Regularized hybrid iterative algorithms for triple hierarchical variational inequalities

Abstract

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let $Fix(T)$ be the fixed point set of a nonexpansive mapping, let $MEP(\Theta ,\phi )$ be the solution set of a mixed equilibrium problem (MEP), and let Γ be the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solution $x^{\ast }\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$ of a variational inequality with a variational inequality constraint over the intersection of $Fix(T)$, $MEP(\Theta ,\phi )$, and Γ. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm.

MSC:49J40, 47J20, 47H10, 65K05, 47H09.

1 Introduction

Let H be a Hilbert space with inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$ over the real scalar field ℝ. Let C be a nonempty closed convex subset of H, and $P_C$ be the metric projection of H onto C. Let $T:C\to C$ be a self-mapping on C. Denote by $Fix(T)$ the set of fixed points of T. We say that T is L-Lipschitzian if there exists a constant $L\ge 0$ such that

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

When $L=1$ or $0\le L<1$, we call T a nonexpansive or a contractive mapping, respectively. We say that a mapping $A:C\to H$ is α-inverse strongly monotone if there exists a constant $\alpha >0$ such that

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

and that A is η-strongly monotone (resp. monotone) if there exists a constant $\eta >0$ (resp. $\eta =0$) such that

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

It is known that T is nonexpansive if and only if $I-T$ is $\frac{1}{2}$-inverse strongly monotone. Moreover, L-Lipschitz continuous mappings are $\frac{1}{L}$-inversely strong monotone (see, e.g., [1]).

Let $f:C\to \mathbb{R}$ be a convex and continuously Frechet differentiable functional. Consider the minimization problem (MP):

$$\underset{x\in C}{min}f(x)$$

(1.1)

(assuming the existence of minimizers). We denote by $\Gamma \ne \mathrm{\varnothing }$ the set of minimizers of problem (1.1). The gradient-projection algorithm (GPA) generates a sequence $\{x_n\}$ determined by the gradient ∇f and the metric projection $P_C$:

$$x_{n+1}:=P_C(x_n-\lambda \mathrm{\nabla }f(x_n)),\mathrm{\forall }n\ge 0.$$

(1.2)

The convergence of algorithm (1.2) depends on ∇f. It is known that if ∇f is η-strongly monotone and L-Lipschitz continuous, then for $0<\lambda <\frac{2\eta }{L^2}$, the operator

$$S:=P_C(I-\lambda \mathrm{\nabla }f)$$

is a contraction. Hence, the sequence $\{x_n\}$ defined by the GPA (1.2) converges in norm to the unique solution of (1.1). If the gradient ∇f is only assumed to be Lipschitz continuous, then $\{x_n\}$ can only be weakly convergent when H is infinite-dimensional (a counterexample to the norm convergence of $\{x_n\}$ is given by Xu [[2], Section 5]).

The regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the regularized minimization problem

$$\underset{x\in C}{min}{f}_{\alpha }(x):=f(x)+\frac{\alpha }{2}{\parallel x\parallel }^2,$$

where $\alpha >0$ is the regularization parameter, and again f is convex with Lipschitz continuous gradient ∇f. While a regularization method provides the possible strong convergence to the minimum-norm solution, its disadvantage is the implicity. Hence explicit iterative methods seem to be attractive. See, e.g., Xu [2, 3].

On the other hand, for a given mapping $A:C\to H$, we consider the variational inequality problem (VIP) of finding $x^{\ast }\in C$ such that

$$〈A{x}^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C.$$

(1.3)

The solution set of VIP (1.3) is denoted by $VI(C,A)$. It is well known that when A is monotone,

$$x\in VI(C,A)\iff x=P_C(x-\lambda Ax),\mathrm{\forall }\lambda >0.$$

Variational inequality theory has been studied quite extensively and has emerged as an important tool in several branches of pure and applied sciences; see, e.g., [1, 4–8] and the references therein.

When C is the fixed point set $Fix(T)$ of a nonexpansive mapping T and $A=I-S$, VIP (1.3) becomes the variational inequality problem of finding $x^{\ast }\in Fix(T)$ such that

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

(1.4)

This problem, introduced by Moudafi and Maingé [9, 10], is called the hierarchical fixed point problem. It is clear that if S has fixed points, then they are solutions of VIP (1.4). If S is contractive, the solution set of VIP (1.4) is a singleton and it is well known as a viscosity problem. This was previously introduced by Moudafi [11] and also developed by Xu [12]. In this case, solving VIP (1.4) is equivalent to finding a fixed point of the nonexpansive mapping $P_{Fix(T)}S$, where $P_{Fix(T)}$ is the metric projection onto the closed and convex set $Fix(T)$. Yao et al. [8] introduced a two-step algorithm to solve VIP (1.4).

Let $\Theta :C\times C\to \mathbb{R}$ be a bifunction and $\phi :C\to \mathbb{R}$ be a function. Consider the mixed equilibrium problem (MEP) of finding $x\in C$ such that

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

(1.5)

which was studied by Ceng and Yao [13]. The solution set of MEP (1.5) is denoted by $MEP(\Theta ,\phi )$. The MEP (1.5) is very general in the sense that it includes, as special cases, fixed point problems, optimization problems, variational inequality problems, minimax problems, Nash equilibrium problems in noncooperative games and others; see, e.g., [13–15].

Recently, Iiduka [16, 17] considered a variational inequality with a variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with bilevel programming problems or hierarchical constrained optimization problems or hierarchical fixed point problems, it is referred to as a triple hierarchical constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such a problem. Since the original problem is a variational inequality, in this paper, we call it a triple hierarchical variational inequality (THVI). Ceng et al. introduced and considered some THVI in [18]. A nice survey article on THVI is [19]. See also [20–22].

Extending the works done in [18], we introduce and study in this paper the following triple hierarchical variational inequality with constraints of minimization and equilibrium problems.

The problem to study

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $f:C\to \mathbb{R}$ be convex and continuously Frechet differentiable with Γ being the set of its minimizers. Let $T:C\to C$ and $S:H\to H$ be both nonexpansive. Let $V:H\to H$ be ρ-contractive, and $F:C\to H$ be κ-Lipschitzian and η-strongly monotone with constants $\rho \in [0,1)$ and $\kappa ,\eta >0$. Suppose $0<\mu <2\eta /{\kappa }^2$ and $0<\gamma \le \tau$ where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$.

Let Ξ denote the solution set of the following hierarchical variational inequality (HVI): find $z^{\ast }\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$ such that

$$〈(\mu F-\gamma S)z^{\ast },z-z^{\ast }〉\ge 0,\mathrm{\forall }z\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma ,$$

where the solution set Ξ is assumed to be nonempty. Consider the following triple hierarchical variational inequality (THVI).

Find $x^{\ast }\in \Xi$ such that

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

(1.6)

Based on the iterative schemes provided by Xu [2] and the two-step iterative scheme provided by Yao et al. [8], by virtue of the viscosity approximation method, hybrid steepest-descent method and the regularization method, we propose the following hybrid iterative algorithm with regularization:

$$\{\begin{array}{l}\Theta (u_n,y)+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\\ y_n={\theta }_n\gamma S{x}_n+(I-{\theta }_n\mu F)P_C(u_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)),\\ x_{n+1}={\beta }_n\gamma V{y}_n+(I-{\beta }_n\mu F)T{P}_C(u_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)),\mathrm{\forall }n\ge 0.\end{array}$$

Here, $0<\lambda <2/L$, $\{r_n\},\{{\alpha }_n\}\subset (0,+\mathrm{\infty })$, and $\{{\beta }_n\},\{{\theta }_n\}\subset (0,1)$. It is shown that under appropriate assumptions, the two iterative sequences $\{x_n\}$ and $\{y_n\}$ converge strongly to the unique solution of the THVI (1.6).

2 Preliminaries

Let K be a nonempty closed convex subset of a real Hilbert space H. We write $x_n\rightharpoonup x$ and $x_n\to x$ to indicate that the sequence $\{x_n\}$ converges weakly and strongly to x, respectively. The weak ω-limit set of the sequence $\{x_n\}$ is denoted by

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

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

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

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

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

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

• (iii) $〈P_{K}x-P_{K}y,x-y〉\ge {\parallel P_{K}x-P_{K}y\parallel }^2$, $\mathrm{\forall }y\in H$.

Hence, $P_K$ is nonexpansive and monotone.

Definition 2.2 A mapping $T:H\to H$ is said to be firmly nonexpansive if $2T-I$ is nonexpansive, or equivalently,

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

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. We call T an averaged mapping if T can be expressed as a proper convex combination of the identity map I and a nonexpansive mapping. In particular, firmly nonexpansive mappings are averaged.

Proposition 2.3 (see [23])

Let $T:H\to H$ be a given mapping.

• (i) T is nonexpansive if and only if the complement $I-T$ is $\frac{1}{2}$-inverse strongly monotone.

• (ii) If T is ν-inverse strongly monotone, then so is γT for all $\gamma >0$.

• (iii) T is averaged if and only if the complement $I-T$ is ν-inverse strongly monotone for some $\nu >1/2$. Indeed, for $\alpha \in (0,1)$, T is α-averaged if and only if $I-T$ is $\frac{1}{2\alpha }$-inverse strongly monotone.

Proposition 2.4 (see [23, 24])

Let $S,T,V:H\to H$.

• (i) If $T=(1-\alpha )S+\alpha V$ for some $\alpha \in (0,1)$ and if S is averaged and V is nonexpansive, then T is averaged.

• (ii) T is firmly nonexpansive if and only if the complement $I-T$ is firmly nonexpansive.

• (iii) If $T=(1-\alpha )S+\alpha V$ for some $\alpha \in (0,1)$ and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.

• (iv) The composition of finitely many averaged mappings is averaged. In particular, if $T_1$ is ${\alpha }_1$-averaged and $T_2$ is ${\alpha }_2$-averaged, where ${\alpha }_1,{\alpha }_2\in (0,1)$, then $T_1{T}_2$ is $({\alpha }_1+{\alpha }_2-{\alpha }_1{\alpha }_2)$-averaged.

• (v) If the mappings $T_1,\dots ,T_N$ are averaged and have a common fixed point, then

  $$\bigcap _{i=1}^{N}Fix(T_i)=Fix(T_1\cdot \cdot \cdot T_N).$$

For solving the equilibrium problem for a bifunction $\Theta :C\times C\to \mathbb{R}$, let us consider the following conditions:

• (A1) $T(x,x)=0$ for all $x?C$;

• (A2) T is monotone, that is, $T(x,y)+T(y,x)=0$ for all $x,y?C$;

• (A3) for each $x,y,z?C$, ${lim}_{t?0}T(tz+(1-t)x,y)=T(x,y)$;

• (A4) for each $x?C$, $y?T(x,y)$ is convex and lower semicontinuous;

• (A5) for each $y?C$, $x?T(x,y)$ is weakly upper semicontinuous;

• (B1) for each $x?H$ and $r>0$, there exist a bounded subset $D_x?C$ and $y_x?C$ such that for any $z?C\backslash D_x$,

  $$T(z,y_x)+f(y_x)-f(z)+\frac{1}{r}{y}_x-z,z-x><0;$$

• (B2) C is a bounded set.

Lemma 2.5 (see [14])

Let C be a nonempty closed convex subset of a real Hilbert space H and $T:C\times C?\mathbf{R}$ be a bifunction satisfying (A1)-(A4). Let $r>0$ and $x?H$. Then there exists $z?C$ such that

$$T(z,y+\frac{1}{r}y-z,z-x>=0,\mathrm{?}y?C.$$

Lemma 2.6 (see [15])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $\Theta :C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A5) and $\phi :C\to \mathbb{R}$ be a proper lower semicontinuous and convex function. For $r>0$ and $x\in H$, define a mapping $T_r:H\to C$ as follows:

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

for all $x\in H$. Assume that either (B1) or (B2) holds. Then $T_r$ is a single-valued firmly nonexpansive map on H, and $Fix(T_r)=MEP(\Theta ,\phi )$ is closed and convex.

Lemma 2.7 (see [25])

Let $\{a_n\}$ be a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-s_n)a_n+s_n{t}_n+{ϵ}_n,\mathrm{\forall }n\ge 0.$$

Here, $0<s_n\le 1$, $0\le {ϵ}_n$, and $t_n\in \mathbb{R}$ for all $n=0,1,2,\dots$ , such that

• (i) ${\sum }_{n=0}^{\mathrm{\infty }}{s}_n=+\mathrm{\infty }$;

• (ii) either ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{t}_n\le 0$ or ${\sum }_{n=0}^{\mathrm{\infty }}{s}_n|t_n|<+\mathrm{\infty }$;

• (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{ϵ}_n<+\mathrm{\infty }$.

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

Lemma 2.8 (Demiclosedness principle; see [1])

Let C be a nonempty closed convex subset of a real Hilbert space H and let $T:C\to C$ be a nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing }$. If $\{x_n\}$ is a sequence in C converging weakly to x and if $\{(I-T)x_n\}$ converges strongly to y, then $(I-T)x=y$; in particular, if $y=0$, then $x\in Fix(T)$.

Lemma 2.9 (see [12])

Let $S:H\to H$ be a nonexpansive mapping and $V:H\to H$ be a ρ-contraction with $\rho \in [0,1)$, respectively.

• (i) $I-S$ is monotone, i.e.,

  $$〈(I-S)x-(I-S)y,x-y〉\ge 0,\mathrm{\forall }x,y\in H.$$

• (ii) $I-V$ is $(1-\rho )$-strongly monotone, i.e.,

  $$〈(I-V)x-(I-V)y,x-y〉\ge (1-\rho ){\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Lemma 2.10 ([26])

Let H be a real Hilbert space. Then, for all $x,y\in H$ and $\lambda \in [0,1]$,

$${\parallel \lambda x+(1-\lambda )y\parallel }^2=\lambda {\parallel x\parallel }^2+(1-\lambda ){\parallel y\parallel }^2-\lambda (1-\lambda ){\parallel x-y\parallel }^2.$$

Lemma 2.11 We have the following inequality in an inner product space X:

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

Notations Let λ be a number in $(0,1]$ and let $\mu ,\kappa ,\eta >0$. Let $F:C\to H$ be κ-Lipschitzian and η-strongly monotone. Associated with a nonexpansive mapping $T:C\to C$, we define the mapping $T^{\lambda }:C\to H$ by

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

Lemma 2.12 (see [[27], Lemma 3.1])

The map $T^{\lambda }$ is a contraction provided $\mu <2\eta /{\kappa }^2$, 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 $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}\in (0,1]$. In particular, if $T=I$ the identity mapping, then

$$\parallel (I-\lambda \mu F)x-(I-\lambda \mu F)y\parallel \le (1-\lambda \tau )\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

A set-valued mapping $\tilde{T}:H\to 2^H$ is called monotone if $〈x-y,f-g〉\ge 0$ for all $x,y\in H$, $f\in \tilde{T}x$ and $g\in \tilde{T}y$. A monotone set-valued mapping $\tilde{T}:H\to 2^H$ is called maximal if its graph $Gph(\tilde{T})$ is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mapping $\tilde{T}:H\to 2^H$ is maximal if and only if for $(x,f)\in H\times H$, $〈x-y,f-g〉\ge 0$ for every $(y,g)\in Gph(\tilde{T})$ implies that $f\in \tilde{T}x$.

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

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

Define

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

Lemma 2.13 (see [28])

Let $A:C\to H$ be a monotone mapping.

• (i) $\tilde{T}$ is maximal monotone;

• (ii) $v\in {\tilde{T}}^{-1}0\iff v\in VI(C,A)$.

3 Main results

Let us consider the following three-step iterative scheme with regularization:

$$\{\begin{array}{l}\Theta (u_n,y)+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C,\\ y_n={\theta }_n\gamma S{x}_n+(I-{\theta }_n\mu F)P_C(u_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)),\\ x_{n+1}={\beta }_n\gamma V{y}_n+(I-{\beta }_n\mu F)T{P}_C(u_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)),\mathrm{\forall }n=0,1,2,\dots .\end{array}$$

(3.1)

Here,

• $V:H\to H$ is a ρ-contraction;

• $S:H\to H$ and $T:C\to C$ are nonexpansive mappings;

• $F:C\to H$ is a κ-Lipschitzian and η-strongly monotone mapping;

• $\Theta :C\times C\to \mathbb{R}$ and $\phi :C\to \mathbb{R}$ are real-valued functions;

• $\mathrm{\nabla }f:C\to H$ is L-Lipschitz continuous with $0<\lambda <\frac{2}{L}$;

• $\{r_n\}$ and $\{{\alpha }_n\}$ are sequences in $(0,+\mathrm{\infty })$ with ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n<+\mathrm{\infty }$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$;

• $\{{\beta }_n\}$ and $\{{\theta }_n\}$ are sequences in $(0,1)$;

• $0<\mu <2\eta /{\kappa }^2$ and $0<\gamma \le \tau$, where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$.

Theorem 3.1 Suppose that $\Theta :C\times C\to \mathbb{R}$ satisfies (A1)-(A5) and that (B1) or (B2) holds. Let $\{x_n\}$ be the bounded sequence generated from any given $x_0\in C$ by (3.1). Assume that

(H1) ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n=+\mathrm{\infty }$, ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\beta }_n}|1-\frac{{\theta }_{n-1}}{{\theta }_n}|=0$;

(H2) ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\beta }_n}|\frac{1}{{\theta }_n}-\frac{1}{{\theta }_{n-1}}|=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\theta }_n}|1-\frac{{\beta }_{n-1}}{{\beta }_n}|=0$;

(H3) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=0$, ${lim}_{n\to \mathrm{\infty }}\frac{{\alpha }_n+{\beta }_n}{{\theta }_n}=0$, and ${lim}_{n\to \mathrm{\infty }}\frac{{\theta }_n^2}{{\beta }_n}=0$;

(H4) ${lim}_{n\to \mathrm{\infty }}\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\beta }_n{\theta }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{|r_n-r_{n-1}|}{{\beta }_n{\theta }_n}=0$.

Then we have the following:

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

• (ii) ${\omega }_w(x_n)\subset Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$;

• (iii) ${\omega }_w(x_n)\subset \Xi$ if $\parallel x_n-y_n\parallel =o({\theta }_n)$ held in addition, i.e., ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_n-y_n\parallel }{{\theta }_n}=0$.

Proof First, let us show that $P_C(I-\lambda \mathrm{\nabla }{f}_{\alpha })$ is ξ-averaged for each $\lambda \in (0,\frac{2}{\alpha +L})$, where

$$\xi =\frac{2+\lambda (\alpha +L)}{4}\in (0,1).$$

The Lipschitz condition implies that the gradient ∇f is $\frac{1}{L}$-inverse strongly monotone [1], that is,

$$〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉\ge \frac{1}{L}{\parallel \mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel }^2.$$

Observe that

$$\begin{array}{c}(\alpha +L)〈\mathrm{\nabla }{f}_{\alpha }(x)-\mathrm{\nabla }{f}_{\alpha }(y),x-y〉\hfill \\ =(\alpha +L)[\alpha {\parallel x-y\parallel }^2+〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉]\hfill \\ ={\alpha }^2{\parallel x-y\parallel }^2+\alpha 〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉+\alpha L{\parallel x-y\parallel }^2+L〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉\hfill \\ \ge {\alpha }^2{\parallel x-y\parallel }^2+2\alpha 〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉+{\parallel \mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel }^2\hfill \\ ={\parallel \alpha (x-y)+\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel }^2\hfill \\ ={\parallel \mathrm{\nabla }{f}_{\alpha }(x)-\mathrm{\nabla }{f}_{\alpha }(y)\parallel }^2.\hfill \end{array}$$

Hence, $\mathrm{\nabla }{f}_{\alpha }=\alpha I+\mathrm{\nabla }f$ is $\frac{1}{\alpha +L}$-inverse strongly monotone. Thus, $\lambda \mathrm{\nabla }{f}_{\alpha }$ is $\frac{1}{\lambda (\alpha +L)}$-inverse strongly monotone by Proposition 2.3(ii). By Proposition 2.3(iii) the complement $I-\lambda \mathrm{\nabla }{f}_{\alpha }$ is $\frac{\lambda (\alpha +L)}{2}$-averaged. Noting that $P_C$ is $\frac{1}{2}$-averaged and utilizing Proposition 2.4(iv), we know that for each $\lambda \in (0,\frac{2}{\alpha +L})$, the map $P_C(I-\lambda \mathrm{\nabla }{f}_{\alpha })$ is ξ-averaged with

$$\xi =\frac{1}{2}+\frac{\lambda (\alpha +L)}{2}-\frac{1}{2}\cdot \frac{\lambda (\alpha +L)}{2}=\frac{2+\lambda (\alpha +L)}{4}\in (0,1).$$

In particular, $P_C(I-\lambda \mathrm{\nabla }{f}_{\alpha })$ is nonexpansive. Furthermore, for $\lambda \in (0,\frac{2}{L})$, utilizing the fact that ${lim}_{n\to \mathrm{\infty }}\frac{2}{{\alpha }_n+L}=\frac{2}{L}$, we may assume

$$0<\lambda <\frac{2}{{\alpha }_n+L},\mathrm{\forall }n\ge 0.$$

Consequently, for each integer $n\ge 0$, $P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})$ is ${\xi }_n$-averaged with

$${\xi }_n=\frac{1}{2}+\frac{\lambda ({\alpha }_n+L)}{2}-\frac{1}{2}\cdot \frac{\lambda ({\alpha }_n+L)}{2}=\frac{2+\lambda ({\alpha }_n+L)}{4}\in (0,1).$$

This immediately implies that $P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})$ is nonexpansive for all $n\ge 0$.

We divide the proof into several steps.

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

For simplicity, put ${\tilde{u}}_n=P_C(u_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_n))$. Then $y_n={\theta }_n\gamma S{x}_n+(I-{\theta }_n\mu F){\tilde{u}}_n$ and $x_{n+1}={\beta }_n\gamma V{y}_n+(I-{\beta }_n\mu F)T{\tilde{u}}_n$ for every $n\ge 0$. We observe that

$$\begin{array}{rcl}\parallel {\tilde{u}}_n-{\tilde{u}}_{n-1}\parallel & \le & \parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_{n-1}\parallel \\ +\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_{n-1}-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}})u_{n-1}\parallel \\ \le & \parallel u_n-u_{n-1}\parallel +\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_{n-1}-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}})u_{n-1}\parallel \\ \le & \parallel u_n-u_{n-1}\parallel +\parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_{n-1}-(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}})u_{n-1}\parallel \\ =& \parallel u_n-u_{n-1}\parallel +\parallel \lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_{n-1})-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}}(u_{n-1})\parallel \\ =& \parallel u_n-u_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel u_{n-1}\parallel ,\mathrm{\forall }n\ge 1.\end{array}$$

(3.2)

Moreover, from (3.1) we have

$$\{\begin{array}{l}{y}_n={\theta }_n\gamma S{x}_n+(I-{\theta }_n\mu F){\tilde{u}}_n,\\ y_{n-1}={\theta }_{n-1}\gamma S{x}_{n-1}+(I-{\theta }_{n-1}\mu F){\tilde{u}}_{n-1},\mathrm{\forall }n\ge 1.\end{array}$$

Thus

$$\begin{array}{rcl}{y}_n-y_{n-1}& =& {\theta }_n(\gamma S{x}_n-\gamma S{x}_{n-1})+({\theta }_n-{\theta }_{n-1})(\gamma S{x}_{n-1}-\mu F{\tilde{u}}_{n-1})\\ +(I-{\theta }_n\mu F){\tilde{u}}_n-(I-{\theta }_n\mu F){\tilde{u}}_{n-1}.\end{array}$$

Utilizing Lemma 2.12 from (3.2) we deduce that

$$\begin{array}{rcl}\parallel y_n-y_{n-1}\parallel & \le & {\theta }_n\parallel \gamma S{x}_n-\gamma S{x}_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{u}}_{n-1}\parallel \\ +\parallel (I-{\theta }_n\mu F){\tilde{u}}_n-(I-{\theta }_n\mu F){\tilde{u}}_{n-1}\parallel \\ \le & {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{u}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )\parallel {\tilde{u}}_n-{\tilde{u}}_{n-1}\parallel \\ \le & {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{u}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )(\parallel u_n-u_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel u_{n-1}\parallel ),\end{array}$$

(3.3)

where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Taking into consideration that $u_n=T_{r_n}{x}_n$ and $u_{n-1}=T_{r_{n-1}}{x}_{n-1}$, we have

$$\Theta (u_n,y)+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C$$

(3.4)

and

$$\Theta (u_{n-1},y)+\phi (y)-\phi (u_{n-1})+\frac{1}{r_{n-1}}〈y-u_{n-1},u_{n-1}-x_{n-1}〉\ge 0,\mathrm{\forall }y\in C.$$

(3.5)

Putting $y=u_{n-1}$ in (3.4) and $y=u_n$ in (3.5), we obtain

$$\Theta (u_n,u_{n-1})+\phi (u_{n-1})-\phi (u_n)+\frac{1}{r_n}〈u_{n-1}-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in C$$

and

$$\Theta (u_{n-1},u_n)+\phi (u_n)-\phi (u_{n-1})+\frac{1}{r_{n-1}}〈u_n-u_{n-1},u_{n-1}-x_{n-1}〉\ge 0,\mathrm{\forall }y\in C.$$

Adding the last two inequalities, by (A2) we get

$$〈u_n-u_{n-1},\frac{u_{n-1}-x_{n-1}}{r_{n-1}}-\frac{u_n-x_n}{r_n}〉\ge 0,$$

and hence

$$〈u_n-u_{n-1},u_{n-1}-u_n+u_n-x_{n-1}-\frac{r_{n-1}}{r_n}(u_n-x_n)〉\ge 0.$$

Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$, we may assume, without loss of generality, that there exists a positive number c such that $r_n\ge c>0$ for all $n\ge 0$. Thus we have

$$\begin{array}{rcl}{\parallel u_n-u_{n-1}\parallel }^2& \le & 〈u_n-u_{n-1},x_n-x_{n-1}+(1-\frac{r_{n-1}}{r_n})(u_n-x_n)〉\\ \le & \parallel u_n-u_{n-1}\parallel [\parallel x_n-x_{n-1}\parallel +|1-\frac{r_{n-1}}{r_n}|\parallel u_n-x_n\parallel ],\end{array}$$

and hence

$$\begin{array}{rcl}\parallel u_n-u_{n-1}\parallel & \le & \parallel x_n-x_{n-1}\parallel +|1-\frac{r_{n-1}}{r_n}|\parallel u_n-x_n\parallel \\ \le & \parallel x_n-x_{n-1}\parallel +\frac{M_0}{c}|r_n-r_{n-1}|.\end{array}$$

(3.6)

Here, $M_0=sup\{\parallel u_n-x_n\parallel :n\ge 0\}<+\mathrm{\infty }$.

Substituting (3.6) into (3.3) we derive

$$\begin{array}{rcl}\parallel y_n-y_{n-1}\parallel & \le & {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{u}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )(\parallel x_n-x_{n-1}\parallel +\frac{M_0}{c}|r_n-r_{n-1}|+\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel u_{n-1}\parallel )\\ \le & (1-{\theta }_n(\tau -\gamma ))\parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{u}}_{n-1}\parallel \\ +\frac{M_0}{c}|r_n-r_{n-1}|+\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel u_{n-1}\parallel \\ \le & \parallel x_n-x_{n-1}\parallel +M_1(|{\theta }_n-{\theta }_{n-1}|+|r_n-r_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|).\end{array}$$

(3.7)

Here, $\parallel \gamma S{x}_n-\mu F{\tilde{u}}_n\parallel +\frac{M_0}{c}+\lambda \parallel u_n\parallel \le M_1$ for some $M_1\ge 0$.

On the other hand, from (3.1) we have

$$\{\begin{array}{l}{x}_{n+1}={\beta }_n\gamma V{y}_n+(I-{\beta }_n\mu F)T{\tilde{u}}_n,\\ x_n={\beta }_{n-1}\gamma V{y}_{n-1}+(I-{\beta }_{n-1}\mu F)T{\tilde{u}}_{n-1},\mathrm{\forall }n\ge 1.\end{array}$$

Simple calculations show that

$$\begin{array}{rl}{x}_{n+1}-x_n=& (I-{\beta }_n\mu F)T{\tilde{u}}_n-(I-{\beta }_n\mu F)T{\tilde{u}}_{n-1}\\ +({\beta }_n-{\beta }_{n-1})(\gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1})+{\beta }_n(\gamma V{y}_n-\gamma V{y}_{n-1}).\end{array}$$

Utilizing Lemma 2.12 from (3.2), (3.6), and (3.7) we deduce that

$$\begin{array}{c}\parallel x_{n+1}-x_n\parallel \hfill \\ \le \parallel (I-{\beta }_n\mu F)T{\tilde{u}}_n-(I-{\beta }_n\mu F)T{\tilde{u}}_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1}\parallel \hfill \\ +{\beta }_n\parallel \gamma V{y}_n-\gamma V{y}_{n-1}\parallel \hfill \\ \le (1-{\beta }_n\tau )\parallel {\tilde{u}}_n-{\tilde{u}}_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1}\parallel +{\beta }_n\gamma \rho \parallel y_n-y_{n-1}\parallel \hfill \\ \le (1-{\beta }_n\tau )(\parallel u_n-u_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel u_{n-1}\parallel )+|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1}\parallel \hfill \\ +{\beta }_n\gamma \rho \parallel y_n-y_{n-1}\parallel \hfill \\ \le (1-{\beta }_n\tau )(\parallel x_n-x_{n-1}\parallel +\frac{M_0}{c}|r_n-r_{n-1}|+\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel u_{n-1}\parallel )\hfill \\ +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1}\parallel +{\beta }_n\gamma \rho [\parallel x_n-x_{n-1}\parallel +M_1(|{\theta }_n-{\theta }_{n-1}|\hfill \\ +|r_n-r_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|\left)\right]\hfill \\ \le (1-{\beta }_n\tau )[\parallel x_n-x_{n-1}\parallel +M_1(|r_n-r_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|)]\hfill \\ +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1}\parallel \hfill \\ +{\beta }_n\gamma \rho [\parallel x_n-x_{n-1}\parallel +M_1(|{\theta }_n-{\theta }_{n-1}|+|r_n-r_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|)]\hfill \\ \le (1-{\beta }_n\tau )\parallel x_n-x_{n-1}\parallel +(1-{\beta }_n\tau )M_1(|r_n-r_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|)\hfill \\ +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1}\parallel \hfill \\ +{\beta }_n\gamma \rho \parallel x_n-x_{n-1}\parallel +{\beta }_n\tau M_1(|{\theta }_n-{\theta }_{n-1}|+|r_n-r_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|)\hfill \\ \le (1-{\beta }_n(\tau -\gamma \rho ))\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{y}_{n-1}-\mu FT{\tilde{u}}_{n-1}\parallel \hfill \\ +M_1(|{\theta }_n-{\theta }_{n-1}|+|r_n-r_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|)\hfill \\ \le (1-{\beta }_n(\tau -\gamma \rho ))\parallel x_n-x_{n-1}\parallel +M(|{\alpha }_n-{\alpha }_{n-1}|+|{\beta }_n-{\beta }_{n-1}|\hfill \\ +|{\theta }_n-{\theta }_{n-1}|+|r_n-r_{n-1}|),\hfill \end{array}$$

where $M_1+\parallel \gamma V{y}_n-\mu FT{\tilde{u}}_n\parallel \le M$, $\mathrm{\forall }n\ge 0$ for some $M\ge 0$. Therefore,

$$\begin{array}{c}\frac{\parallel x_{n+1}-x_n\parallel }{{\theta }_n}\hfill \\ \le (1-{\beta }_n(\tau -\gamma \rho ))\frac{\parallel x_n-x_{n-1}\parallel }{{\theta }_n}\hfill \\ +M(\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\theta }_n}+\frac{|{\beta }_n-{\beta }_{n-1}|}{{\theta }_n}+\frac{|{\theta }_n-{\theta }_{n-1}|}{{\theta }_n}+\frac{|r_n-r_{n-1}|}{{\theta }_n})\hfill \\ =(1-(\tau -\gamma \rho ){\beta }_n)\frac{\parallel x_n-x_{n-1}\parallel }{{\theta }_{n-1}}+(1-(\tau -\gamma \rho ){\beta }_n)\parallel x_n-x_{n-1}\parallel (\frac{1}{{\theta }_n}-\frac{1}{{\theta }_{n-1}})\hfill \\ +M(\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\theta }_n}+\frac{|{\beta }_n-{\beta }_{n-1}|}{{\theta }_n}+\frac{|{\theta }_n-{\theta }_{n-1}|}{{\theta }_n}+\frac{|r_n-r_{n-1}|}{{\theta }_n})\hfill \\ \le (1-(\tau -\gamma \rho ){\beta }_n)\frac{\parallel x_n-x_{n-1}\parallel }{{\theta }_{n-1}}+(\tau -\gamma \rho ){\beta }_n\cdot \frac{1}{\tau -\gamma \rho }\{\parallel x_n-x_{n-1}\parallel \frac{1}{{\beta }_n}|\frac{1}{{\theta }_n}-\frac{1}{{\theta }_{n-1}}|\hfill \\ +M(\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\beta }_n{\theta }_n}+\frac{|{\beta }_n-{\beta }_{n-1}|}{{\beta }_n{\theta }_n}+\frac{|{\theta }_n-{\theta }_{n-1}|}{{\beta }_n{\theta }_n}+\frac{|r_n-r_{n-1}|}{{\beta }_n{\theta }_n})\}\hfill \\ \le (1-(\tau -\gamma \rho ){\beta }_n)\frac{\parallel x_n-x_{n-1}\parallel }{{\theta }_{n-1}}+(\tau -\gamma \rho ){\beta }_n\cdot \frac{\tilde{M}}{\tau -\gamma \rho }\left\{\frac{1}{{\beta }_n}\right|\frac{1}{{\theta }_n}-\frac{1}{{\theta }_{n-1}}|\hfill \\ +\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\beta }_n{\theta }_n}+\frac{1}{{\theta }_n}|1-\frac{{\beta }_{n-1}}{{\beta }_n}|+\frac{1}{{\beta }_n}|1-\frac{{\theta }_{n-1}}{{\theta }_n}|+\frac{|r_n-r_{n-1}|}{{\beta }_n{\theta }_n}\},\hfill \end{array}$$

(3.8)

where $M+\parallel x_n-x_{n-1}\parallel \le \tilde{M}$, $\mathrm{\forall }n\ge 1$ for some $\tilde{M}\ge 0$. From (H1), (H2), and (H4), it follows that ${\sum }_{n=0}^{\mathrm{\infty }}(\tau -\gamma \rho ){\beta }_n=+\mathrm{\infty }$ and

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}\frac{\tilde{M}}{\tau -\gamma \rho }\left\{\frac{1}{{\beta }_n}\right|\frac{1}{{\theta }_n}-\frac{1}{{\theta }_{n-1}}|+\frac{|{\alpha }_n-{\alpha }_{n-1}|}{{\beta }_n{\theta }_n}+\frac{1}{{\theta }_n}|1-\frac{{\beta }_{n-1}}{{\beta }_n}|\hfill \\ +\frac{1}{{\beta }_n}|1-\frac{{\theta }_{n-1}}{{\theta }_n}|+\frac{|r_n-r_{n-1}|}{{\beta }_n{\theta }_n}\}=0.\hfill \end{array}$$

Applying Lemma 2.7 to (3.8), we immediately conclude that

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

In particular, from (H3) it follows that

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

Step 2. ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel y_n-{\tilde{u}}_n\parallel =0$.

By the firm nonexpansivity of $T_{r_n}$, if $v\in MEP(\Theta ,\phi )=\text{Fix}(T_{r_n})$, we have

$$\begin{array}{rl}{\parallel u_n-v\parallel }^2& ={\parallel T_{r_n}{x}_n-T_{r_n}v\parallel }^2\\ \le 〈T_{r_n}{x}_n-T_{r_n}v,x_n-v〉\\ =\frac{1}{2}\{{\parallel u_n-v\parallel }^2+{\parallel x_n-v\parallel }^2-{\parallel T_{r_n}{x}_n-T_{r_n}v-(x_n-v)\parallel }^2\}.\end{array}$$

This immediately yields

$${\parallel u_n-v\parallel }^2\le {\parallel x_n-v\parallel }^2-{\parallel x_n-u_n\parallel }^2.$$

(3.9)

Let $p\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$. We have

$$\begin{array}{rcl}\parallel {\tilde{u}}_n-p\parallel & =& \parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-P_C(I-\lambda \mathrm{\nabla }f)p\parallel \\ \le & \parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p\parallel \\ +\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-P_C(I-\lambda \mathrm{\nabla }f)p\parallel \\ \le & \parallel u_n-p\parallel +\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-P_C(I-\lambda \mathrm{\nabla }f)p\parallel \\ \le & \parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel .\end{array}$$

(3.10)

Note that

$$\begin{array}{rl}{y}_n-p& ={\theta }_n\gamma S{x}_n-{\theta }_n\mu Fp+(I-{\theta }_n\mu F){\tilde{u}}_n-(I-{\theta }_n\mu F)p\\ ={\theta }_n(\gamma S{x}_n-\mu Fp)+(1-{\theta }_n)({\tilde{u}}_n-p)+{\theta }_n[(I-\mu F){\tilde{u}}_n-(I-\mu F)p]\\ ={\theta }_n(\gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p)+(1-{\theta }_n)({\tilde{u}}_n-p).\end{array}$$

Hence we have

$$y_n-{\tilde{u}}_n={\theta }_n(\gamma S{x}_n+(I-\mu F){\tilde{u}}_n-{\tilde{u}}_n).$$

By Lemmas 2.10 and 2.12, we have from (3.9) and (3.10) that

$$\begin{array}{c}{\parallel y_n-p\parallel }^2\hfill \\ ={\parallel {\theta }_n(\gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p)+(1-{\theta }_n)({\tilde{u}}_n-p)\parallel }^2\hfill \\ ={\theta }_n{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+(1-{\theta }_n){\parallel {\tilde{u}}_n-p\parallel }^2\hfill \\ -{\theta }_n(1-{\theta }_n){\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-{\tilde{u}}_n\parallel }^2\hfill \\ ={\theta }_n{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+(1-{\theta }_n){\parallel {\tilde{u}}_n-p\parallel }^2-\frac{1-{\theta }_n}{{\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2\hfill \\ \le {\theta }_n{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\parallel {\tilde{u}}_n-p\parallel }^2-\frac{1-{\theta }_n}{{\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2.\hfill \end{array}$$

(3.11)

Furthermore, utilizing Lemmas 2.11 and 2.12 we have from (3.9) and (3.10) that

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ ={\parallel {\beta }_n\gamma V{y}_n-{\beta }_n\mu FTp+(I-{\beta }_n\mu F)T{\tilde{u}}_n-(I-{\beta }_n\mu F)Tp\parallel }^2\hfill \\ \le {(\parallel {\beta }_n\gamma V{y}_n-{\beta }_n\mu FTp\parallel +\parallel (I-{\beta }_n\mu F)T{\tilde{u}}_n-(I-{\beta }_n\mu F)Tp\parallel )}^2\hfill \\ \le {({\beta }_n\parallel \gamma V{y}_n-\mu Fp\parallel +(1-{\beta }_n\tau )\parallel {\tilde{u}}_n-p\parallel )}^2\hfill \\ \le {\beta }_n\frac{1}{\tau }{\parallel \gamma V{y}_n-\mu Fp\parallel }^2+(1-{\beta }_n\tau ){\parallel {\tilde{u}}_n-p\parallel }^2\hfill \\ \le {\beta }_n\frac{1}{\tau }[{\parallel \gamma V{y}_n-\gamma Vp\parallel }^2+2〈\gamma Vp-\mu Fp,\gamma V{y}_n-\mu Fp〉]+(1-{\beta }_n\tau ){\parallel {\tilde{u}}_n-p\parallel }^2\hfill \\ \le {\beta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel y_n-p\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel +(1-{\beta }_n\tau ){\parallel {\tilde{u}}_n-p\parallel }^2\hfill \\ \le {\beta }_n\frac{{\gamma }^2{\rho }^2}{\tau }[{\theta }_n{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\parallel {\tilde{u}}_n-p\parallel }^2-\frac{1-{\theta }_n}{{\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2]\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel +(1-{\beta }_n\tau ){\parallel {\tilde{u}}_n-p\parallel }^2\hfill \\ =(1-{\beta }_n(\tau -\frac{{\gamma }^2{\rho }^2}{\tau })){\parallel {\tilde{u}}_n-p\parallel }^2+{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ -\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel {\tilde{u}}_n-p\parallel }^2+{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ -\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {(\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )}^2+{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ -\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel u_n-p\parallel }^2+\lambda {\alpha }_n\parallel p\parallel (2\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )\hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2-\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel x_n-u_n\parallel }^2+\lambda {\alpha }_n\parallel p\parallel (2\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )\hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2-\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

(3.12)

It turns out therefore that

$$\begin{array}{c}{\parallel x_n-u_n\parallel }^2+\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2+\lambda {\alpha }_n\parallel p\parallel (2\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )\hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel +\lambda {\alpha }_n\parallel p\parallel (2\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )\hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

Then it is clear that

$$\begin{array}{rcl}{\parallel x_n-u_n\parallel }^2& \le & (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel +\lambda {\alpha }_n\parallel p\parallel (2\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )\\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\end{array}$$

Since ${\alpha }_n\to 0$, ${\beta }_n\to 0$, ${\theta }_n\to 0$, and $\parallel x_{n+1}-x_n\parallel \to 0$, we conclude that

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

Furthermore,

$$\begin{array}{c}\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2\hfill \\ \le (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel +\lambda {\alpha }_n\parallel p\parallel (2\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )\hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

This yields

$$\begin{array}{c}\frac{{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau }{\parallel y_n-{\tilde{u}}_n\parallel }^2\hfill \\ \le (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\frac{{\theta }_n}{{\beta }_n}\parallel x_n-x_{n+1}\parallel +\frac{{\alpha }_n{\theta }_n}{{\beta }_n}\lambda \parallel p\parallel (2\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )\hfill \\ +{\theta }_n^2\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\theta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

Since $\frac{\parallel x_{n+1}-x_n\parallel }{{\theta }_n}\to 0$, $\frac{{\alpha }_n+{\beta }_n}{{\theta }_n}\to 0$, and $\frac{{\theta }_n^2}{{\beta }_n}\to 0$ as $n\to \mathrm{\infty }$, we have

$$\underset{n\to \mathrm{\infty }}{lim}\frac{{\theta }_n}{{\beta }_n}\parallel x_n-x_{n+1}\parallel =\underset{n\to \mathrm{\infty }}{lim}\frac{\parallel x_n-x_{n+1}\parallel }{{\theta }_n}\cdot \frac{{\theta }_n^2}{{\beta }_n}=0$$

and

$$\underset{n\to \mathrm{\infty }}{lim}\frac{{\alpha }_n{\theta }_n}{{\beta }_n}=\underset{n\to \mathrm{\infty }}{lim}\frac{{\alpha }_n}{{\theta }_n}\cdot \frac{{\theta }_n^2}{{\beta }_n}=0.$$

Therefore, from the last inequality we have

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

Step 3. ${lim}_{n\to \mathrm{\infty }}\parallel u_n-{\tilde{u}}_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$.

Let $p\in Fix(T)\cap Fix(\Gamma )\cap \Xi$. Utilizing Lemmas 2.6 and 2.11 we have from (3.12) that

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ \le {\parallel {\tilde{u}}_n-p\parallel }^2+{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ -\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-P_C(I-\lambda \mathrm{\nabla }f)p\parallel }^2+{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel (I-\lambda \mathrm{\nabla }f)u_n-(I-\lambda \mathrm{\nabla }f)p-\lambda {\alpha }_n{u}_n\parallel }^2+{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel (I-\lambda \mathrm{\nabla }f)u_n-(I-\lambda \mathrm{\nabla }f)p\parallel }^2-2\lambda {\alpha }_n〈u_n,(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p〉\hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel u_n-p\parallel }^2+\lambda (\lambda -\frac{2}{L}){\parallel \mathrm{\nabla }f(u_n)-\mathrm{\nabla }f(p)\parallel }^2\hfill \\ +2\lambda {\alpha }_n\parallel u_n\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel x_n-p\parallel }^2+\lambda (\lambda -\frac{2}{L}){\parallel \mathrm{\nabla }f(u_n)-\mathrm{\nabla }f(p)\parallel }^2\hfill \\ +2\lambda {\alpha }_n\parallel u_n\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

Hence,

$$\begin{array}{c}\lambda (\frac{2}{L}-\lambda ){\parallel \mathrm{\nabla }f(u_n)-\mathrm{\nabla }f(p)\parallel }^2\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2+2\lambda {\alpha }_n\parallel u_n\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel +2\lambda {\alpha }_n\parallel u_n\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

Since ${\alpha }_n\to 0$, ${\beta }_n\to 0$, ${\theta }_n\to 0$, and $\parallel x_{n+1}-x_n\parallel \to 0$, it follows from $0<\lambda <\frac{2}{L}$ that ${lim}_{n\to \mathrm{\infty }}\parallel \mathrm{\nabla }f(u_n)-\mathrm{\nabla }f(p)\parallel =0$, and hence

$$\underset{n\to \mathrm{\infty }}{lim}\parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel =0.$$

Furthermore, from the firm nonexpansiveness of $P_C$ we obtain

$$\begin{array}{rcl}{\parallel {\tilde{u}}_n-p\parallel }^2& =& {\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-P_C(I-\lambda \mathrm{\nabla }f)p\parallel }^2\\ \le & 〈(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p,{\tilde{u}}_n-p〉\\ =& \frac{1}{2}\{{\parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel }^2+{\parallel {\tilde{u}}_n-p\parallel }^2\\ -{\parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p-({\tilde{u}}_n-p)\parallel }^2\}\\ \le & \frac{1}{2}\{{\parallel u_n-p\parallel }^2+2\lambda \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \\ +{\parallel {\tilde{u}}_n-p\parallel }^2-{\parallel u_n-{\tilde{u}}_n\parallel }^2+2\lambda 〈u_n-{\tilde{u}}_n,\mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)〉\\ -{\lambda }^2{\parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel }^2\}.\end{array}$$

Consequently,

$$\begin{array}{c}{\parallel {\tilde{u}}_n-p\parallel }^2\hfill \\ \le {\parallel u_n-p\parallel }^2-{\parallel u_n-{\tilde{u}}_n\parallel }^2+2\lambda \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +2\lambda 〈u_n-{\tilde{u}}_n,\mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)〉-{\lambda }^2{\parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel }^2.\hfill \end{array}$$

Thus, from (3.12) we have

$$\begin{array}{c}{\parallel x_{n+1}-p\parallel }^2\hfill \\ \le {\parallel {\tilde{u}}_n-p\parallel }^2+{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ -\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2+{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel u_n-p\parallel }^2-{\parallel u_n-{\tilde{u}}_n\parallel }^2+2\lambda \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +2\lambda 〈u_n-{\tilde{u}}_n,\mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)〉-{\lambda }^2{\parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel }^2\hfill \\ +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2-\frac{{\beta }_n{\gamma }^2{\rho }^2(1-{\theta }_n)}{\tau {\theta }_n}{\parallel y_n-{\tilde{u}}_n\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel u_n-{\tilde{u}}_n\parallel }^2+2\lambda \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +2\lambda \parallel u_n-{\tilde{u}}_n\parallel \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

This implies that

$$\begin{array}{c}{\parallel u_n-{\tilde{u}}_n\parallel }^2\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2+2\lambda \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +2\lambda \parallel u_n-{\tilde{u}}_n\parallel \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel \hfill \\ \le (\parallel x_n-p\parallel +\parallel x_{n+1}-p\parallel )\parallel x_n-x_{n+1}\parallel \hfill \\ +2\lambda \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel \parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n-(I-\lambda \mathrm{\nabla }f)p\parallel \hfill \\ +2\lambda \parallel u_n-{\tilde{u}}_n\parallel \parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel +{\beta }_n{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }{\parallel \gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ +{\beta }_n\frac{2}{\tau }\parallel \gamma Vp-\mu Fp\parallel \parallel \gamma V{y}_n-\mu Fp\parallel .\hfill \end{array}$$

Since ${\theta }_n\to 0$, ${\beta }_n\to 0$, $\parallel x_n-x_{n+1}\parallel \to 0$, and $\parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel \to 0$, it follows that

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

This, together with $\parallel x_n-u_n\parallel \to 0$ and $\parallel y_n-{\tilde{u}}_n\parallel \to 0$ (due to Step 2), implies that

$$\parallel x_n-y_n\parallel \le \parallel x_n-u_n\parallel +\parallel u_n-{\tilde{u}}_n\parallel +\parallel {\tilde{u}}_n-y_n\parallel \to 0\text{as }n\to \mathrm{\infty },$$

and thus

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

Step 4. ${\omega }_w(x_n)\subset Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$; moreover, if $\parallel x_n-y_n\parallel =o({\theta }_n)$ in addition, then ${\omega }_w(x_n)\subset \Xi$.

Let $p^{\ast }\in {\omega }_w(x_n)$. Then there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\rightharpoonup p^{\ast }$. Since

$$\begin{array}{rl}{x}_{n+1}-{\tilde{u}}_n& ={\beta }_n\gamma S{y}_n+(I-{\beta }_n\mu F)T{\tilde{u}}_n-{\tilde{u}}_n\\ ={\beta }_n(\gamma S{y}_n-\mu FT{\tilde{u}}_n)+T{\tilde{u}}_n-{\tilde{u}}_n,\end{array}$$

we have

$$\begin{array}{rl}\parallel T{\tilde{u}}_n-{\tilde{u}}_n\parallel & =\parallel x_{n+1}-{\tilde{u}}_n-{\beta }_n(\gamma S{y}_n-\mu FT{\tilde{u}}_n)\parallel \\ \le \parallel x_{n+1}-{\tilde{u}}_n\parallel +{\beta }_n\parallel \gamma S{y}_n-\mu FT{\tilde{u}}_n\parallel \\ \le \parallel x_{n+1}-x_n\parallel +\parallel x_n-u_n\parallel +\parallel u_n-{\tilde{u}}_n\parallel +{\beta }_n\parallel \gamma S{y}_n-\mu FT{\tilde{u}}_n\parallel .\end{array}$$

Hence from $\parallel x_{n+1}-x_n\parallel \to 0$, $\parallel x_n-u_n\parallel \to 0$, $\parallel u_n-{\tilde{u}}_n\parallel \to 0$, and ${\beta }_n\to 0$, we get

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

Since $\parallel x_n-u_n\parallel \to 0$ and $\parallel u_n-{\tilde{u}}_n\parallel \to 0$, we have ${\tilde{u}}_{n_i}\rightharpoonup p^{\ast }$. Utilizing Lemma 2.8 we derive $p^{\ast }\in Fix(T)$.

Let us show that $p^{\ast }\in MEP(\Theta ,\phi )$. As a matter of fact, since $u_n=T_{r_n}{x}_n$, for any $y\in C$ we have

$$\Theta (u_n,y)+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0.$$

It follows from (A2) that

$$\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge \Theta (y,u_n).$$

Replacing n by $n_i$, we have

$$\phi (y)-\phi (u_{n_i})+\frac{1}{r_{n_i}}〈y-u_{n_i},u_{n_i}-x_{n_i}〉\ge \Theta (y,u_{n_i}).$$

Since $\frac{u_{n_i}-x_{n_i}}{r_{n_i}}\to 0$ and $u_{n_i}\rightharpoonup p^{\ast }$, it follows from (A4) that

$$0\ge -\phi (y)+\phi \left(p^{\ast }\right)+\Theta (y,p^{\ast }),\mathrm{\forall }y\in C.$$

Put $z_t=ty+(1-t)p^{\ast }$ for all $t\in (0,1]$ and $y\in C$. We have $z_t\in C$ and

$$0\ge -\phi (z_t)+\phi \left(p^{\ast }\right)+\Theta (z_t,p^{\ast }).$$

(3.13)

Utilizing (A1), (A4), and (3.13), we have

$$\begin{array}{rl}0& =\Theta (z_t,z_t)+\phi (z_t)-\phi (z_t)\\ \le t\Theta (z_t,y)+(1-t)\Theta (z_t,p^{\ast })+t\phi (y)+(1-t)\phi \left(p^{\ast }\right)-\phi (z_t)\\ \le t(\Theta (z_t,y)+\phi (y)-\phi (z_t))+(1-t)(\Theta (z_t,p^{\ast })+\phi \left(p^{\ast }\right)-\phi (z_t))\\ \le t(\Theta (z_t,y)+\phi (y)-\phi (z_t)),\end{array}$$

and hence

$$0\le \Theta (z_t,y)+\phi (y)-\phi (z_t).$$

(3.14)

Letting $t\to 0$ in (3.14) and utilizing (A3), we get, for each $y\in C$,

$$0\le \Theta (p^{\ast },y)+\phi (y)-\phi \left(p^{\ast }\right).$$

Hence, $p^{\ast }\in MEP(\Theta ,\phi )$.

Let us show that $p^{\ast }\in \Gamma$. From $\parallel x_n-u_n\parallel \to 0$ and $\parallel u_n-{\tilde{u}}_n\parallel \to 0$, we know that $u_{n_i}\rightharpoonup p^{\ast }$ and ${\tilde{u}}_{n_i}\rightharpoonup p^{\ast }$. Define

$$\tilde{T}v=\{\begin{array}{ll}\mathrm{\nabla }f(v)+N_{C}v,& \text{if }v\in C,\\ \mathrm{\varnothing },& \text{if }v\notin C,\end{array}$$

where

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

Then $\tilde{T}$ is maximal monotone and $0\in \tilde{T}v$ if and only if $v\in VI(C,\mathrm{\nabla }f)$; see [28] for more details. Let $(v,w)\in Gph(\tilde{T})$. Then we have

$$w\in \tilde{T}v=\mathrm{\nabla }f(v)+N_{C}v$$

and hence,

$$w-\mathrm{\nabla }f(v)\in N_{C}v.$$

Therefore,

$$〈v-u,w-\mathrm{\nabla }f(v)〉\ge 0,\mathrm{\forall }u\in C.$$

On the other hand, from

$${\tilde{u}}_n=P_C(u_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_n))\text{and}v\in C,$$

we have

$$〈u_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-{\tilde{u}}_n,{\tilde{u}}_n-v〉\ge 0,$$

and hence

$$〈v-{\tilde{u}}_n,\frac{{\tilde{u}}_n-u_n}{\lambda }+\mathrm{\nabla }{f}_{{\alpha }_n}(u_n)〉\ge 0.$$

Therefore, from

$$w-\mathrm{\nabla }f(v)\in N_C(v)\text{and}{\tilde{u}}_{n_i}\in C,$$

we have

$$\begin{array}{rcl}〈v-{\tilde{u}}_{n_i},w〉& \ge & 〈v-{\tilde{u}}_{n_i},\mathrm{\nabla }f(v)〉\\ \ge & 〈v-{\tilde{u}}_{n_i},\mathrm{\nabla }f(v)〉-〈v-{\tilde{u}}_{n_i},\frac{{\tilde{u}}_{n_i}-u_{n_i}}{\lambda }+\mathrm{\nabla }{f}_{{\alpha }_{n_i}}(u_{n_i})〉\\ =& 〈v-{\tilde{u}}_{n_i},\mathrm{\nabla }f(v)〉-〈v-{\tilde{u}}_{n_i},\frac{{\tilde{u}}_{n_i}-u_{n_i}}{\lambda }+\mathrm{\nabla }f(u_{n_i})〉-{\alpha }_{n_i}〈v-{\tilde{u}}_{n_i},u_{n_i}〉\\ =& 〈v-{\tilde{u}}_{n_i},\mathrm{\nabla }f(v)-\mathrm{\nabla }f({\tilde{u}}_{n_i})〉+〈v-{\tilde{u}}_{n_i},\mathrm{\nabla }f({\tilde{u}}_{n_i})-\mathrm{\nabla }f(u_{n_i})〉\\ -〈v-{\tilde{u}}_{n_i},\frac{{\tilde{u}}_{n_i}-u_{n_i}}{\lambda }〉-{\alpha }_{n_i}〈v-{\tilde{u}}_{n_i},u_{n_i}〉\\ \ge & 〈v-{\tilde{u}}_{n_i},\mathrm{\nabla }f({\tilde{u}}_{n_i})-\mathrm{\nabla }f(u_{n_i})〉-〈v-{\tilde{u}}_{n_i},\frac{{\tilde{u}}_{n_i}-u_{n_i}}{\lambda }〉-{\alpha }_{n_i}〈v-{\tilde{u}}_{n_i},u_{n_i}〉.\end{array}$$

Hence, we obtain

$$〈v-p^{\ast },w〉\ge 0\text{as }i\to \mathrm{\infty }.$$

Since $\tilde{T}$ is maximal monotone, we have $p^{\ast }\in {\tilde{T}}^{-1}0$, and hence, $p^{\ast }\in VI(C,\mathrm{\nabla }f)$, which leads to $p^{\ast }\in \Gamma$. Consequently, $p^{\ast }\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$. This shows that ${\omega }_w(x_n)\subset Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$.

Utilizing Lemmas 2.11 and 2.12, we have for every $p\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$,

$$\begin{array}{c}{\parallel y_n-p\parallel }^2\hfill \\ ={\parallel {\theta }_n\gamma S{x}_n+(I-\mu F){\tilde{u}}_n-p\parallel }^2\hfill \\ ={\parallel {\theta }_n(\gamma Sp-\mu Fp)+{\theta }_n(\gamma S{x}_n-\gamma Sp)+(I-{\theta }_n\mu F){\tilde{u}}_n-(I-{\theta }_n\mu F)p\parallel }^2\hfill \\ \le {\parallel {\theta }_n(\gamma S{x}_n-\gamma Sp)+(I-{\theta }_n\mu F){\tilde{u}}_n-(I-{\theta }_n\mu F)p\parallel }^2+2{\theta }_n〈\gamma Sp-\mu Fp,y_n-p〉\hfill \\ \le {[\parallel {\theta }_n(\gamma S{x}_n-\gamma Sp)\parallel +\parallel (I-{\theta }_n\mu F){\tilde{u}}_n-(I-{\theta }_n\mu F)p\parallel ]}^2+2{\theta }_n〈\gamma Sp-\mu Fp,y_n-p〉\hfill \\ \le {[{\theta }_n\gamma \parallel x_n-p\parallel +(1-{\theta }_n\tau )\parallel {\tilde{u}}_n-p\parallel ]}^2+2{\theta }_n〈\gamma Sp-\mu Fp,y_n-p〉\hfill \\ \le {[{\theta }_n\gamma \parallel x_n-p\parallel +(1-{\theta }_n\tau )(\parallel u_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )]}^2+2{\theta }_n〈\gamma Sp-\mu Fp,y_n-p〉\hfill \\ \le {[{\theta }_n\gamma \parallel x_n-p\parallel +(1-{\theta }_n\tau )(\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )]}^2+2{\theta }_n〈\gamma Sp-\mu Fp,y_n-p〉\hfill \\ \le {[(1-{\theta }_n(\tau -\gamma ))\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel ]}^2+2{\theta }_n〈\gamma Sp-\mu Fp,y_n-p〉\hfill \\ \le {(\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )}^2+2{\theta }_n〈\gamma Sp-\mu Fp,y_n-p〉.\hfill \end{array}$$

(3.15)

Suppose now that $\parallel x_n-y_n\parallel =o({\theta }_n)$ in addition. It follows from (3.15) that

$$\begin{array}{r}2〈\gamma Sp-\mu Fp,y_n-p〉\\ \le \frac{1}{{\theta }_n}[{(\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel )}^2-{\parallel y_n-p\parallel }^2]\\ \le \frac{1}{{\theta }_n}(\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel +\parallel y_n-p\parallel )(\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel -\parallel y_n-p\parallel )\\ \le \frac{1}{{\theta }_n}(\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel +\parallel y_n-p\parallel )(\parallel x_n-y_n\parallel +\lambda {\alpha }_n\parallel p\parallel ).\end{array}$$

This, together with $\frac{{\alpha }_n}{{\theta }_n}\to 0$ and $\frac{\parallel x_n-y_n\parallel }{{\theta }_n}\to 0$, leads to

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma Sp-\mu Fp,y_n-p〉\le 0.$$

Observe that

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma Sp-\mu Fp,x_n-p〉\hfill \\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(〈\gamma Sp-\mu Fp,x_n-y_n〉+〈\gamma Sp-\mu Fp,y_n-p〉)\hfill \\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma Sp-\mu Fp,y_n-p〉\le 0.\hfill \end{array}$$

So, it follows from $x_{n_i}\rightharpoonup p^{\ast }$ that

$$〈\gamma Sp-\mu Fp,p^{\ast }-p〉\le 0,\mathrm{\forall }p\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma .$$

Note also that $0<\gamma \le \tau$ and

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

It is clear that

$$〈(\mu F-\gamma S)x-(\mu F-\gamma S)y,x-y〉\ge (\mu \eta -\gamma ){\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Hence, it follows from $0<\gamma \le \tau \le \mu \eta$ that $\mu F-\gamma S$ is monotone. Since $p^{\ast }\in {\omega }_w(x_n)\subset Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$, by Minty’s lemma [1] we have

$$〈\gamma S{p}^{\ast }-\mu F{p}^{\ast },p-p^{\ast }〉\le 0,\mathrm{\forall }p\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma ;$$

that is, $p^{\ast }\in \Xi$. This shows that ${\omega }_w(x_n)\subset \Xi$. □

Theorem 3.2 Assuming the conditions in Theorem  3.1. We have:

• (i) $\{x_n\}$ and $\{y_n\}$ both converge strongly to an element $x^{\ast }\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$, which is a unique solution of the variational inequality

  $$〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x-x^{\ast }〉\le 0,\mathrm{\forall }x\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma .$$

• (ii) $\{x_n\}$ and $\{y_n\}$ both converge strongly to a unique solution of THVI (1.6) if $\parallel x_n-y_n\parallel =o({\theta }_n)$ in addition.

Proof Utilizing Lemmas 2.11 and 2.12 we get from (3.15)

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

(3.16)

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

Note that $\mu F-\gamma V:H\to H$ is $(\mu \kappa +\gamma \rho )$-Lipschitzian and $(\mu \eta -\gamma \rho )$-strongly monotone, namely

$$\parallel (\mu F-\gamma V)x-(\mu F-\gamma V)y\parallel \le (\mu \kappa +\gamma \rho )\parallel x-y\parallel ,\mathrm{\forall }x,y\in H$$

and

$$〈(\mu F-\gamma V)x-(\mu F-\gamma V)y,x-y〉\ge (\mu \eta -\gamma \rho ){\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Hence there exists a unique solution $x^{\ast }\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$ of the variational inequality problem

$$〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x-x^{\ast }〉\le 0,\mathrm{\forall }x\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma .$$

(3.17)

Since the sequence $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_n-x^{\ast }〉=\underset{i\to \mathrm{\infty }}{lim}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_{n_i}-x^{\ast }〉.$$

(3.18)

Also, since H is reflexive and $\{x_n\}$ is bounded, without loss of generality we may assume that $x_{n_i}\rightharpoonup \overline{x}\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$ (due to Theorem 3.1(i)). Taking into consideration that $x^{\ast }$ is the unique solution of VIP (3.17), we obtain from (3.18)

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_{n+1}-x^{\ast }〉\hfill \\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_n-x^{\ast }〉+〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_{n+1}-x_n〉)\hfill \\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_n-x^{\ast }〉=\underset{i\to \mathrm{\infty }}{lim}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_{n_i}-x^{\ast }〉\hfill \\ =〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },\overline{x}-x^{\ast }〉\le 0.\hfill \end{array}$$

(3.19)

Putting $p=x^{\ast }$, from (3.16) we conclude that

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

(3.20)

Since ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n=+\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n<+\mathrm{\infty }$, and ${\theta }_n\to 0$ as $n\to \mathrm{\infty }$, it follows from (3.19) that ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n\frac{{\tau }^2-{\gamma }^2{\rho }^2}{\tau }=+\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n\lambda \parallel x^{\ast }\parallel (2\parallel x_n-x^{\ast }\parallel +\lambda {\alpha }_n\parallel x^{\ast }\parallel )<+\mathrm{\infty }$, and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{\tau }{{\tau }^2-{\gamma }^2{\rho }^2}\{2{\theta }_n\frac{{\gamma }^2{\rho }^2}{\tau }\parallel \gamma S{x}^{\ast }-\mu F{x}^{\ast }\parallel \parallel y_n-x^{\ast }\parallel +2〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_{n+1}-x^{\ast }〉\}\le 0.$$

Applying Lemma 2.7 to (3.20), we get

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

This, together with $\parallel x_n-y_n\parallel \to 0$, implies that

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

From now on, we suppose that $\parallel x_n-y_n\parallel =o({\theta }_n)$. Then by Theorem 3.1(ii) we know that ${\omega }_w(x_n)\subset \Xi$. Since $\mu F-\gamma V:H\to H$ is $(\mu \kappa +\gamma \rho )$-Lipschitzian and $(\mu \eta -\gamma \rho )$-strongly monotone, there exists a unique solution $x^{\ast }\in \Xi$ of the variational inequality problem

$$〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x-x^{\ast }〉\le 0,\mathrm{\forall }x\in \Xi .$$

(3.21)

Since the sequence $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_n-x^{\ast }〉=\underset{i\to \mathrm{\infty }}{lim}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_{n_i}-x^{\ast }〉.$$

(3.22)

Again, since H is reflexive and $\{x_n\}$ is bounded, without loss of generality we may assume that $x_{n_i}\rightharpoonup \overline{x}\in \Xi$ (due to Theorem 3.1(ii)). Taking into account that $x^{\ast }$ is the unique solution of VIP (3.21), we deduce from (3.22) that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x_{n+1}-x^{\ast }〉\le 〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },\overline{x}-x^{\ast }〉\le 0.$$

Putting $p=x^{\ast }$, from (3.16) we immediately infer that

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

Repeating the same arguments as above, we can readily see that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x^{\ast }\parallel =0,$$

which, together with $\parallel x_n-y_n\parallel \to 0$, yields

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

This completes the proof. □

Remark 3.3 Our iterative algorithm (3.1) is very different from Xu’s iterative ones in [2], and Yao et al.’s iterative one in [8]. Here, the two-step iterative scheme in [8] for two nonexpansive mappings and the gradient-projection iterative schemes in [2] for MP (1.1) are extended to develop our three-step iterative scheme (3.1) with regularization for the THVI (1.6). It is worth pointing out that without assuming the conditions that $\parallel x_n-y_n\parallel =o({\theta }_n)$ and that $\parallel x-Tx\parallel \ge kDist(x,Fix(T))$, $\mathrm{\forall }x\in C$ for some constant $k>0$, our three-step iterative scheme (3.1) converges strongly to an element $x^{\ast }\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$, which is a unique solution of the variational inequality

$$〈\gamma V{x}^{\ast }-\mu F{x}^{\ast },x-x^{\ast }〉\le 0,\mathrm{\forall }x\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma .$$

See Theorem 3.2(i).

Remark 3.4 As an example, we consider the following sequences:

• (a) ${\alpha }_n=\frac{1}{n^{1+s+t}}$, ${\beta }_n=\frac{1}{n^s}$, and ${\theta }_n=\frac{1}{n^t}$ where $t\in (0,\frac{1}{3}]$ and $s\in (t,2t)$ or $t\in (\frac{1}{3},\frac{1}{2})$, $s\in (t,1-t)$;

• (b) $r_n=\frac{1}{2}+\frac{1}{n^{1+s+t}}$.

They satisfy the hypotheses on the parameter sequences in Theorems 3.1 and 3.2.

Remark 3.5 Our Theorems 3.1 and 3.2 improve, extend, supplement, and develop [[8], Theorems 3.1 and 3.2] and [[2], Theorems 5.2 and 6.1] in the following aspects:

• (a) Our THVI (1.6) with the unique solution $x^{\ast }\in \Xi$ satisfying

  $$x^{\ast }=P_{Fix(T)\cap Fix(\Gamma )\cap \Xi }(I-(\mu F-\gamma S))x^{\ast }$$

  is more general than the problem of finding an element $\tilde{x}\in C$ satisfying $\tilde{x}=P_{Fix(T)}S\tilde{x}$ in [8] and the problem of finding an element $\tilde{x}\in \Xi$ in [2].

• (b) Our three-step iterative algorithm (3.1) for THVI (1.6) is more flexible, more advantageous and more subtle than Xu’s iterative ones in [2] and than Yao et al.’s two-step iterative one in [8], because, e.g., it drops the requirement of $\parallel x-Tx\parallel \ge kDist(x,Fix(T))$, $\mathrm{\forall }x\in C$ for some $k>0$ in [[8], Theorem 3.2(v)].

• (c) The arguments and techniques in our Theorems 3.1 and 3.2 are very different from the ones in [[8], Theorems 3.1 and 3.2] and in [[2], Theorems 5.2 and 6.1] because we utilize the properties of resolvent operators and maximal monotone mappings (Lemmas 2.5, 2.6 and 2.13), the convergence criteria of real sequences (Lemma 2.7), and the contractive coefficient estimates for the contractions associated with nonexpansive mappings (Lemma 2.12).

• (d) Compared with the proofs of [[2], Theorems 5.2 and 6.1], the proofs of our Theorems 3.1 and 3.2 derive ${lim}_{n\to \mathrm{\infty }}\parallel u_n-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})u_n\parallel =0$ via the argument showing ${lim}_{n\to \mathrm{\infty }}\parallel \mathrm{\nabla }{f}_{{\alpha }_n}(u_n)-\mathrm{\nabla }f(p)\parallel =0$, $\mathrm{\forall }p\in Fix(T)\cap MEP(\Theta ,\phi )\cap \Gamma$ (see Step 3 in the proof of Theorem 3.1).