n+(1-{\beta }_n)w_n$ for all $n\ge 0$. It follows that

$$\begin{array}{r}{w}_{n+1}-w_n\\ =\frac{y_{n+1}-{\beta }_{n+1}{x}_{n+1}}{1-{\beta }_{n+1}}-\frac{y_n-{\beta }_n{x}_n}{1-{\beta }_n}\\ =\frac{{\gamma }_{n+1}G{v}_{n+1}+{\delta }_{n+1}TG{v}_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_{n}G{v}_n+{\delta }_{n}TG{v}_n}{1-{\beta }_n}\\ =\frac{{\gamma }_{n+1}(G{v}_{n+1}-G{v}_n)+{\delta }_{n+1}(TG{v}_{n+1}-TG{v}_n)}{1-{\beta }_{n+1}}\\ +(\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n})G{v}_n+(\frac{{\delta }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\delta }_n}{1-{\beta }_n})TG{v}_n.\end{array}$$

(3.9)

Since $({\gamma }_n+{\delta }_n)\xi \le {\gamma }_n$ for all $n\ge 0$, utilizing Lemma 2.2, we have

$$\parallel {\gamma }_{n+1}(G{v}_{n+1}-G{v}_n)+{\delta }_{n+1}(TG{v}_{n+1}-TG{v}_n)\parallel \le ({\gamma }_{n+1}+{\delta }_{n+1})\parallel G{v}_{n+1}-G{v}_n\parallel .$$

(3.10)

Hence it follows from (3.7)-(3.10) that

$$\begin{array}{c}\parallel w_{n+1}-w_n\parallel \hfill \\ \le \frac{\parallel {\gamma }_{n+1}(G{v}_{n+1}-G{v}_n)+{\delta }_{n+1}(TG{v}_{n+1}-TG{v}_n)\parallel }{1-{\beta }_{n+1}}\hfill \\ +|\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}|\parallel G{v}_n\parallel +|\frac{{\delta }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\delta }_n}{1-{\beta }_n}|\parallel TG{v}_n\parallel \hfill \\ \le \frac{({\gamma }_{n+1}+{\delta }_{n+1})}{1-{\beta }_{n+1}}\parallel G{v}_{n+1}-G{v}_n\parallel +|\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}|(\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel )\hfill \\ =\parallel G{v}_{n+1}-G{v}_n\parallel +|\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}|(\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel )\hfill \\ \le \parallel v_{n+1}-v_n\parallel +|\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}|(\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel )\hfill \\ \le {\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+\parallel u_{n+1}-u_n\parallel +|\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}|(\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel )\hfill \\ \le {\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+{\tilde{M}}_1\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|+\parallel x_{n+1}-x_n\parallel \hfill \\ +|\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}|(\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel ).\hfill \end{array}$$

(3.11)

In the meantime, simple calculation shows that

$$y_{n+1}-y_n={\beta }_n(x_{n+1}-x_n)+(1-{\beta }_n)(w_{n+1}-w_n)+({\beta }_{n+1}-{\beta }_n)(x_{n+1}-w_{n+1}).$$

So, it follows from (3.11) that

$$\begin{array}{c}\parallel y_{n+1}-y_n\parallel \hfill \\ \le {\beta }_n\parallel x_{n+1}-x_n\parallel +(1-{\beta }_n)\parallel w_{n+1}-w_n\parallel +|{\beta }_{n+1}-{\beta }_n|\parallel x_{n+1}-w_{n+1}\parallel \hfill \\ \le {\beta }_n\parallel x_{n+1}-x_n\parallel +(1-{\beta }_n)[{\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+{\tilde{M}}_1\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|+\parallel x_{n+1}-x_n\parallel \hfill \\ +|\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}|(\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel )]+|{\beta }_{n+1}-{\beta }_n|\parallel x_{n+1}-w_{n+1}\parallel \hfill \\ \le \parallel x_{n+1}-x_n\parallel +{\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+{\tilde{M}}_1\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|\hfill \\ +\frac{|{\gamma }_{n+1}-{\gamma }_n|(1-{\beta }_n)+{\gamma }_n|{\beta }_{n+1}-{\beta }_n|}{1-{\beta }_{n+1}}(\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel )\hfill \\ +|{\beta }_{n+1}-{\beta }_n|\parallel x_{n+1}-w_{n+1}\parallel \hfill \\ \le \parallel x_{n+1}-x_n\parallel +{\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+{\tilde{M}}_1\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|\hfill \\ +|{\gamma }_{n+1}-{\gamma }_n|\frac{\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel }{1-b}+|{\beta }_{n+1}-{\beta }_n|(\parallel x_{n+1}-w_{n+1}\parallel +\frac{\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel }{1-b})\hfill \\ \le \parallel x_{n+1}-x_n\parallel +{\tilde{M}}_2(\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|\hfill \\ +\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|+|{\gamma }_{n+1}-{\gamma }_n|+|{\beta }_{n+1}-{\beta }_n|),\hfill \end{array}$$

(3.12)

where ${sup}_{n\ge 0}\{\parallel x_{n+1}-w_{n+1}\parallel +\frac{\parallel G{v}_n\parallel +\parallel TG{v}_n\parallel }{1-b}+{\tilde{M}}_0+{\tilde{M}}_1\}\le {\tilde{M}}_2$ for some ${\tilde{M}}_2>0$.

On the other hand, we define $z_n:={\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n$ for all $n\ge 0$. Then it is well known that $x_{n+1}={\lambda }_n\gamma z_n+(I-{\lambda }_n\mu F)y_n$ for all $n\ge 0$. Simple calculations show that

$$\{\begin{array}{l}{z}_{n+1}-z_n=({\alpha }_{n+1}-{\alpha }_n)(V{x}_n-S{x}_n)+{\alpha }_{n+1}(V{x}_{n+1}-V{x}_n)\\ \phantom{z_{n+1}-z_n=}+(1-{\alpha }_{n+1})(S{x}_{n+1}-S{x}_n),\\ x_{n+2}-x_{n+1}=({\lambda }_{n+1}-{\lambda }_n)(\gamma z_n-\mu F{y}_n)+{\lambda }_{n+1}\gamma (z_{n+1}-z_n)\\ \phantom{x_{n+2}-x_{n+1}=}+(I-{\lambda }_{n+1}\mu F)y_{n+1}-(I-{\lambda }_{n+1}\mu F)y_n.\end{array}$$

Since V is a ρ-contraction with coefficient $\rho \in [0,1)$ and S is a nonexpansive mapping, we conclude that

$$\begin{array}{rcl}\parallel z_{n+1}-z_n\parallel & \le & |{\alpha }_{n+1}-{\alpha }_n|\parallel V{x}_n-S{x}_n\parallel +{\alpha }_{n+1}\parallel V{x}_{n+1}-V{x}_n\parallel \\ +(1-{\alpha }_{n+1})\parallel S{x}_{n+1}-S{x}_n\parallel \\ \le & |{\alpha }_{n+1}-{\alpha }_n|\parallel V{x}_n-S{x}_n\parallel +{\alpha }_{n+1}\rho \parallel x_{n+1}-x_n\parallel +(1-{\alpha }_{n+1})\parallel x_{n+1}-x_n\parallel \\ =& (1-{\alpha }_{n+1}(1-\rho ))\parallel x_{n+1}-x_n\parallel +|{\alpha }_{n+1}-{\alpha }_n|\parallel V{x}_n-S{x}_n\parallel ,\end{array}$$

which together with (3.12) and $0\le \gamma <\tau$ implies that

$$\begin{array}{c}\parallel x_{n+2}-x_{n+1}\parallel \hfill \\ \le |{\lambda }_{n+1}-{\lambda }_n|\parallel \gamma z_n-\mu F{y}_n\parallel +{\lambda }_{n+1}\gamma \parallel z_{n+1}-z_n\parallel \hfill \\ +\parallel (I-{\lambda }_{n+1}\mu F)y_{n+1}-(I-{\lambda }_{n+1}\mu F)y_n\parallel \hfill \\ \le |{\lambda }_{n+1}-{\lambda }_n|\parallel \gamma z_n-\mu F{y}_n\parallel +{\lambda }_{n+1}\gamma \parallel z_{n+1}-z_n\parallel +(1-{\lambda }_{n+1}\tau )\parallel y_{n+1}-y_n\parallel \hfill \\ \le |{\lambda }_{n+1}-{\lambda }_n|\parallel \gamma z_n-\mu F{y}_n\parallel +{\lambda }_{n+1}\gamma [(1-{\alpha }_{n+1}(1-\rho ))\parallel x_{n+1}-x_n\parallel \hfill \\ +|{\alpha }_{n+1}-{\alpha }_n|\parallel V{x}_n-S{x}_n\parallel ]+(1-{\lambda }_{n+1}\tau )[\parallel x_{n+1}-x_n\parallel +{\tilde{M}}_2(\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|\hfill \\ +\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|+|{\gamma }_{n+1}-{\gamma }_n|+|{\beta }_{n+1}-{\beta }_n|\left)\right]\hfill \\ \le (1-{\lambda }_{n+1}(\tau -\gamma ))\parallel x_{n+1}-x_n\parallel +|{\lambda }_{n+1}-{\lambda }_n|\parallel \gamma z_n-\mu F{y}_n\parallel +|{\alpha }_{n+1}-{\alpha }_n|\parallel V{x}_n-S{x}_n\parallel \hfill \\ +{\tilde{M}}_2(\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|+|{\gamma }_{n+1}-{\gamma }_n|+|{\beta }_{n+1}-{\beta }_n|)\hfill \\ \le (1-{\lambda }_{n+1}(\tau -\gamma ))\parallel x_{n+1}-x_n\parallel +{\tilde{M}}_3\{\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|\hfill \\ +|{\lambda }_{n+1}-{\lambda }_n|+|{\alpha }_{n+1}-{\alpha }_n|+|{\beta }_{n+1}-{\beta }_n|+|{\gamma }_{n+1}-{\gamma }_n|\},\hfill \end{array}$$

where ${sup}_{n\ge 0}\{\parallel \gamma z_n-\mu F{y}_n\parallel +\parallel V{x}_n-S{x}_n\parallel +{\tilde{M}}_2\}\le {\tilde{M}}_3$ for some ${\tilde{M}}_3>0$. Consequently,

$$\begin{array}{c}\frac{\parallel x_{n+1}-x_n\parallel }{{\alpha }_n}\hfill \\ \le (1-{\lambda }_n(\tau -\gamma ))\frac{\parallel x_n-x_{n-1}\parallel }{{\alpha }_n}+{\tilde{M}}_3\{\sum _{i=1}^N\frac{|{\lambda }_{i,n}-{\lambda }_{i,n-1}|}{{\alpha }_n}+\sum _{k=1}^M\frac{|r_{k,n}-r_{k,n-1}|}{{\alpha }_n}\hfill \\ +\frac{|{\lambda }_n-{\lambda }_{n-1}|}{{\alpha }_n}+\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\alpha }_n}+\frac{|{\beta }_n-{\beta }_{n-1}|}{{\alpha }_n}+\frac{|{\gamma }_n-{\gamma }_{n-1}|}{{\alpha }_n}\}\hfill \\ =(1-{\lambda }_n(\tau -\gamma ))\frac{\parallel x_n-x_{n-1}\parallel }{{\alpha }_{n-1}}+(1-{\lambda }_n(\tau -\gamma ))\parallel x_n-x_{n-1}\parallel (\frac{1}{{\alpha }_n}-\frac{1}{{\alpha }_{n-1}})\hfill \\ +{\tilde{M}}_3\{\sum _{i=1}^N\frac{|{\lambda }_{i,n}-{\lambda }_{i,n-1}|}{{\alpha }_n}+\sum _{k=1}^M\frac{|r_{k,n}-r_{k,n-1}|}{{\alpha }_n}+\frac{|{\lambda }_n-{\lambda }_{n-1}|}{{\alpha }_n}\hfill \\ +\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\alpha }_n}+\frac{|{\beta }_n-{\beta }_{n-1}|}{{\alpha }_n}+\frac{|{\gamma }_n-{\gamma }_{n-1}|}{{\alpha }_n}\}\hfill \\ \le (1-{\lambda }_n(\tau -\gamma ))\frac{\parallel x_n-x_{n-1}\parallel }{{\alpha }_{n-1}}+{\lambda }_n(\tau -\gamma )\cdot \frac{{\tilde{M}}_4}{\tau -\gamma }\{\frac{1}{{\lambda }_n}|\frac{1}{{\alpha }_n}-\frac{1}{{\alpha }_{n-1}}|\hfill \\ +\sum _{i=1}^N\frac{|{\lambda }_{i,n}-{\lambda }_{i,n-1}|}{{\lambda }_n{\alpha }_n}+\sum _{k=1}^M\frac{|r_{k,n}-r_{k,n-1}|}{{\lambda }_n{\alpha }_n}+\frac{1}{{\alpha }_n}|1-\frac{{\lambda }_{n-1}}{{\lambda }_n}|\hfill \\ +\frac{1}{{\lambda }_n}|1-\frac{{\alpha }_{n-1}}{{\alpha }_n}|+\frac{|{\beta }_n-{\beta }_{n-1}|}{{\lambda }_n{\alpha }_n}+\frac{|{\gamma }_n-{\gamma }_{n-1}|}{{\lambda }_n{\alpha }_n}\},\hfill \end{array}$$

(3.13)

where ${sup}_{n\ge 1}\{\parallel x_n-x_{n-1}\parallel +{\tilde{M}}_3\}\le {\tilde{M}}_4$ for some ${\tilde{M}}_4>0$. From conditions (i)-(iv) it follows that ${\sum }_{n=0}^{\mathrm{\infty }}{\lambda }_n(\tau -\gamma )=\mathrm{\infty }$ and

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}\frac{{\tilde{M}}_4}{\tau -\gamma }\left\{\frac{1}{{\lambda }_n}\right|\frac{1}{{\alpha }_n}-\frac{1}{{\alpha }_{n-1}}|+\sum _{i=1}^N\frac{|{\lambda }_{i,n}-{\lambda }_{i,n-1}|}{{\lambda }_n{\alpha }_n}+\sum _{k=1}^M\frac{|r_{k,n}-r_{k,n-1}|}{{\lambda }_n{\alpha }_n}\hfill \\ +\frac{1}{{\alpha }_n}|1-\frac{{\lambda }_{n-1}}{{\lambda }_n}|+\frac{1}{{\lambda }_n}|1-\frac{{\alpha }_{n-1}}{{\alpha }_n}|+\frac{|{\beta }_n-{\beta }_{n-1}|}{{\lambda }_n{\alpha }_n}+\frac{|{\gamma }_n-{\gamma }_{n-1}|}{{\lambda }_n{\alpha }_n}\}=0.\hfill \end{array}$$

(3.14)

Thus, utilizing Lemma 2.12, we immediately conclude that

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

So, from ${\alpha }_n\to 0$ it follows that

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

Step 3. We prove that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u_n\parallel =0$, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-v_n\parallel =0$, ${lim}_{n\to \mathrm{\infty }}\parallel v_n-G{v}_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel v_n-T{v}_n\parallel =0$.

Indeed, utilizing Lemmas 2.8 and 2.9(b), from (3.1), (3.4)-(3.5), and $0\le \gamma <\tau$ we deduce that

$$\begin{array}{c}{\parallel y_n-p\parallel }^2\hfill \\ ={\parallel {\beta }_n{x}_n+{\gamma }_{n}G{v}_n+{\delta }_{n}TG{v}_n-p\parallel }^2\hfill \\ ={\parallel {\beta }_n(x_n-p)+(1-{\beta }_n)(\frac{{\gamma }_{n}G{v}_n+{\delta }_{n}TG{v}_n}{1-{\beta }_n}-p)\parallel }^2\hfill \\ ={\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel \frac{{\gamma }_{n}G{v}_n+{\delta }_{n}TG{v}_n}{1-{\beta }_n}-p\parallel }^2\hfill \\ -{\beta }_n(1-{\beta }_n){\parallel \frac{{\gamma }_{n}G{v}_n+{\delta }_{n}TG{v}_n}{1-{\beta }_n}-x_n\parallel }^2\hfill \\ ={\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel \frac{{\gamma }_n(G{v}_n-p)+{\delta }_n(TG{v}_n-p)}{1-{\beta }_n}\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel \frac{y_n-x_n}{1-{\beta }_n}\parallel }^2\hfill \\ \le {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)\frac{{({\gamma }_n+{\delta }_n)}^2{\parallel G{v}_n-p\parallel }^2}{{(1-{\beta }_n)}^2}-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ ={\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel G{v}_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ \le {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ \le {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel x_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ ={\parallel x_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2,\hfill \end{array}$$

(3.15)

and hence

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ ={\parallel {\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)+(I-{\lambda }_n\mu F)y_n-p\parallel }^2\hfill \\ ={\parallel {\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)-{\lambda }_n\mu Fp+(I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel }^2\hfill \\ ={\parallel {\lambda }_n[{\alpha }_n(\gamma V{x}_n-\mu Fp)+(1-{\alpha }_n)(\gamma S{x}_n-\mu Fp)]+(I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel }^2\hfill \\ =\parallel {\lambda }_n[{\alpha }_n(\gamma V{x}_n-\gamma Vp)+(1-{\alpha }_n)(\gamma S{x}_n-\gamma Sp)]+(I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\hfill \\ {+{\lambda }_n[{\alpha }_n(\gamma Vp-\mu Fp)+(1-{\alpha }_n)(\gamma Sp-\mu Fp)]\parallel }^2\hfill \\ \le {\parallel {\lambda }_n[{\alpha }_n(\gamma V{x}_n-\gamma Vp)+(1-{\alpha }_n)(\gamma S{x}_n-\gamma Sp)]+(I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel }^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ \le {[{\lambda }_n\parallel {\alpha }_n(\gamma V{x}_n-\gamma Vp)+(1-{\alpha }_n)(\gamma S{x}_n-\gamma Sp)\parallel +\parallel (I-{\lambda }_n\mu F)y_n-(I-{\lambda }_n\mu F)p\parallel ]}^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ \le {[{\lambda }_n({\alpha }_n\gamma \rho \parallel x_n-p\parallel +(1-{\alpha }_n)\gamma \parallel x_n-p\parallel )+(1-{\lambda }_n\tau )\parallel y_n-p\parallel ]}^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2(1-{\alpha }_n){\lambda }_n〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ ={[{\lambda }_n(1-{\alpha }_n(1-\rho ))\gamma \parallel x_n-p\parallel +(1-{\lambda }_n\tau )\parallel y_n-p\parallel ]}^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ \le {[{\lambda }_n\gamma \parallel x_n-p\parallel +(1-{\lambda }_n\tau )\parallel y_n-p\parallel ]}^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ ={[{\lambda }_n\tau \cdot \frac{\gamma }{\tau }\parallel x_n-p\parallel +(1-{\lambda }_n\tau )\parallel y_n-p\parallel ]}^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ \le {\lambda }_n\frac{{\gamma }^2}{\tau }\parallel x_n-p\parallel +(1-{\lambda }_n\tau ){\parallel y_n-p\parallel }^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ \le {\lambda }_n\frac{{\gamma }^2}{\tau }\parallel x_n-p\parallel +(1-{\lambda }_n\tau )[{\parallel x_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2]\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ =(1-{\lambda }_n\frac{{\tau }^2-{\gamma }^2}{\tau }){\parallel x_n-p\parallel }^2-\frac{{\beta }_n(1-{\lambda }_n\tau )}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ \le {\parallel x_n-p\parallel }^2-\frac{{\beta }_n(1-{\lambda }_n\tau )}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ +2{\lambda }_n{\alpha }_n\parallel \gamma Vp-\mu Fp\parallel \parallel x_{n+1}-p\parallel +2{\lambda }_n\parallel \gamma Sp-\mu Fp\parallel \parallel x_{n+1}-p\parallel ,\hfill \end{array}$$

(3.16)

which together with $\{{\beta }_n\}\subset [a,b]\subset (0,1)$, immediately yields

$$\begin{array}{rcl}\frac{a(1-{\lambda }_n\tau )}{1-a}{\parallel y_n-x_n\parallel }^2& \le & \frac{{\beta }_n(1-{\lambda }_n\tau )}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2+2{\lambda }_n{\alpha }_n\parallel \gamma Vp-\mu Fp\parallel \parallel x_{n+1}-p\parallel \\ +2{\lambda }_n\parallel \gamma Sp-\mu Fp\parallel \parallel x_{n+1}-p\parallel \\ \le & \parallel x_n-x_{n+1}\parallel (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\\ +2{\lambda }_n{\alpha }_n\parallel \gamma Vp-\mu Fp\parallel \parallel x_{n+1}-p\parallel \\ +2{\lambda }_n\parallel \gamma Sp-\mu Fp\parallel \parallel x_{n+1}-p\parallel .\end{array}$$

Since ${\lambda }_n\to 0$, ${\alpha }_n\to 0$, $\parallel x_{n+1}-x_n\parallel \to 0$, and $\{x_n\}$ is bounded, we have

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

(3.17)

Observe that

$$\begin{array}{rl}{\parallel {\Delta }_n^k{x}_n-p\parallel }^2& ={\parallel T_{r_{k,n}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n}{A}_k){\Delta }_n^{k-1}{x}_n-T_{r_{k,n}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n}{A}_k)p\parallel }^2\\ \le {\parallel (I-r_{k,n}{A}_k){\Delta }_n^{k-1}{x}_n-(I-r_{k,n}{A}_k)p\parallel }^2\\ \le {\parallel {\Delta }_n^{k-1}{x}_n-p\parallel }^2+r_{k,n}(r_{k,n}-2{\mu }_k){\parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel }^2\\ \le {\parallel x_n-p\parallel }^2+r_{k,n}(r_{k,n}-2{\mu }_k){\parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel }^2\end{array}$$

(3.18)

and

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

(3.19)

for $i\in \{1,2,\dots ,N\}$ and $k\in \{1,2,\dots ,M\}$. Combining (3.15), (3.18), and (3.19), we get

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel {\Lambda }_n^i{u}_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel u_n-p\parallel }^2+{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2]\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel {\Delta }_n^k{x}_n-p\parallel }^2+{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2]\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel x_n-p\parallel }^2+r_{k,n}(r_{k,n}-2{\mu }_k){\parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel }^2\\ +{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2]\\ =& {\parallel x_n-p\parallel }^2+(1-{\beta }_n)[r_{k,n}(r_{k,n}-2{\mu }_k){\parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel }^2\\ +{\lambda }_{i,n}({\lambda }_{i,n}-2{\eta }_i){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2],\end{array}$$

which immediately leads to

$$\begin{array}{c}(1-{\beta }_n)[r_{k,n}(2{\mu }_k-r_{k,n}){\parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel }^2+{\lambda }_{i,n}(2{\eta }_i-{\lambda }_{i,n}){\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel }^2]\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2\hfill \\ \le \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel ).\hfill \end{array}$$

Since $\parallel x_n-y_n\parallel \to 0$, $\{{\beta }_n\}\subset [a,b]\subset (0,1)$, $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$, $\{r_{k,n}\}\subset [c_k,d_k]\subset (0,2{\mu }_k)$, $i\in \{1,2,\dots ,N\}$, $k\in \{1,2,\dots ,M\}$, and $\{x_n\}$, $\{y_n\}$ are bounded sequences, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel =0\text{and}\underset{n\to \mathrm{\infty }}{lim}\parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel =0$$

(3.20)

for all $k\in \{1,2,\dots ,M\}$ and $i\in \{1,2,\dots ,N\}$.

Furthermore, by Proposition 1.1(ii) and Lemma 2.9(a), we have

$$\begin{array}{c}{\parallel {\Delta }_n^k{x}_n-p\parallel }^2\hfill \\ ={\parallel T_{r_{k,n}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n}{A}_k){\Delta }_n^{k-1}{x}_n-T_{r_{k,n}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n}{A}_k)p\parallel }^2\hfill \\ \le 〈(I-r_{k,n}{A}_k){\Delta }_n^{k-1}{x}_n-(I-r_{k,n}{A}_k)p,{\Delta }_n^k{x}_n-p〉\hfill \\ =\frac{1}{2}({\parallel (I-r_{k,n}{A}_k){\Delta }_n^{k-1}{x}_n-(I-r_{k,n}{A}_k)p\parallel }^2+{\parallel {\Delta }_n^k{x}_n-p\parallel }^2\hfill \\ -{\parallel (I-r_{k,n}{A}_k){\Delta }_n^{k-1}{x}_n-(I-r_{k,n}{A}_k)p-({\Delta }_n^k{x}_n-p)\parallel }^2)\hfill \\ \le \frac{1}{2}({\parallel {\Delta }_n^{k-1}{x}_n-p\parallel }^2+{\parallel {\Delta }_n^k{x}_n-p\parallel }^2-{\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n-r_{k,n}(A_k{\Delta }_n^{k-1}{x}_n-A_{k}p)\parallel }^2),\hfill \end{array}$$

which implies that

$$\begin{array}{c}{\parallel {\Delta }_n^k{x}_n-p\parallel }^2\hfill \\ \le {\parallel {\Delta }_n^{k-1}{x}_n-p\parallel }^2-{\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n-r_{k,n}(A_k{\Delta }_n^{k-1}{x}_n-A_{k}p)\parallel }^2\hfill \\ ={\parallel {\Delta }_n^{k-1}{x}_n-p\parallel }^2-{\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel }^2-r_{k,n}^2{\parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel }^2\hfill \\ +2r_{k,n}〈{\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n,A_k{\Delta }_n^{k-1}{x}_n-A_{k}p〉\hfill \\ \le {\parallel {\Delta }_n^{k-1}{x}_n-p\parallel }^2-{\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel }^2+2r_{k,n}\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel \parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel \hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel }^2+2r_{k,n}\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel \parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel .\hfill \end{array}$$

(3.21)

By Lemma 2.9(a) and Lemma 2.3, we obtain

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

which immediately leads to

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

(3.22)

Combining (3.15) and (3.22), we conclude

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel {\Lambda }_n^i{u}_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel x_n-p\parallel }^2-{\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel }^2\\ +2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel ]\\ \le & {\parallel x_n-p\parallel }^2-(1-{\beta }_n){\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel }^2\\ +2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel ,\end{array}$$

which yields

$$\begin{array}{c}(1-{\beta }_n){\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel }^2\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2+2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel \hfill \\ \le \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel )+2{\lambda }_{i,n}\parallel {\Lambda }_n^{i-1}{u}_n-{\Lambda }_n^i{u}_n\parallel \parallel B_i{\Lambda }_n^{i-1}{u}_n-B_{i}p\parallel .\hfill \end{array}$$

Since $\{{\beta }_n\}\subset [a,b]\subset (0,1)$, $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$, $i=1,2,\dots ,N$, and $\{u_n\}$, $\{x_n\}$, and $\{y_n\}$ are bounded sequences, we deduce from (3.20) and $\parallel x_n-y_n\parallel \to 0$ that

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

(3.23)

Also, combining (3.3), (3.15), and (3.21), we deduce

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel v_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel u_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel {\Delta }_n^k{x}_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel x_n-p\parallel }^2-{\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel }^2\\ +2r_{k,n}\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel \parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel ]\\ \le & {\parallel x_n-p\parallel }^2-(1-{\beta }_n){\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel }^2\\ +2r_{k,n}\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel \parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel ,\end{array}$$

which yields

$$\begin{array}{c}(1-{\beta }_n){\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel }^2\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2+2r_{k,n}\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel \parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel \hfill \\ \le \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel )+2r_{k,n}\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel \parallel A_k{\Delta }_n^{k-1}{x}_n-A_{k}p\parallel .\hfill \end{array}$$

Since $\{{\beta }_n\}\subset [a,b]\subset (0,1)$, $\{r_{k,n}\}\subset [c_k,d_k]\subset (0,2{\mu }_k)$ for $k=1,2,\dots ,M$, and $\{x_n\}$, $\{y_n\}$ are bounded sequences, we deduce from (3.20) and $\parallel x_n-y_n\parallel \to 0$ that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel {\Delta }_n^{k-1}{x}_n-{\Delta }_n^k{x}_n\parallel =0,\mathrm{\forall }k\in \{1,2,\dots ,M\}.$$

(3.24)

Hence from (3.23) and (3.24), we get

$$\begin{array}{rl}\parallel x_n-u_n\parallel & =\parallel {\Delta }_n^0{x}_n-{\Delta }_n^M{x}_n\parallel \\ \le \parallel {\Delta }_n^0{x}_n-{\Delta }_n^1{x}_n\parallel +\parallel {\Delta }_n^1{x}_n-{\Delta }_n^2{x}_n\parallel +\cdots +\parallel {\Delta }_n^{M-1}{x}_n-{\Delta }_n^M{x}_n\parallel \\ \to 0\text{as }n\to \mathrm{\infty }\end{array}$$

(3.25)

and

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

(3.26)

respectively. Thus, from (3.25) and (3.26), we obtain

$$\parallel x_n-v_n\parallel \le \parallel x_n-u_n\parallel +\parallel u_n-v_n\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

(3.27)

On the other hand, for simplicity, we write $\tilde{p}=P_C(I-{\nu }_2{F}_2)p$, ${\tilde{v}}_n=P_C(I-{\nu }_2{F}_2)v_n$, and $k_n=G{v}_n=P_C(I-{\nu }_1{F}_1){\tilde{v}}_n$ for all $n\ge 1$. Then

$$p=Gp=P_C(I-{\nu }_1{F}_1)\tilde{p}=P_C(I-{\nu }_1{F}_1)P_C(I-{\nu }_2{F}_2)p.$$

We now show that ${lim}_{n\to \mathrm{\infty }}\parallel G{v}_n-v_n\parallel =0$, i.e., ${lim}_{n\to \mathrm{\infty }}\parallel k_n-v_n\parallel =0$. As a matter of fact, for $p\in \Omega$, it follows from (3.4), (3.5), and (3.15) that

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel G{v}_n-p\parallel }^2-\frac{{\beta }_n}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel G{v}_n-p\parallel }^2\\ =& {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel k_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2+{\nu }_1({\nu }_1-2{\zeta }_1){\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel }^2]\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel v_n-p\parallel }^2+{\nu }_2({\nu }_2-2{\zeta }_2){\parallel F_2{v}_n-F_{2}p\parallel }^2\\ +{\nu }_1({\nu }_1-2{\zeta }_1){\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel }^2]\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel x_n-p\parallel }^2+{\nu }_2({\nu }_2-2{\zeta }_2){\parallel F_2{v}_n-F_{2}p\parallel }^2\\ +{\nu }_1({\nu }_1-2{\zeta }_1){\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel }^2]\\ =& {\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\nu }_2({\nu }_2-2{\zeta }_2){\parallel F_2{v}_n-F_{2}p\parallel }^2\\ +{\nu }_1({\nu }_1-2{\zeta }_1){\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel }^2],\end{array}$$

(3.28)

which immediately yields

$$\begin{array}{r}(1-{\beta }_n)[{\nu }_2(2{\zeta }_2-{\nu }_2){\parallel F_2{v}_n-F_{2}p\parallel }^2+{\nu }_1(2{\zeta }_1-{\nu }_1){\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel }^2]\\ \le {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2\\ \le \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel ).\end{array}$$

Since $\parallel x_n-y_n\parallel \to 0$, $\{{\beta }_n\}\subset [a,b]\subset (0,1)$, ${\nu }_j\in (0,2{\zeta }_j)$, $j=1,2$, and $\{x_n\}$, $\{y_n\}$ are bounded sequences, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel F_2{v}_n-F_{2}p\parallel =0\text{and}\underset{n\to \mathrm{\infty }}{lim}\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel =0.$$

(3.29)

Also, in terms of the firm nonexpansivity of $P_C$ and the ${\zeta }_j$-inverse-strong monotonicity of $F_j$ for $j=1,2$, we obtain from ${\nu }_j\in (0,2{\zeta }_j)$, $j=1,2$,

$$\begin{array}{rcl}{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2& =& {\parallel P_C(I-{\nu }_2{F}_2)v_n-P_C(I-{\nu }_2{F}_2)p\parallel }^2\\ \le & 〈(I-{\nu }_2{F}_2)v_n-(I-{\nu }_2{F}_2)p,{\tilde{v}}_n-\tilde{p}〉\\ =& \frac{1}{2}[{\parallel (I-{\nu }_2{F}_2)v_n-(I-{\nu }_2{F}_2)p\parallel }^2+{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2\\ -{\parallel (I-{\nu }_2{F}_2)v_n-(I-{\nu }_2{F}_2)p-({\tilde{v}}_n-\tilde{p})\parallel }^2]\\ \le & \frac{1}{2}[{\parallel v_n-p\parallel }^2+{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2-{\parallel (v_n-{\tilde{v}}_n)-{\nu }_2(F_2{v}_n-F_{2}p)-(p-\tilde{p})\parallel }^2]\\ =& \frac{1}{2}[{\parallel v_n-p\parallel }^2+{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2-{\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel }^2\\ +2{\nu }_2〈(v_n-{\tilde{v}}_n)-(p-\tilde{p}),F_2{v}_n-F_{2}p〉-{\nu }_2^2{\parallel F_2{v}_n-F_{2}p\parallel }^2]\end{array}$$

and

$$\begin{array}{rcl}{\parallel k_n-p\parallel }^2& =& {\parallel P_C(I-{\nu }_1{F}_1){\tilde{v}}_n-P_C(I-{\nu }_1{F}_1)\tilde{p}\parallel }^2\\ \le & 〈(I-{\nu }_1{F}_1){\tilde{v}}_n-(I-{\nu }_1{F}_1)\tilde{p},k_n-p〉\\ =& \frac{1}{2}[{\parallel (I-{\nu }_1{F}_1){\tilde{v}}_n-(I-{\nu }_1{F}_1)\tilde{p}\parallel }^2+{\parallel k_n-p\parallel }^2\\ -{\parallel (I-{\nu }_1{F}_1){\tilde{v}}_n-(I-{\nu }_1{F}_1)\tilde{p}-(k_n-p)\parallel }^2]\\ \le & \frac{1}{2}[{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2+{\parallel k_n-p\parallel }^2-{\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel }^2\\ +2{\nu }_1〈F_1{\tilde{v}}_n-F_1\tilde{p},({\tilde{v}}_n-k_n)+(p-\tilde{p})〉-{\nu }_1^2{\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel }^2]\\ \le & \frac{1}{2}[{\parallel v_n-p\parallel }^2+{\parallel w_n-p\parallel }^2-{\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel }^2\\ +2{\nu }_1〈F_1{\tilde{v}}_n-F_1\tilde{p},({\tilde{v}}_n-k_n)+(p-\tilde{p})〉].\end{array}$$

Thus, we have

$$\begin{array}{rcl}{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2& \le & {\parallel v_n-p\parallel }^2-{\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel }^2\\ +2{\nu }_2〈(v_n-{\tilde{v}}_n)-(p-\tilde{p}),F_2{v}_n-F_{2}p〉-{\nu }_2^2{\parallel F_2{v}_n-F_{2}p\parallel }^2\end{array}$$

(3.30)

and

$$\begin{array}{rcl}{\parallel k_n-p\parallel }^2& \le & {\parallel v_n-p\parallel }^2-{\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel }^2\\ +2{\nu }_1\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel \parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel .\end{array}$$

(3.31)

Consequently, from (3.4), (3.28), and (3.30), it follows that

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2+{\nu }_1({\nu }_1-2{\zeta }_1){\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel }^2]\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel {\tilde{v}}_n-\tilde{p}\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel v_n-p\parallel }^2-{\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel }^2\\ +2{\nu }_2〈(v_n-{\tilde{v}}_n)-(p-\tilde{p}),F_2{v}_n-F_{2}p〉-{\nu }_2^2{\parallel F_2{v}_n-F_{2}p\parallel }^2]\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel x_n-p\parallel }^2-{\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel }^2\\ +2{\nu }_2\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel \parallel F_2{v}_n-F_{2}p\parallel ]\\ \le & {\parallel x_n-p\parallel }^2-(1-{\beta }_n){\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel }^2\\ +2{\nu }_2\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel \parallel F_2{v}_n-F_{2}p\parallel ,\end{array}$$

which hence leads to

$$\begin{array}{c}(1-{\beta }_n){\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel }^2\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2+2{\nu }_2\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel \parallel F_2{v}_n-F_{2}p\parallel \hfill \\ \le \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel )+2{\nu }_2\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel \parallel F_2{v}_n-F_{2}p\parallel .\hfill \end{array}$$

Since $\parallel x_n-y_n\parallel \to 0$, $\{{\beta }_n\}\subset [a,b]\subset (0,1)$, ${\nu }_2\in (0,2{\zeta }_2)$, and $\{x_n\}$, $\{y_n\}$, $\{v_n\}$, $\{{\tilde{v}}_n\}$ are bounded sequences, we obtain from (3.29)

$$\underset{n\to \mathrm{\infty }}{lim}\parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel =0.$$

(3.32)

Furthermore, from (3.4), (3.28), and (3.31), it follows that

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n){\parallel k_n-p\parallel }^2\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel v_n-p\parallel }^2-{\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel }^2\\ +2{\nu }_1\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel \parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel ]\\ \le & {\beta }_n{\parallel x_n-p\parallel }^2+(1-{\beta }_n)[{\parallel x_n-p\parallel }^2-{\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel }^2\\ +2{\nu }_1\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel \parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel ]\\ =& {\parallel x_n-p\parallel }^2-(1-{\beta }_n){\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel }^2\\ +2{\nu }_1\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel \parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel ,\end{array}$$

which hence yields

$$\begin{array}{c}(1-{\beta }_n){\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel }^2\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2+2{\nu }_1\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel \parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel \hfill \\ \le \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel )+2{\nu }_1\parallel F_1{\tilde{v}}_n-F_1\tilde{p}\parallel \parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel .\hfill \end{array}$$

Since $\parallel x_n-y_n\parallel \to 0$, $\{{\beta }_n\}\subset [a,b]\subset (0,1)$, ${\nu }_1\in (0,2{\zeta }_1)$, and $\{x_n\}$, $\{y_n\}$, $\{k_n\}$, $\{{\tilde{v}}_n\}$ are bounded sequences, we obtain from (3.29)

$$\underset{n\to \mathrm{\infty }}{lim}\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel =0.$$

(3.33)

Note that

$$\parallel v_n-k_n\parallel \le \parallel (v_n-{\tilde{v}}_n)-(p-\tilde{p})\parallel +\parallel ({\tilde{v}}_n-k_n)+(p-\tilde{p})\parallel .$$

Hence from (3.32) and (3.33), we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel v_n-G{v}_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel v_n-k_n\parallel =0.$$

(3.34)

Also, observe that

$$y_n-x_n={\gamma }_n(G{v}_n-x_n)+{\delta }_n(TG{v}_n-x_n),\mathrm{\forall }n\ge 0.$$

Hence we find that

$$\begin{array}{rl}{\delta }_n\parallel TG{v}_n-v_n\parallel & \le {\delta }_n\parallel TG{v}_n-x_n\parallel +{\delta }_n\parallel x_n-v_n\parallel \\ =\parallel y_n-x_n-{\gamma }_n(G{v}_n-x_n)\parallel +{\delta }_n\parallel x_n-v_n\parallel \\ \le \parallel y_n-x_n\parallel +{\gamma }_n\parallel G{v}_n-x_n\parallel +{\delta }_n\parallel x_n-v_n\parallel \\ \le \parallel y_n-x_n\parallel +{\gamma }_n\parallel G{v}_n-v_n\parallel +{\gamma }_n\parallel v_n-x_n\parallel +{\delta }_n\parallel x_n-v_n\parallel \\ =\parallel y_n-x_n\parallel +{\gamma }_n\parallel G{v}_n-v_n\parallel +({\gamma }_n+{\delta }_n)\parallel x_n-v_n\parallel \\ \le \parallel y_n-x_n\parallel +\parallel G{v}_n-v_n\parallel +\parallel x_n-v_n\parallel .\end{array}$$

So, from ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n>0$, (3.17), (3.27), and (3.34), it follows that

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

(3.35)

In addition, noticing that

$$\begin{array}{rl}\parallel T{v}_n-v_n\parallel & \le \parallel T{v}_n-TG{v}_n\parallel +\parallel TG{v}_n-v_n\parallel \\ \le \parallel v_n-G{v}_n\parallel +\parallel TG{v}_n-v_n\parallel ,\end{array}$$

we know from (3.34) and (3.35) that

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

(3.36)

Step 4. We prove that ${\omega }_w(x_n)\subset \Omega$.

Indeed, since H is reflexive and $\{x_n\}$ is bounded, there exists at least a weak convergence subsequence of $\{x_n\}$. Hence it is well known that ${\omega }_w(x_n)\ne \mathrm{\varnothing }$. Now, take an arbitrary $w\in {\omega }_w(x_n)$. Then there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\rightharpoonup w$. From (3.23)-(3.25), and (3.27), we have $u_{n_i}\rightharpoonup w$, $v_{n_i}\rightharpoonup w$, ${\Lambda }_{n_i}^m{u}_{n_i}\rightharpoonup w$, and ${\Delta }_{n_i}^k{x}_{n_i}\rightharpoonup w$, where $m\in \{1,2,\dots ,N\}$ and $k\in \{1,2,\dots ,M\}$. Utilizing Lemma 2.1(ii), we deduce from $v_{n_i}\rightharpoonup w$ and (3.36) that $w\in Fix(T)$. In the meantime, utilizing Lemma 2.10, we obtain from $v_{n_i}\rightharpoonup w$ and (3.34) $w\in GSVI(G)$. 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 2.6 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

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

In particular,

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

Since $\parallel {\Lambda }_n^m{u}_n-{\Lambda }_n^{m-1}{u}_n\parallel \to 0$ (due to (3.23)) 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 $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$ 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 prove that $w\in {\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,A_k)$. Since ${\Delta }_n^k{x}_n=T_{r_{k,n}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n}{A}_k){\Delta }_n^{k-1}{x}_n$, $n\ge 1$, $k\in \{1,2,\dots ,M\}$, we have

$$\begin{array}{r}{\Theta }_k({\Delta }_n^k{x}_n,y)+{\phi }_k(y)-{\phi }_k\left({\Delta }_n^k{x}_n\right)+〈A_k{\Delta }_n^{k-1}{x}_n,y-{\Delta }_n^k{x}_n〉\\ +\frac{1}{r_{k,n}}〈y-{\Delta }_n^k{x}_n,{\Delta }_n^k{x}_n-{\Delta }_n^{k-1}{x}_n〉\ge 0.\end{array}$$

By (A2), we have

$${\phi }_k(y)-{\phi }_k\left({\Delta }_n^k{x}_n\right)+〈A_k{\Delta }_n^{k-1}{x}_n,y-{\Delta }_n^k{x}_n〉+\frac{1}{r_{k,n}}〈y-{\Delta }_n^k{x}_n,{\Delta }_n^k{x}_n-{\Delta }_n^{k-1}{x}_n〉\ge {\Theta }_k(y,{\Delta }_n^k{x}_n).$$

Let $z_t=ty+(1-t)w$ for all $t\in (0,1]$ and $y\in C$. This implies that $z_t\in C$. Then we have

$$\begin{array}{c}〈z_t-{\Delta }_n^k{x}_n,A_k{z}_t〉\hfill \\ \ge {\phi }_k\left({\Delta }_n^k{x}_n\right)-{\phi }_k(z_t)+〈z_t-{\Delta }_n^k{x}_n,A_k{z}_t〉-〈z_t-{\Delta }_n^k{x}_n,A_k{\Delta }_n^{k-1}{x}_n〉\hfill \\ -〈z_t-{\Delta }_n^k{x}_n,\frac{{\Delta }_n^k{x}_n-{\Delta }_n^{k-1}{x}_n}{r_{k,n}}〉+{\Theta }_k(z_t,{\Delta }_n^k{x}_n)\hfill \\ ={\phi }_k\left({\Delta }_n^k{x}_n\right)-{\phi }_k(z_t)+〈z_t-{\Delta }_n^k{x}_n,A_k{z}_t-A_k{\Delta }_n^k{x}_n〉+〈z_t-{\Delta }_n^k{x}_n,A_k{\Delta }_n^k{x}_n-A_k{\Delta }_n^{k-1}{x}_n〉\hfill \\ -〈z_t-{\Delta }_n^k{x}_n,\frac{{\Delta }_n^k{x}_n-{\Delta }_n^{k-1}{x}_n}{r_{k,n}}〉+{\Theta }_k(z_t,{\Delta }_n^k{x}_n).\hfill \end{array}$$

(3.37)

By (3.24), we have $\parallel A_k{\Delta }_n^k{x}_n-A_k{\Delta }_n^{k-1}{x}_n\parallel \to 0$ as $n\to \mathrm{\infty }$. Furthermore, by the monotonicity of $A_k$, we obtain $〈z_t-{\Delta }_n^k{x}_n,A_k{z}_t-A_k{\Delta }_n^k{x}_n〉\ge 0$. Then by (A4) we obtain

$$〈z_t-w,A_k{z}_t〉\ge {\phi }_k(w)-{\phi }_k(z_t)+{\Theta }_k(z_t,w).$$

(3.38)

Utilizing (A1), (A4), and (3.26), we obtain

$$\begin{array}{rl}0& ={\Theta }_k(z_t,z_t)+{\phi }_k(z_t)-{\phi }_k(z_t)\\ \le t{\Theta }_k(z_t,y)+(1-t){\Theta }_k(z_t,w)+t{\phi }_k(y)+(1-t){\phi }_k(w)-{\phi }_k(z_t)\\ \le t[{\Theta }_k(z_t,y)+{\phi }_k(y)-{\phi }_k(z_t)]+(1-t)〈z_t-w,A_k{z}_t〉\\ =t[{\Theta }_k(z_t,y)+{\phi }_k(y)-{\phi }_k(z_t)]+(1-t)t〈y-w,A_k{z}_t〉,\end{array}$$

and hence

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

Letting $t\to 0$, we have, for each $y\in C$,

$$0\le {\Theta }_k(w,y)+{\phi }_k(y)-{\phi }_k(w)+〈y-w,A_{k}w〉.$$

This implies that $w\in GMEP({\Theta }_k,{\phi }_k,A_k)$, and hence, $w\in {\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,A_k)$. Thus, $w\in \Omega ={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)\cap {\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,A_k)\cap {\bigcap }_{m=1}^N\mathrm{I}(B_m,R_m)$. Consequently, $w\in {\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,A_k)\cap {\bigcap }_{m=1}^N\mathrm{I}(B_m,R_m)\cap GSVI(G)\cap Fix(T)=:\Omega$. This shows that ${\omega }_w(x_n)\subset \Omega$.

Step 5. We prove that ${\omega }_w(x_n)\subset \Xi$.

Indeed, take an arbitrary $w\in {\omega }_w(x_n)$. Then there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\rightharpoonup w$. Utilizing (3.16), we obtain, for all $p\in \Omega$,

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ \le (1-{\lambda }_n\frac{{\tau }^2-{\gamma }^2}{\tau }){\parallel x_n-p\parallel }^2-\frac{{\beta }_n(1-{\lambda }_n\tau )}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma Vp-\mu Fp),x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉\hfill \\ \le {\parallel x_n-p\parallel }^2+2{\lambda }_n{\alpha }_n〈(\gamma V-\mu F)p,x_{n+1}-p〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma Sp-\mu Fp),x_{n+1}-p〉,\hfill \end{array}$$

which implies that

$$\begin{array}{c}〈(\mu F-\gamma S)p,x_n-p〉\hfill \\ \le 〈(\mu F-\gamma S)p,x_n-x_{n+1}〉+〈(\mu F-\gamma S)p,x_{n+1}-p〉\hfill \\ \le \parallel (\mu F-\gamma S)p\parallel \parallel x_n-x_{n+1}\parallel +\frac{{\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2}{2{\lambda }_n(1-{\alpha }_n)}\hfill \\ +\frac{{\alpha }_n}{1-{\alpha }_n}〈(\gamma V-\mu F)p,x_{n+1}-p〉\hfill \\ \le \parallel (\mu F-\gamma S)p\parallel \parallel x_n-x_{n+1}\parallel +\frac{\parallel x_n-x_{n+1}\parallel (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )}{2{\lambda }_n(1-{\alpha }_n)}\hfill \\ +\frac{{\alpha }_n}{1-{\alpha }_n}\parallel (\gamma V-\mu F)p\parallel \parallel x_{n+1}-p\parallel .\hfill \end{array}$$

(3.39)

Since ${\alpha }_n\to 0$, $\parallel x_n-x_{n+1}\parallel \to 0$ and

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\parallel x_n-x_{n+1}\parallel }{{\lambda }_n}=\underset{n\to \mathrm{\infty }}{lim}\frac{\parallel x_n-x_{n+1}\parallel }{{\alpha }_n}\cdot \frac{{\alpha }_n}{{\lambda }_n}=0,$$

from (3.39), we conclude that

$$\begin{array}{rl}〈(\mu F-\gamma S)p,w-p〉& =\underset{i\to \mathrm{\infty }}{lim}〈(\mu F-\gamma S)p,x_{n_i}-p〉\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\mu F-\gamma S)p,x_n-p〉\\ \le 0,\mathrm{\forall }p\in \Omega ,\end{array}$$

that is,

$$〈(\mu F-\gamma S)p,w-p〉\le 0,\mathrm{\forall }p\in \Omega .$$

(3.40)

Since $\mu F-\gamma S$ is $(\mu \eta -\gamma )$-strongly monotone and $(\mu \kappa +\gamma )$-Lipschitz continuous, by Minty’s lemma [29] we know that (3.40) is equivalent to the VIP

$$〈(\mu F-\gamma S)w,p-w〉\ge 0,\mathrm{\forall }p\in \Omega .$$

(3.41)

This shows that $w\in VI(\Omega ,\mu F-\gamma S)$. Taking into account $\{x^{\ast }\}=VI(\Omega ,\mu F-\gamma S)$, we know that $w=x^{\ast }$. Thus, ${\omega }_w(x_n)=\{x^{\ast }\}$; that is, $x_n\rightharpoonup x^{\ast }$.

Next we prove that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x^{\ast }\parallel =0$. As a matter of fact, utilizing (3.16) with $p=x^{\ast }$, we get

$$\begin{array}{c}{\parallel x_{n+1}-x^{\ast }\parallel }^2\hfill \\ \le (1-{\lambda }_n\frac{{\tau }^2-{\gamma }^2}{\tau }){\parallel x_n-x^{\ast }\parallel }^2-\frac{{\beta }_n(1-{\lambda }_n\tau )}{1-{\beta }_n}{\parallel y_n-x_n\parallel }^2\hfill \\ +2{\lambda }_n{\alpha }_n〈(\gamma V{x}^{\ast }-\mu F{x}^{\ast }),x_{n+1}-x^{\ast }〉+2{\lambda }_n(1-{\alpha }_n)〈(\gamma S{x}^{\ast }-\mu F{x}^{\ast }),x_{n+1}-x^{\ast }〉\hfill \\ \le (1-{\lambda }_n\frac{{\tau }^2-{\gamma }^2}{\tau }){\parallel x_n-x^{\ast }\parallel }^2+2{\lambda }_n{\alpha }_n\parallel (\gamma V-\mu F)x^{\ast }\parallel \parallel x_{n+1}-x^{\ast }\parallel \hfill \\ +2{\lambda }_n(1-{\alpha }_n)〈(\gamma S{x}^{\ast }-\mu F{x}^{\ast }),x_{n+1}-x^{\ast }〉\hfill \\ =(1-{\lambda }_n\frac{{\tau }^2-{\gamma }^2}{\tau }){\parallel x_n-x^{\ast }\parallel }^2+{\lambda }_n\frac{{\tau }^2-{\gamma }^2}{\tau }\cdot \frac{2\tau }{{\tau }^2-{\gamma }^2}[{\alpha }_n\parallel (\gamma V-\mu F)x^{\ast }\parallel \parallel x_{n+1}-x^{\ast }\parallel \hfill \\ +(1-{\alpha }_n)〈(\gamma S{x}^{\ast }-\mu F{x}^{\ast }),x_{n+1}-x^{\ast }〉].\hfill \end{array}$$

(3.42)

Since ${\sum }_{n=0}^{\mathrm{\infty }}{\lambda }_n=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}〈(\gamma S{x}^{\ast }-\mu F{x}^{\ast }),x^{\ast }-x_{n+1}〉=0$ (due to $x_n\rightharpoonup x^{\ast }$), we deduce that ${\sum }_{n=0}^{\mathrm{\infty }}{\lambda }_n\frac{{\tau }^2-{\gamma }^2}{\tau }=\mathrm{\infty }$, and

$$\underset{n\to \mathrm{\infty }}{lim}\frac{2\tau }{{\tau }^2-{\gamma }^2}[{\alpha }_n\parallel (\gamma V-\mu F)x^{\ast }\parallel \parallel x_{n+1}-x^{\ast }\parallel +(1-{\alpha }_n)〈(\gamma S{x}^{\ast }-\mu F{x}^{\ast }),x_{n+1}-x^{\ast }〉]=0.$$

Therefore, applying Lemma 2.12 to (3.42) we infer that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x^{\ast }\parallel =0$. This completes the proof. □

Putting $M=N=1$, $F_2\equiv 0$, and $F_1=\Gamma$ an inverse-strongly monotone mapping on C in Algorithm 3.1, we have the following algorithm.

Algorithm 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, $\Theta :C\times C\to \mathbf{R}$ be a bifunction from satisfying conditions (A1)-(A4), $\phi :C\to \mathbf{R}\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and $A:H\to H$ be ${\mu }_0$-inverse-strongly monotone. Let $R:C\to 2^H$ be a maximal monotone mapping, $B:C\to H$ be ${\eta }_0$-inverse-strongly monotone, $T:C\to C$ be a ξ-strictly pseudocontractive mapping, $S:H\to H$ is a nonexpansive mapping and $V:H\to H$ be a ρ-contraction with coefficient $\rho \in [0,1)$. Let $\Gamma :C\to H$ be ζ-inverse-strongly monotone, and $F:H\to H$ be κ-Lipschitzian and η-strongly monotone with positive constants $\kappa ,\eta >0$ such that $0\le \gamma <\tau$ and $0<\mu <\frac{2\eta }{{\kappa }^2}$ where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Assume that $\Omega :=GMEP(\Theta ,\phi ,A)\cap \mathrm{I}(B,R)\cap VI(C,\Gamma )\cap Fix(T)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\},\{{\lambda }_n\}\subset (0,1]$, $\{{\beta }_n\},\{{\gamma }_n\},\{{\delta }_n\}\subset [0,1]$, $\{{\rho }_n\}\subset (0,2\alpha ]$, $\{r_n\}\subset [c,d]\subset (0,2{\mu }_0)$ and $\{{\rho }_n\}\subset [e,f]\subset (0,2{\eta }_0)$. For arbitrarily given $x_0\in H$, let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{l}\Theta (u_n,y)+\phi (y)-\phi (u_n)+〈A{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\\ v_n=J_{R,{\rho }_n}(I-{\rho }_{n}B)u_n,\\ y_n={\beta }_n{x}_n+{\gamma }_n{P}_C(I-\nu \Gamma )v_n+{\delta }_{n}T{P}_C(I-\nu \Gamma )v_n,\\ x_{n+1}={\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)+(I-{\lambda }_n\mu F)y_n,\mathrm{\forall }n\ge 0,\end{array}$$

(3.43)

where $\nu \in (0,2\zeta )$.

From Theorem 3.1, we have following result.

Corollary 3.1 In addition to assumption of Algorithm  3.2, suppose that

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

• (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\alpha }_n}{{\lambda }_n}<\mathrm{\infty }$, ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\lambda }_n}|\frac{1}{{\alpha }_n}-\frac{1}{{\alpha }_{n-1}}|=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\alpha }_n}|1-\frac{{\lambda }_{n-1}}{{\lambda }_n}|=0$;

• (iii) ${lim}_{n\to \mathrm{\infty }}\frac{|{\beta }_n-{\beta }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{|{\gamma }_n-{\gamma }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$;

• (iv) ${lim}_{n\to \mathrm{\infty }}\frac{|r_n-r_{n-1}|}{{\lambda }_n{\alpha }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{|{\rho }_n-{\rho }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$;

• (v) ${\beta }_n+{\gamma }_n+{\delta }_n=1$ and $({\gamma }_n+{\delta }_n)\xi \le {\gamma }_n$ for all $n\ge 0$;

• (vi) $\{{\beta }_n\}\subset [a,b]\subset (0,1)$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n>0$.

Then

• (a) ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\alpha }_n}=0$;

• (b) ${\omega }_w(x_n)\subset \Omega$;

• (c) $\{x_n\}$ converges strongly to a point $x^{\ast }\in \Omega$, which is a unique solution of HVIP (1.9), i.e.,

  $$〈(\mu F-\gamma S)x^{\ast },p-x^{\ast }〉\ge 0,\mathrm{\forall }p\in \Omega .$$

Proof Since $F_2\equiv 0$ and $F_1=\Gamma$ a ζ-inverse-strongly monotone mapping on C, it is easy to see that $GSVI(G)=VI(C,\Gamma )$. Thus, in terms of Theorem 3.1, we derive the desired result. □

Putting $\Gamma =I-\Phi$, where $\Phi :C\to C$ is a ${\xi }_0$-strictly pseudocontractive mapping on C, in Algorithm 3.2, we obtain the following algorithm.

Algorithm 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, $\Theta :C\times C\to \mathbf{R}$ be a bifunction satisfying conditions (A1)-(A4), $\phi :C\to \mathbf{R}\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and $A:H\to H$ be ${\mu }_0$-inverse-strongly monotone. Let $R:C\to 2^H$ be a maximal monotone mapping, $B:C\to H$ be ${\eta }_0$-inverse-strongly monotone, $T:C\to C$ be a ξ-strictly pseudocontractive mapping, $S:H\to H$ is a nonexpansive mapping and $V:H\to H$ be a ρ-contraction with coefficient $\rho \in [0,1)$. Let $\Phi :C\to C$ be a ${\xi }_0$-strictly pseudocontractive mapping, and $F:H\to H$ be κ-Lipschitzian and η-strongly monotone with positive constants $\kappa ,\eta >0$ such that $0\le \gamma <\tau$ and $0<\mu <\frac{2\eta }{{\kappa }^2}$ where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Assume that $\Omega :=GMEP(\Theta ,\phi ,A)\cap \mathrm{I}(B,R)\cap Fix(\Phi )\cap Fix(T)\ne \mathrm{\varnothing }$. Let $\{{\alpha }_n\},\{{\lambda }_n\}\subset (0,1]$, $\{{\beta }_n\},\{{\gamma }_n\},\{{\delta }_n\}\subset [0,1]$, $\{{\rho }_n\}\subset (0,2\alpha ]$, $\{r_n\}\subset [c,d]\subset (0,2{\mu }_0)$ and $\{{\rho }_n\}\subset [e,f]\subset (0,2{\eta }_0)$. For arbitrarily given $x_0\in H$, let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{l}\Theta (u_n,y)+\phi (y)-\phi (u_n)+〈A{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\\ v_n=J_{R,{\rho }_n}(I-{\rho }_{n}B)u_n,\\ y_n={\beta }_n{x}_n+{\gamma }_n(I-\nu (I-\Phi ))v_n+{\delta }_{n}T(I-\nu (I-\Phi ))v_n,\\ x_{n+1}={\lambda }_n\gamma ({\alpha }_{n}V{x}_n+(1-{\alpha }_n)S{x}_n)+(I-{\lambda }_n\mu F)y_n,\mathrm{\forall }n\ge 0,\end{array}$$

(3.44)

where $\nu \in (0,1-{\xi }_0)$.

Corollary 3.2 In addition to assumption of Algorithm  3.3, suppose that

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

• (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\alpha }_n}{{\lambda }_n}<\mathrm{\infty }$, ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\lambda }_n}|\frac{1}{{\alpha }_n}-\frac{1}{{\alpha }_{n-1}}|=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\alpha }_n}|1-\frac{{\lambda }_{n-1}}{{\lambda }_n}|=0$;

• (iii) ${lim}_{n\to \mathrm{\infty }}\frac{|{\beta }_n-{\beta }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{|{\gamma }_n-{\gamma }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$;

• (iv) ${lim}_{n\to \mathrm{\infty }}\frac{|r_n-r_{n-1}|}{{\lambda }_n{\alpha }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{|{\rho }_n-{\rho }_{n-1}|}{{\lambda }_n{\alpha }_n}=0$;

• (v) ${\beta }_n+{\gamma }_n+{\delta }_n=1$ and $({\gamma }_n+{\delta }_n)\xi \le {\gamma }_n$ for all $n\ge 0$;

• (vi) $\{{\beta }_n\}\subset [a,b]\subset (0,1)$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n>0$.

Then

• (a) ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\alpha }_n}=0$;

• (b) ${\omega }_w(x_n)\subset \Omega$;

• (c) $\{x_n\}$ converges strongly to a point $x^{\ast }\in \Omega$, which is a unique solution of HVIP (1.9), i.e.,

  $$〈(\mu F-\gamma S)x^{\ast },p-x^{\ast }〉\ge 0,\mathrm{\forall }p\in \Omega .$$

Proof Since $\Phi :C\to C$ is a ${\xi }_0$-strictly pseudocontractive mapping on C, it is well known that for constant ${\xi }_0\in [0,1)$,

$$〈\Phi x-\Phi y,x-y〉\le {\parallel x-y\parallel }^2-\frac{1-{\xi }_0}{2}{\parallel (I-\Phi )x-(I-\Phi )y\parallel }^2,\mathrm{\forall }x,y\in C.$$

It is clear that in this case the mapping $\Gamma =I-\Phi$ is $\frac{1-{\xi }_0}{2}$-inverse-strongly monotone. Moreover, we have, for $\nu \in (0,1-{\xi }_0)$,

$$\begin{array}{rl}{y}_n& ={\beta }_n{x}_n+{\gamma }_n{P}_C(I-\nu \Gamma )v_n+{\delta }_{n}T{P}_C(I-\nu \Gamma )v_n\\ ={\beta }_n{x}_n+{\gamma }_n(I-\nu (I-\Phi ))v_n+{\delta }_{n}T(I-\nu (I-\Phi ))v_n.\end{array}$$

Now let us show $Fix(\Phi )=VI(C,\Gamma )$. In fact, we have, for $\lambda >0$,

$$\begin{array}{rl}u\in VI(C,\Gamma )& ⟺〈\Gamma u,y-u〉\ge 0,\mathrm{\forall }y\in C\\ ⟺〈u-\lambda \Gamma u-u,u-y〉\ge 0,\mathrm{\forall }y\in C\\ ⟺u=P_C(u-\lambda \Gamma u)\\ ⟺u=P_C(u-\lambda u+\lambda \Phi u)\\ ⟺〈u-\lambda u+\lambda \Phi u-u,u-y〉\ge 0,\mathrm{\forall }y\in C\\ ⟺〈u-\Phi u,u-y〉\le 0,\mathrm{\forall }y\in C\\ ⟺u=\Phi u\\ ⟺u\in Fix(\Phi ).\end{array}$$

Consequently,

$$\begin{array}{rl}\Omega & =GMEP(\Theta ,\phi ,A)\cap \mathrm{I}(B,R)\cap VI(C,\Gamma )\cap Fix(T)\\ =GMEP(\Theta ,\phi ,A)\cap \mathrm{I}(B,R)\cap Fix(\Phi )\cap Fix(T).\end{array}$$

Therefore, by Corollary 3.1, we derive the desired result. □

Remark 3.1 Our results generalize and improve results in [12, 20] and the references therein.