Regularized gradient-projection methods for equilibrium and constrained convex minimization problems

Abstract

In this article, based on Marino and Xu’s method, an iterative method which combines the regularized gradient-projection algorithm (RGPA) and the averaged mappings approach is proposed for finding a common solution of equilibrium and constrained convex minimization problems. Under suitable conditions, it is proved that the sequences generated by implicit and explicit schemes converge strongly. The results of this paper extend and improve some existing results.

MSC:58E35, 47H09, 65J15.

1 Introduction

Let H be a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the induced norm $\parallel \cdot \parallel$. Let C be a nonempty, closed and convex subset of H. We need some nonlinear operators which are introduced below.

Let $T,A:H\to H$ be nonlinear operators.

• T is nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in H$.

• T is Lipschitz continuous if there exists a constant $L>0$ such that $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel$ for all $x,y\in H$.

• A is a strongly positive bounded linear operator if there exists a constant $\overline{\gamma }$ such that $〈Ax,x〉\ge \overline{\gamma }{\parallel x\parallel }^2$ for all $x\in H$.

• $A:H\to H$ is monotone if $〈x-y,Ax-Ay〉\ge 0$ for all $x,y\in H$.

• Given is a number $\eta >0$, $A:H\to H$ is η-strongly monotone if $〈x-y,Ax-Ay〉\ge \eta {\parallel x-y\parallel }^2$ for all $x,y\in H$.

• Given is a number $\upsilon >0$. $A:H\to H$ is υ-inverse strongly monotone (υ-ism) if $〈x-y,Ax-Ay〉\ge \upsilon {\parallel Ax-Ay\parallel }^2$ for all $x,y\in H$.

It is known that inverse strongly monotone operators have been studied widely (see [1–3]) and applied to solve practical problems in various fields; for instance, in traffic assignment problems (see [4, 5]).

• T is firmly nonexpansive if and only if $2T-I$ is nonexpansive or, equivalently, $〈x-y,Tx-Ty〉\ge {\parallel Tx-Ty\parallel }^2$ for all $x,y\in H$.

• $T:H\to H$ is said to be an averaged mapping if $T=(1-\alpha )I+\alpha S$, where α is a number in $(0,1)$ and $S:H\to H$ is nonexpansive. In particular, projections are $(1/2)$-averaged mappings.

Averaged mappings have received many investigations, see ([6–10]).

Let ϕ be a bifunction of $C\times C$ into ℝ, where ℝ is the set of real numbers. The equilibrium problem for $\varphi :C\times C\to \mathbb{R}$ is to find $x\in C$ such that

$$\varphi (x,y)\ge 0\text{for all }y\in C.$$

(1.1)

The set of solutions of (1.1) is denoted by $EP(\varphi )$. Given a mapping $T:C\to H$, let $\varphi (x,y)=〈Tx,y-x〉$ for all $x,y\in C$. Then $z\in EP(\varphi )$ if and only if $〈Tz,y-z〉\ge 0$ for all $y\in C$, i.e., z is a solution of the variational inequality. Numerous problems in physics, optimization and economics reduce to finding a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [11–15].

In 2000, Moudafi [16] introduced the viscosity approximation method for nonexpansive mappings. Let h be a contraction on H, starting with an arbitrary initial $x_0\in H$, define a sequence $\{x_n\}$ recursively by

$$x_{n+1}={\alpha }_{n}h(x_n)+(1-{\alpha }_n)T{x}_n,n\ge 0,$$

(1.2)

where $\{{\alpha }_n\}$ is a sequence in $(0,1)$. Xu [17] proved that under certain conditions on $\{{\alpha }_n\}$, the sequence $\{x_n\}$ generated by (1.2) converges strongly to the unique solution $x^{\ast }\in F(T)$ of the variational inequality

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

We use $F(T)$ to denote the set of fixed points of the mapping T; that is, $F(T)=\{x\in H:x=Tx\}$.

In 2006, Marino and Xu [18] introduced a general iterative method for nonexpansive mappings. Let h be a contraction on H with a coefficient $\rho \in (0,1)$, and let A be a strongly positive bounded linear operator on H with a constant $\overline{\gamma }>0$. Starting with an arbitrary initial guess $x_0\in H$, define a sequence $\{x_n\}$ recursively by

$$x_{n+1}={\alpha }_n\gamma h(x_n)+(I-{\alpha }_{n}A)T{x}_n,n\ge 0,$$

(1.3)

where $\{{\alpha }_n\}$ is a sequence in $(0,1)$, and $0<\gamma <\overline{\gamma }/\rho$ is a constant. It is proved that the sequence $\{x_n\}$ converges strongly to the unique solution $x^{\ast }\in F(T)$ of the variational inequality

$$〈(\gamma h-A)x^{\ast },x-x^{\ast }〉\le 0\mathrm{\forall }x\in F(T).$$

For finding the common solution of $EP(\varphi )$ and a fixed point problem, Takahashi and Takahashi [11] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: $x_1\in H$ and

$$\{\begin{array}{c}\varphi (u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}h(x_n)+(1-{\alpha }_n)S{u}_n\hfill \end{array}$$

(1.4)

for all $n\in \mathbb{N}$, where $\{{\alpha }_n\}\subset (0,1)$ and $\{r_n\}\subset (0,\mathrm{\infty })$ satisfy some appropriate conditions. Further, they proved $\{x_n\}$ and $\{u_n\}$ converge strongly to $z\in F(S)\cap EP(\varphi )$, where $z=P_{F(S)\cap EP(\varphi )}h(z)$.

On the other hand, let $f:C\to \mathbb{R}$ be a convex function, and consider the following minimization problem:

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

(1.5)

Assume that the constrained convex minimization problem (1.5) is solvable, and let U denote the set of solutions of (1.5). Then the gradient-projection algorithm (GPA) generates a sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ according to the recursive formula

$$x_{n+1}={Proj}_C(I-{\gamma }_n\mathrm{\nabla }f)x_n,$$

where the parameters ${\gamma }_n$ are real positive numbers, and ${Proj}_C$ is the metric projection from H onto C. It is known that the convergence of the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ depends on the behavior of the gradient ∇f. If the gradient ∇f is only assumed to be inverse strongly monotone, then ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ can only be convergent weakly to a minimizer of (1.5). If the gradient ∇f is Lipschitz continuous and strongly monotone, then the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ can be convergent strongly to a minimizer of (1.5) provided the parameters ${\gamma }_n$ satisfy appropriate conditions.

As everyone knows, Xu [9] gave an averaged mapping approach to the gradient-projection method, and he constructed a counterexample which shows that the sequence generated by the gradient-projection method does not converge strongly in the infinite-dimensional space. Moreover, he presented two modifications of the gradient-projection method which are shown to have strong convergence.

In 2011, motivated by Xu, Ceng [19] proposed the following iterative algorithm:

$$x_{n+1}={Proj}_C[s_n\gamma V{x}_n+(I-s_n\mu F)T_n{x}_n],n\ge 0,$$

(1.6)

where $V:C\to H$ is an l-Lipschitzian mapping with a constant $l>0$, and $F:C\to H$ is a k-Lipschitzian and η-strongly monotone operator with constants $k,\eta >0$. Let $0<\mu <2\eta /k^2$, $0\le \gamma l<\tau$, and $\tau =1-\sqrt{1-\mu (2\eta -\mu k^2)}$. Let $T_n$ and $s_n$ satisfy $s_n=\frac{2-{\lambda }_{n}L}{4}$, ${Proj}_C(I-{\lambda }_n\mathrm{\nabla }f)=s_{n}I+(1-s_n)T_n$. Under suitable conditions, it is proved that the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ generated by (1.6) converges strongly to a minimizer $x^{\ast }$ of (1.5).

In 2012, Tian and Liu [20] introduced the following iterative method in a Hilbert space: $x_1\in C$ and

$$\{\begin{array}{c}\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_n\gamma V{u}_n+(I-{\alpha }_{n}A)T_n{u}_n\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(1.7)

where $u_n=Q_{{\beta }_n}{x}_n$, ${Proj}_C(I-{\lambda }_n\mathrm{\nabla }f)=s_{n}I+(1-s_n)T_n$, $s_n=\frac{2-{\lambda }_{n}L}{4}$ and $\{{\lambda }_n\}\subset (0,2/L)$, and $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{s_n\}$ satisfy appropriate conditions. Further, they proved the sequence $\{x_n\}$ converges strongly to a point $q\in U\cap EP(\varphi )$, which solves the variational inequality

$$〈(A-\gamma V)q,q-z〉\le 0,z\in U\cap EP(\varphi ).$$

It is the first time that the equilibrium and constrained convex minimization problems have been solved.

Since, in general, the minimization problem (1.5) has more than one solution, so regularization is needed. Now we consider the following 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, f is a convex function with a $1/L$-ism continuous gradient ∇f. Then the regularized GPA generates a sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ by the following recursive formula:

$$x_{n+1}={Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\alpha }_n})x_n={Proj}_C[x_n-\gamma (\mathrm{\nabla }f+{\alpha }_{n}I)(x_n)],$$

(1.8)

where the parameter ${\alpha }_n>0$, γ is a constant with $0<\gamma <2/L$, and ${Proj}_C$ is the metric projection from H onto C. We all know that the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ generated by algorithm (1.8) converges weakly to a minimizer of (1.5) in the setting of infinite-dimensional spaces (see [21]).

In this paper, motivated and inspired by the above results, we introduce a new iterative method: $x_1\in H$ and

$$\{\begin{array}{c}\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(1.9)

for finding a element of $U\cap EP(\varphi )$, where $u_n=Q_{{\beta }_n}{x}_n$, ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})={\theta }_{n}I+(1-{\theta }_n)T_{{\lambda }_n}$, ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$, $\gamma \in (0,2/L)$. Under appropriate conditions, it is proved that the sequence $\{x_n\}$ generated by (1.9) converges strongly to a point $z\in U\cap EP(\varphi )$, which solves the variational inequality

$$〈(A-rV)z,z-x〉\le 0,x\in U\cap EP(\varphi ).$$

2 Preliminaries

In this section we introduce some useful properties and lemmas which will be used in the proofs for the main results in the next section.

Some properties of averaged mappings are gathered in the proposition below.

Proposition 2.1 [7, 8]

Let the operators $S,T,V:H\to H$ be given:

• (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) The composition of finitely many averaged mappings is averaged. That is, if each of the mappings ${\{T_i\}}_{i=1}^N$ is averaged, then so is the composite $T_1\cdots T_N$. 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 the composite $T_1{T}_2$ is α-averaged, where $\alpha ={\alpha }_1+{\alpha }_2-{\alpha }_1{\alpha }_2$.

• (iii) If the mappings ${\{T_i\}}_{i=1}^N$ are averaged and have a common fixed point, then

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

Here the notation $Fix(T)$ denotes the set of fixed points of the mapping T; that is, $Fix(T):=\{x\in H:Tx=x\}$.

The following proposition gathers some results on the relationship between averaged mappings and inverse strongly monotone operators.

Proposition 2.2 [7, 22]

Let $T:H\to H$ be given. We have:

• (i) T is nonexpansive if and only if the complement $I-T$ is ($1/2$)-ism;

• (ii) If T is υ-ism, then for $\gamma >0$, γT is ($\upsilon /\gamma$)-ism;

• (iii) T is averaged if and only if the complement $I-T$ is υ-ism for some $\upsilon >1/2$; indeed, for $\alpha \in (0,1)$, T is α-averaged if and only if $I-T$ is ($1/2\alpha$)-ism.

Lemma 2.1 [9]

Assume that ${\{a_n\}}_{n=0}^{\mathrm{\infty }}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\gamma }_n{\delta }_n+{\beta }_n,n\ge 0,$$

where ${\{{\gamma }_n\}}_{n=0}^{\mathrm{\infty }}$ and ${\{{\beta }_n\}}_{n=0}^{\mathrm{\infty }}$ are sequences in $(0,1)$ and ${\{{\delta }_n\}}_{n=0}^{\mathrm{\infty }}$ is a sequence in ℝ such that

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

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

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

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

The so-called demiclosed principle for nonexpansive mappings will often be used.

Lemma 2.2 (Demiclosed principle [23])

Let C be a closed and convex subset of a Hilbert space H and let $T:C\to C$ be a nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing }$. If ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ is a sequence in C weakly converging to x and if ${\{(I-T)x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to y, then $(I-T)x=y$. In particular, if $y=0$, then $x\in Fix(T)$.

Lemma 2.3 [18]

Let H be a Hilbert space, let C be a closed and convex subset of H, let $V:C\to H$ be a Lipschitzian operator with a coefficient $l>0$, and let $A:C\to H$ be a strongly positive bounded linear operator with a coefficient $\overline{\gamma }>0$. Then, for $0<r<\overline{\gamma }/l$,

$$\begin{array}{c}〈x-y,(A-rV)x-(A-rV)y〉\hfill \\ \ge (\overline{\gamma }-rl){\parallel x-y\parallel }^2\mathrm{\forall }x,y\in C.\hfill \end{array}$$

That is, $A-rV$ is strongly monotone with a coefficient $\overline{\gamma }-rl$.

Recall the metric (nearest point) projection ${Proj}_C$ from a real Hilbert space H to a closed and convex subset C of H is defined as follows: given $x\in H$, ${Proj}_{C}x$ is the unique point in C with the property

$$\parallel x-{Proj}_{C}x\parallel =inf\{\parallel x-y\parallel :y\in C\}.$$

${Proj}_C$ is characterized as follows.

Lemma 2.4 Let C be a closed and convex subset of a real Hilbert space H. Given $x\in H$ and $y\in C$. Then $y={Proj}_{C}x$ if and only if the following inequality holds:

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

Lemma 2.5 [18]

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

For solving the equilibrium problem for a bifunction $\varphi :C\times C\to \mathbb{R}$, let us assume that ϕ satisfies the following conditions:

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

(A2) ϕ is monotone, i.e., $\varphi (x,y)+\varphi (y,x)\le 0$ for all $x,y\in C$;

(A3) for each $x,y,z\in C$, ${lim}_{t\downarrow 0}\varphi (tz+(1-t)x,y)\le \varphi (x,y)$;

(A4) for each $x\in C$, $y\mapsto \varphi (x,y)$ is convex and lower semicontinuous.

Lemma 2.6 [24]

Let C be a nonempty, closed and convex subset of H and let ϕ be a bifunction of $C\times C$ into ℝ satisfying (A1)-(A4). Let $r>0$ and $x\in H$. Then there exists $z\in C$ such that

$$\varphi (z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0\mathit{\text{for all}}y\in C.$$

Lemma 2.7 [13]

Assume that $\varphi :C\times C\to \mathbb{R}$ satisfies (A1)-(A4). For $r>0$ and $x\in H$, define a mapping $Q_r:H\to C$ as follows:

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

Then the following hold:

• (1) $Q_r$ is single-valued;

• (2) $Q_r$ is firmly nonexpansive, i.e., ${\parallel Q_{r}x-Q_{r}y\parallel }^2\le 〈Q_{r}x-Q_{r}y,x-y〉$ for any $x,y\in H$;

• (3) $F(Q_r)=EP(\varphi )$;

• (4) $EP(\varphi )$ is closed and convex.

We adopt the following notation:

• $x_n\to x$ means that $x_n\to x$ strongly;

• $x_n\rightharpoonup x$ means that $x_n\to x$ weakly.

3 Main results

Recall that throughout this paper we always denote U as the solution set of the constrained convex minimization problem (1.5), and denote $EP(\varphi )$ as the solution set of the equilibrium problem (1.1).

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let $V:C\to H$ be Lipschitzian with a constant $l>0$, and $A:C\to H$ be a strongly positive bounded linear operator with a coefficient $\overline{\gamma }>0$, and $0<r<\overline{\gamma }/l$. Suppose that ∇f is $1/L$-ism continuous. Let $Q_{{\beta }_n}$ be a mapping defined as in Lemma 2.7. We now consider the following mapping $S_n$ on H defined by

$$S_n(x)={\alpha }_{n}rV{Q}_{{\beta }_n}(x)+(I-{\alpha }_{n}A)T_{{\lambda }_n}{Q}_{{\beta }_n}(x)\mathrm{\forall }x\in H,n\in \mathbb{N},$$

where ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})={\theta }_{n}I+(1-{\theta }_n)T_{{\lambda }_n}$, ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$, and $\gamma \in (0,2/L)$, $\{{\alpha }_n\}\subset (0,1)$. It is easy to see that $S_n$ is a contraction. Indeed, by Lemma 2.5 and Lemma 2.7, we have for each $x,y\in H$

$$\begin{array}{c}\parallel S_{n}x-S_{n}y\parallel \hfill \\ =\parallel ({\alpha }_{n}rV{Q}_{{\beta }_n}x-{\alpha }_{n}rV{Q}_{{\beta }_n}y)+(I-{\alpha }_{n}A)T_{{\lambda }_n}{Q}_{{\beta }_n}x-(I-{\alpha }_{n}A)T_{{\lambda }_n}{Q}_{{\beta }_n}y\parallel \hfill \\ \le {\alpha }_{n}rl\parallel x-y\parallel +(1-{\alpha }_n\overline{\gamma })\parallel x-y\parallel \hfill \\ =(1-{\alpha }_n(\overline{\gamma }-rl))\parallel x-y\parallel .\hfill \end{array}$$

Since $0<1-{\alpha }_n(\overline{\gamma }-rl)<1$, it follows that $S_n$ is a contraction. Therefore, by the Banach contraction principle, $S_n$ has a unique fixed point $x_n\in H$ such that

$$x_n={\alpha }_{n}rV{Q}_{{\beta }_n}(x_n)+(I-{\alpha }_{n}A)T_{{\lambda }_n}{Q}_{{\beta }_n}(x_n).$$

Note that $x_n$ indeed depends on V as well, but we will suppress this dependence of $x_n$ on V for simplicity of notation throughout the rest of this paper.

The following theorem summarizes the properties of the sequence $\{x_n\}$.

Theorem 3.1 Let C be a nonempty, closed and convex subset of a Hilbert space H. Let ϕ be a bifunction from $C\times C\to \mathbb{R}$ satisfying (A1)-(A4), and let $f:C\to \mathbb{R}$ be a real-valued convex function, and assume that the gradient ∇f is $1/L$-ism with a constant $L>0$. Assume that $U\cap EP(\varphi )\ne \mathrm{\varnothing }$. Let $V:C\to H$ be Lipschitzian with a constant $l>0$, and let $A:C\to H$ be a strongly positive bounded linear operator with a coefficient $\overline{\gamma }>0$, and $0<r<\overline{\gamma }/l$. Let the sequences $\{u_n\}$ and $\{x_n\}$ be generated by

$$\{\begin{array}{c}\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C,\hfill \\ x_n={\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(3.1)

where $u_n=Q_{{\beta }_n}{x}_n$, ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})={\theta }_{n}I+(1-{\theta }_n)T_{{\lambda }_n}$, ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$ and $\gamma \in (0,2/L)$. Let $\{{\beta }_n\}$, $\{{\alpha }_n\}$ and $\{{\lambda }_n\}$ satisfy the following conditions:

• (i) $\{{\beta }_n\}\subset (0,\mathrm{\infty })$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$;

• (ii) $\{{\alpha }_n\}\subset (0,1)$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$;

• (iii) $\{{\lambda }_n\}\subset (0,2/\gamma -L)$, ${\lambda }_n=o({\alpha }_n)$.

Then the sequence $\{x_n\}$ converges strongly to a point $z\in U\cap EP(\varphi )$, which solves the variational inequality

$$〈(A-rV)z,z-x〉\le 0,x\in U\cap EP(\varphi ).$$

(3.2)

Equivalently, we have $z=P_{U\cap EP(\varphi )}(I-A+rV)(z)$.

Proof It is well known that $\hat{x}\in C$ solves the minimization problem (1.5) if and only if for each fixed $0<\gamma <2/L$, $\hat{x}$ solves the fixed-point equation

$$\hat{x}={Proj}_C(I-\gamma \mathrm{\nabla }f)\hat{x}=\frac{2-\gamma L}{4}\hat{x}+\frac{2+\gamma L}{4}T\hat{x}.$$

It is clear that $\hat{x}=T\hat{x}$, i.e., $\hat{x}\in U=Fix(T)$.

First, we assume that ${\alpha }_n\in (0,{\parallel A\parallel }^{-1})$. By Lemma 2.5, we obtain $\parallel I-{\alpha }_{n}A\parallel \le 1-{\alpha }_n\overline{\gamma }$. Let $p\in U\cap EP(\varphi )$, then from $u_n=Q_{{\beta }_n}{x}_n$, we have

$$\parallel u_n-p\parallel =\parallel Q_{{\beta }_n}{x}_n-Q_{{\beta }_n}p\parallel \le \parallel x_n-p\parallel$$

(3.3)

for all $n\in \mathbb{N}$. Thus, we have from (3.3)

$$\begin{array}{c}\parallel x_n-p\parallel \hfill \\ =\parallel {\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n-p\parallel \hfill \\ \le \parallel {\alpha }_{n}rV{u}_n-{\alpha }_{n}A(p)\parallel +\parallel I-{\alpha }_{n}A\parallel \cdot \parallel T_{{\lambda }_n}{u}_n-p\parallel \hfill \\ \le \parallel {\alpha }_{n}rV{u}_n-{\alpha }_{n}rV(p)\parallel +\parallel {\alpha }_{n}rV(p)-{\alpha }_{n}A(p)\parallel \hfill \\ +(1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}(p)+T_{{\lambda }_n}(p)-Tp\parallel \hfill \\ \le {\alpha }_{n}rl\parallel u_n-p\parallel +{\alpha }_n\parallel rV(p)-A(p)\parallel \hfill \\ +(1-{\alpha }_n\overline{\gamma })\parallel u_n-p\parallel +(1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}(p)-Tp\parallel \hfill \\ \le (1-{\alpha }_n(\overline{\gamma }-rl))\parallel x_n-p\parallel +{\alpha }_n\parallel rV(p)-A(p)\parallel +(1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}(p)-Tp\parallel .\hfill \end{array}$$

It follows that

$$\parallel x_n-p\parallel \le \frac{1}{\overline{\gamma }-rl}\parallel rV(p)-A(p)\parallel +\frac{1-{\alpha }_n\overline{\gamma }}{{\alpha }_n(\overline{\gamma }-rl)}\parallel T_{{\lambda }_n}(p)-Tp\parallel .$$

(3.4)

For $x\in C$, note that

$${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})x={\theta }_{n}x+(1-{\theta }_n)T_{{\lambda }_n}x$$

and

$${Proj}_C(I-\gamma \mathrm{\nabla }f)x=\theta x+(1-\theta )Tx,$$

where ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$ and $\theta =\frac{2-\gamma L}{4}$.

Then we get

$$\begin{array}{rcl}\parallel ({\theta }_n-\theta )x+T_{{\lambda }_n}x-Tx+\theta Tx-{\theta }_n{T}_{{\lambda }_n}x\parallel & =& \parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})x-{Proj}_C(I-\gamma \mathrm{\nabla }f)x\parallel \\ \le & \gamma {\lambda }_n\parallel x\parallel .\end{array}$$

Since ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$ and $\theta =\frac{2-\gamma L}{4}$, there exists $M(x)>0$ such that

$$\parallel T_{{\lambda }_n}x-Tx\parallel \le \frac{{\lambda }_n\gamma (5\parallel x\parallel +\parallel Tx\parallel )}{2+\gamma (L+{\lambda }_n)}\le {\lambda }_{n}M(x),$$

(3.5)

where $M(x)=\gamma (5\parallel x\parallel +\parallel Tx\parallel )$.

It follows from (3.4) and (3.5) that

$$\parallel x_n-p\parallel \le \frac{1}{\overline{\gamma }-rl}\parallel rV(p)-A(p)\parallel +\frac{M(p)}{\overline{\gamma }-rl}\cdot \frac{{\lambda }_n}{{\alpha }_n}.$$

Since ${\lambda }_n=o({\alpha }_n)$, there exists a real number $M^{\mathrm{\prime }}>0$ such that $\frac{{\lambda }_n}{{\alpha }_n}\le M^{\mathrm{\prime }}$, and

$$\begin{array}{c}\parallel x_n-p\parallel \hfill \\ \le \frac{1}{\overline{\gamma }-rl}\parallel rV(p)-A(p)\parallel +\frac{M^{\mathrm{\prime }}M(p)}{\overline{\gamma }-rl}\hfill \\ =\frac{\parallel rV(p)-A(p)\parallel +M^{\mathrm{\prime }}M(p)}{\overline{\gamma }-rl}.\hfill \end{array}$$

Hence $\{x_n\}$ is bounded and we also obtain that $\{u_n\}$ is bounded.

Next, we show that $\parallel x_n-u_n\parallel \to 0$.

Indeed, for any $p\in U\cap EP(\varphi )$, by Lemma 2.7, we have

$$\begin{array}{c}{\parallel u_n-p\parallel }^2\hfill \\ ={\parallel Q_{{\beta }_n}{x}_n-Q_{{\beta }_n}p\parallel }^2\hfill \\ \le 〈x_n-p,u_n-p〉\hfill \\ =\frac{1}{2}({\parallel x_n-p\parallel }^2+{\parallel u_n-p\parallel }^2-{\parallel u_n-x_n\parallel }^2).\hfill \end{array}$$

This implies that

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

(3.6)

Then from (3.5), we derive that

$$\begin{array}{c}{\parallel x_n-p\parallel }^2\hfill \\ ={\parallel {\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n-p\parallel }^2\hfill \\ ={\parallel (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-p)+{\alpha }_{n}rV{u}_n-{\alpha }_{n}A(p)\parallel }^2\hfill \\ \le {\parallel (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-p)\parallel }^2\hfill \\ +2\parallel (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-p)\parallel \cdot \parallel {\alpha }_{n}rV{u}_n-{\alpha }_{n}A(p)\parallel +{\parallel {\alpha }_{n}rV{u}_n-{\alpha }_{n}A(p)\parallel }^2\hfill \\ \le {(1-{\alpha }_n\overline{\gamma })}^2{\parallel T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}p+T_{{\lambda }_n}p-Tp\parallel }^2\hfill \\ +2(1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}{u}_n-p\parallel \parallel {\alpha }_{n}rV{u}_n-{\alpha }_{n}A(p)\parallel +{\alpha }_n^2{\parallel rV{u}_n-A(p)\parallel }^2\hfill \\ \le {(1-{\alpha }_n\overline{\gamma })}^2{(\parallel u_n-p\parallel +\parallel T_{{\lambda }_n}p-Tp\parallel )}^2\hfill \\ +2(1-{\alpha }_n\overline{\gamma })(\parallel u_n-p\parallel +\parallel T_{{\lambda }_n}p-Tp\parallel )\cdot ({\alpha }_{n}rl\parallel u_n-p\parallel +{\alpha }_n\parallel rV(p)-A(p)\parallel )\hfill \\ +{\alpha }_n^2{\parallel rV{u}_n-A(p)\parallel }^2\hfill \\ \le ({(1-{\alpha }_n\overline{\gamma })}^2+2(1-{\alpha }_n\overline{\gamma }){\alpha }_{n}rl){\parallel u_n-p\parallel }^2\hfill \\ +{(1-{\alpha }_n\overline{\gamma })}^2{\lambda }_n^2\cdot {(M(p))}^2+2{(1-{\alpha }_n\overline{\gamma })}^2\parallel u_n-p\parallel {\lambda }_{n}M(p)\hfill \\ +2(1-{\alpha }_n\overline{\gamma })\parallel u_n-p\parallel {\alpha }_n\parallel rV(p)-A(p)\parallel +2(1-{\alpha }_n\overline{\gamma }){\lambda }_{n}M(p){\alpha }_{n}rl\parallel u_n-p\parallel \hfill \\ +2(1-{\alpha }_n\overline{\gamma }){\lambda }_{n}M(p){\alpha }_n\parallel rV(p)-A(p)\parallel +{\alpha }_n^2{(rl\parallel u_n-p\parallel +\parallel rV(p)-A(p)\parallel )}^2.\hfill \end{array}$$

It follows from (3.6) that

$$\begin{array}{c}({(1-{\alpha }_n\overline{\gamma })}^2+2(1-{\alpha }_n\overline{\gamma }){\alpha }_{n}rl){\parallel u_n-x_n\parallel }^2\hfill \\ \le ({(1-{\alpha }_n\overline{\gamma })}^2+2(1-{\alpha }_n\overline{\gamma }){\alpha }_{n}rl-1){\parallel x_n-p\parallel }^2\hfill \\ +{\lambda }_n^2\cdot {(M(p))}^2+2\parallel u_n-p\parallel ({\lambda }_{n}M(p)+{\alpha }_n\parallel rV(p)-A(p)\parallel )\hfill \\ +2{\alpha }_n{\lambda }_{n}M(p)(rl\parallel u_n-p\parallel +\parallel rV(p)-A(p)\parallel )\hfill \\ +{\alpha }_n^2{(rl\parallel u_n-p\parallel +\parallel rV(p)-A(p)\parallel )}^2.\hfill \end{array}$$

Since both $\{x_n\}$ and $\{u_n\}$ are bounded and ${\alpha }_n\to 0$, ${\lambda }_n\to 0$, it follows that $\parallel u_n-x_n\parallel \to 0$.

We claim that $\parallel x_n-T_{{\lambda }_n}{x}_n\parallel \to 0$. Indeed,

$$\begin{array}{c}\parallel x_n-T_{{\lambda }_n}{x}_n\parallel \hfill \\ =\parallel x_n-T_{{\lambda }_n}{u}_n+T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}{x}_n\parallel \hfill \\ \le \parallel x_n-T_{{\lambda }_n}{u}_n\parallel +\parallel T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}{x}_n\parallel \hfill \\ \le {\alpha }_n\parallel rV{u}_n-A{T}_{{\lambda }_n}{u}_n\parallel +\parallel u_n-x_n\parallel .\hfill \end{array}$$

Since ${\alpha }_n\to 0$ and $\parallel u_n-x_n\parallel \to 0$, we obtain that

$$\parallel x_n-T_{{\lambda }_n}{x}_n\parallel \to 0.$$

Thus,

$$\begin{array}{c}\parallel u_n-T_{{\lambda }_n}{u}_n\parallel \hfill \\ =\parallel u_n-x_n+x_n-T_{{\lambda }_n}{x}_n+T_{{\lambda }_n}{x}_n-T_{{\lambda }_n}{u}_n\parallel \hfill \\ \le \parallel u_n-x_n\parallel +\parallel x_n-T_{{\lambda }_n}{x}_n\parallel +\parallel T_{{\lambda }_n}{x}_n-T_{{\lambda }_n}{u}_n\parallel \hfill \\ \le \parallel u_n-x_n\parallel +\parallel x_n-T_{{\lambda }_n}{x}_n\parallel +\parallel x_n-u_n\parallel \hfill \end{array}$$

and

$$\parallel x_n-T_{{\lambda }_n}{u}_n\parallel \le \parallel u_n-x_n\parallel +\parallel T_{{\lambda }_n}{u}_n-u_n\parallel ,$$

we have $\parallel u_n-T_{{\lambda }_n}{u}_n\parallel \to 0$ and $\parallel x_n-T_{{\lambda }_n}{u}_n\parallel \to 0$.

Since $\{u_n\}$ is bounded, without loss of generality, we can assume that $u_{n_i}\rightharpoonup z$. Next, we show that $z\in U\cap EP(\varphi )$.

By (3.5), we have

$$\begin{array}{c}\parallel u_n-T{u}_n\parallel \hfill \\ \le \parallel u_n-T_{{\lambda }_n}{u}_n\parallel +\parallel T_{{\lambda }_n}{u}_n-T{u}_n\parallel \hfill \\ \le \parallel u_n-T_{{\lambda }_n}{u}_n\parallel +{\lambda }_{n}M(u_n)\to 0.\hfill \end{array}$$

So, by Lemma 2.2, we get $z\in Fix(T)=U$.

Next, we show that $z\in EP(\varphi )$. Since $u_n=Q_{{\beta }_n}{x}_n$, for any $y\in C$, we obtain

$$\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0.$$

From (A2), we have

$$\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge \varphi (y,u_n)$$

and hence

$$〈y-u_{n_i},\frac{u_{n_i}-x_{n_i}}{{\beta }_{n_i}}〉\ge \varphi (y,u_{n_i}).$$

Since $\frac{u_{n_i}-x_{n_i}}{{\beta }_{n_i}}\to 0$ and $u_{n_i}\rightharpoonup z$, it follows from (A4) that $\varphi (y,z)\le 0$ for any $y\in C$.

Let $z_t=ty+(1-t)z$, $\mathrm{\forall }t\in (0,1]$, $y\in C$, then we have $z_t\in C$ and hence $\varphi (z_t,z)\le 0$.

Thus, from (A1) and (A4), we have

$$\begin{array}{rcl}0& =& \varphi (z_t,z_t)\\ \le & t\varphi (z_t,y)+(1-t)\varphi (z_t,z)\\ \le & t\varphi (z_t,y)\end{array}$$

and hence $\varphi (z_t,y)\ge 0$. From (A3), we have $\varphi (z,y)\ge 0$ for any $y\in C$, hence $z\in EP(\varphi )$. Therefore, $z\in U\cap EP(\varphi )$.

On the other hand, we note that

$$\begin{array}{rcl}{x}_n-z& =& {\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n-z\\ =& (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-z)+{\alpha }_{n}rV{u}_n-{\alpha }_{n}A(z)\\ =& (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-z)+{\alpha }_{n}r(V{u}_n-Vz)+{\alpha }_n(rV(z)-A(z)).\end{array}$$

Hence, we obtain from (3.3) and (3.5) that

$$\begin{array}{rcl}{\parallel x_n-z\parallel }^2& =& 〈(I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-z),x_n-z〉\\ +{\alpha }_{n}r〈V{u}_n-Vz,x_n-z〉+{\alpha }_n〈rV(z)-A(z),x_n-z〉\\ \le & (1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}{u}_n-z\parallel \parallel x_n-z\parallel +{\alpha }_{n}rl\parallel u_n-z\parallel \parallel x_n-z\parallel \\ +{\alpha }_n〈rV(z)-A(z),x_n-z〉\\ \le & (1-{\alpha }_n\overline{\gamma })(\parallel T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}z\parallel +\parallel T_{{\lambda }_n}z-Tz\parallel )\parallel x_n-z\parallel \\ +{\alpha }_{n}rl\parallel u_n-z\parallel \parallel x_n-z\parallel +{\alpha }_n〈rV(z)-A(z),x_n-z〉\\ \le & (1-{\alpha }_n\overline{\gamma }){\parallel x_n-z\parallel }^2+(1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}z-Tz\parallel \parallel x_n-z\parallel \\ +{\alpha }_{n}rl{\parallel x_n-z\parallel }^2+{\alpha }_n〈rV(z)-A(z),x_n-z〉\\ \le & (1-{\alpha }_n(\overline{\gamma }-rl)){\parallel x_n-z\parallel }^2\\ +(1-{\alpha }_n\overline{\gamma }){\lambda }_{n}M(z)\parallel x_n-z\parallel +{\alpha }_n〈rV(z)-A(z),x_n-z〉.\end{array}$$

It follows that

$${\parallel x_n-z\parallel }^2\le \frac{1}{\overline{\gamma }-rl}\frac{{\lambda }_n}{{\alpha }_n}M(z)\parallel x_n-z\parallel +\frac{1}{\overline{\gamma }-rl}〈rV(z)-A(z),x_n-z〉.$$

In particular,

$${\parallel x_{n_i}-z\parallel }^2\le \frac{1}{\overline{\gamma }-rl}\frac{{\lambda }_{n_i}}{{\alpha }_{n_i}}M(z)\parallel x_{n_i}-z\parallel +\frac{1}{\overline{\gamma }-rl}〈rV(z)-A(z),x_{n_i}-z〉.$$

(3.7)

Since $x_{n_i}\rightharpoonup z$ and ${\lambda }_n=o({\alpha }_n)$, it follows from (3.7) that $x_{n_i}\to z$ as $i\to \mathrm{\infty }$.

Next, we show that z solves the variational inequality (3.2). Observe that $x_n={\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n$.

Hence, we conclude that

$$(A-rV)x_n=-\frac{1}{{\alpha }_n}(I-{\alpha }_{n}A)(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x_n+r(V{u}_n-V{x}_n).$$

Since $T_{{\lambda }_n}{Q}_{{\beta }_n}$ is nonexpansive, we have $I-T_{{\lambda }_n}{Q}_{{\beta }_n}$ is monotone. Note that for any given $x\in U\cap EP(\varphi )$,

$$\begin{array}{c}〈(A-rV)x_n,x_n-x〉\hfill \\ =-\frac{1}{{\alpha }_n}〈(I-{\alpha }_{n}A)(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x_n,x_n-x〉+r〈V{u}_n-V{x}_n,x_n-x〉\hfill \\ =-\frac{1}{{\alpha }_n}〈(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x_n-(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x,x_n-x〉\hfill \\ -\frac{1}{{\alpha }_n}〈(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x,x_n-x〉+〈A(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x_n,x_n-x〉\hfill \\ +r〈V{u}_n-V{x}_n,x_n-x〉\hfill \\ \le -\frac{1}{{\alpha }_n}〈(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x,x_n-x〉+〈A(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x_n,x_n-x〉\hfill \\ +r〈V{u}_n-V{x}_n,x_n-x〉\hfill \\ \le \frac{1}{{\alpha }_n}\parallel x-T_{{\lambda }_n}x\parallel \parallel x_n-x\parallel +\parallel A(I-T_{{\lambda }_n}{Q}_{{\beta }_n})x_n\parallel \parallel x_n-x\parallel \hfill \\ +rl\parallel u_n-x_n\parallel \parallel x_n-x\parallel \hfill \\ \le \frac{{\lambda }_n}{{\alpha }_n}M(x)\parallel x_n-x\parallel +\parallel A\parallel \parallel x_n-T_{{\lambda }_n}{Q}_{{\beta }_n}{x}_n\parallel \parallel x_n-x\parallel \hfill \\ +rl\parallel u_n-x_n\parallel \parallel x_n-x\parallel .\hfill \end{array}$$

Now, replacing n with $n_i$ in the above inequality, and letting $i\to \mathrm{\infty }$, since ${\lambda }_n=o({\alpha }_n)$, $\parallel x_n-T_{{\lambda }_n}{u}_n\parallel \to 0$, and $\parallel u_n-x_n\parallel \to 0$, we have

$$〈(A-rV)z,z-x〉\le 0.$$

It follows that $z\in U\cap EP(\varphi )$ is a solution of the variational inequality (3.2). Further, by the uniqueness of the solution of the variational inequality (3.2), we conclude that $x_n\to z$ as $n\to \mathrm{\infty }$.

The variational inequality (3.2) can be rewritten as

$$〈(I-A+rV)z-z,z-x〉\ge 0\mathrm{\forall }x\in U\cap EP(\varphi ).$$

By Lemma 2.4, it is equivalent to the following fixed-point equation:

$$P_{U\cap EP(\varphi )}(I-A+rV)z=z.$$

This completes the proof. □

Theorem 3.2 Let C be a nonempty, closed and convex subset of Hilbert space H. Let ϕ be a bifunction from $C\times C\to \mathbb{R}$ satisfying (A1)-(A4), and let $f:C\to \mathbb{R}$ be a real-valued convex function, and assume that the gradient ∇f is $1/L$-ism with a constant $L>0$. Assume that $U\cap EP(\varphi )\ne \mathrm{\varnothing }$. Let $V:C\to H$ be Lipschitzian with a constant $l>0$, and let $A:C\to H$ be a strongly positive bounded linear operator with a coefficient $\overline{\gamma }>0$, and $0<r<\overline{\gamma }/l$. Let the sequences $\{u_n\}$ and $\{x_n\}$ be generated by $x_1\in H$ and

$$\{\begin{array}{c}\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(3.8)

where $u_n=Q_{{\beta }_n}{x}_n$, ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})={\theta }_{n}I+(1-{\theta }_n)T_{{\lambda }_n}$, ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$ and $\gamma \in (0,2/L)$. Let $\{{\beta }_n\}$, $\{{\alpha }_n\}$ and $\{{\lambda }_n\}$ satisfy the following conditions:

(C1) $\{{\beta }_n\}\subset (0,\mathrm{\infty })$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$;

(C2) $\{{\alpha }_n\}\subset (0,1)$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$;

(C3) $\{{\lambda }_n\}\subset (0,2/\gamma -L)$, ${\lambda }_n=o({\alpha }_n)$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n+1}-{\lambda }_n|<\mathrm{\infty }$.

Then the sequence $\{x_n\}$ converges strongly to a point $z\in U\cap EP(\varphi )$, which solves the variational inequality (3.2).

Proof It is well known that:

• (a) $\hat{x}\in C$ solves the minimization problem (1.5) if and only if for each fixed $0<\gamma <2/L$, $\hat{x}$ solves the fixed-point equation

  $$\hat{x}={Proj}_C(I-\gamma \mathrm{\nabla }f)\hat{x}=\frac{2-\gamma L}{4}\hat{x}+\frac{2+\gamma L}{4}T\hat{x}.$$

It is clear that $\hat{x}=T\hat{x}$, i.e., $\hat{x}\in U=Fix(T)$.

• (b) The gradient ∇f is $1/L$-ism.

• (c) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})$ is $\frac{2+\gamma (L+{\lambda }_n)}{4}$ averaged for $\gamma \in (0,2/L)$, in particular, the following relation holds:

  $${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})=\frac{2-\gamma (L+{\lambda }_n)}{4}I+\frac{2+\gamma (L+{\lambda }_n)}{4}{T}_{{\lambda }_n}={\theta }_{n}I+(1-{\theta }_n)T_{{\lambda }_n}.$$

Since ${\alpha }_n\to 0$, we may assume that ${\alpha }_n\in (0,{\parallel A\parallel }^{-1})$. Now, we first show that $\{x_n\}$ is bounded. Indeed, pick $p\in U\cap EP(\varphi )$, since $u_n=Q_{{\beta }_n}{x}_n$, by Lemma 2.7, we know that

$$\parallel u_n-p\parallel =\parallel Q_{{\beta }_n}{x}_n-Q_{{\beta }_n}p\parallel \le \parallel x_n-p\parallel .$$

(3.9)

Thus, we derive from (3.5) that

$$\begin{array}{rcl}\parallel x_{n+1}-p\parallel & =& \parallel {\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n-p\parallel \\ =& \parallel (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-p)+{\alpha }_{n}rV{u}_n-{\alpha }_{n}A(p)\parallel \\ \le & (1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}{u}_n-p\parallel +{\alpha }_n\parallel rV{u}_n-A(p)\parallel \\ \le & (1-{\alpha }_n\overline{\gamma })(\parallel T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}p\parallel +\parallel T_{{\lambda }_n}p-Tp\parallel )\\ +{\alpha }_n(\parallel rV{u}_n-rV(p)\parallel +\parallel rV(p)-A(p)\parallel )\\ \le & (1-{\alpha }_n\overline{\gamma })(\parallel u_n-p\parallel +{\lambda }_{n}M(p))\\ +{\alpha }_n(rl\parallel u_n-p\parallel +\parallel rV(p)-A(p)\parallel )\\ =& (1-{\alpha }_n(\overline{\gamma }-rl))\parallel u_n-p\parallel +(1-{\alpha }_n\overline{\gamma }){\lambda }_{n}M(p)+{\alpha }_n\parallel rV(p)-A(p)\parallel \\ \le & (1-{\alpha }_n(\overline{\gamma }-rl))\parallel x_n-p\parallel +{\lambda }_{n}M(p)+{\alpha }_n\parallel rV(p)-A(p)\parallel \\ =& (1-{\alpha }_n(\overline{\gamma }-rl))\parallel x_n-p\parallel +{\alpha }_n(\overline{\gamma }-rl)[\frac{{\lambda }_n}{{\alpha }_n}\frac{M(p)}{\overline{\gamma }-rl}+\frac{\parallel rV(p)-A(p)\parallel }{\overline{\gamma }-rl}].\end{array}$$

Since ${\lambda }_n=o({\alpha }_n)$, there exists a real number $M^{\mathrm{\prime }}>0$ such that $\frac{{\lambda }_n}{{\alpha }_n}\le M^{\mathrm{\prime }}$. Thus,

$$\parallel x_{n+1}-p\parallel \le (1-{\alpha }_n(\overline{\gamma }-rl))\parallel x_n-p\parallel +{\alpha }_n(\overline{\gamma }-rl)\frac{M^{\mathrm{\prime }}M(p)+\parallel rV(p)-A(p)\parallel }{\overline{\gamma }-rl}.$$

By induction, we have

$$\parallel x_n-p\parallel \le max\{\parallel x_1-p\parallel ,\frac{M^{\mathrm{\prime }}M(p)+\parallel rV(p)-A(p)\parallel }{\overline{\gamma }-rl}\},n\ge 1.$$

Hence $\{x_n\}$ is bounded. From (3.9), we also derive that $\{u_n\}$ is bounded.

Next, we show that $\parallel x_{n+1}-x_n\parallel \to 0$.

Indeed, since

$${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})=\frac{2-\gamma (L+{\lambda }_n)}{4}I+\frac{2+\gamma (L+{\lambda }_n)}{4}{T}_{{\lambda }_n},$$

we have

$$T_{{\lambda }_n}=\frac{4{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})-[2-\gamma (L+{\lambda }_n)]I}{2+\gamma (L+{\lambda }_n)}.$$

So, we obtain that

$$\begin{array}{c}\parallel T_{{\lambda }_n}(u_{n-1})-T_{{\lambda }_{n-1}}(u_{n-1})\parallel \hfill \\ =\parallel \frac{4{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})-[2-\gamma (L+{\lambda }_n)]I}{2+\gamma (L+{\lambda }_n)}{u}_{n-1}\hfill \\ -\frac{4{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_{n-1}})-[2-\gamma (L+{\lambda }_{n-1})]I}{2+\gamma (L+{\lambda }_{n-1})}{u}_{n-1}\parallel \hfill \\ \le \parallel \frac{4{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})}{2+\gamma (L+{\lambda }_n)}{u}_{n-1}-\frac{4{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_{n-1}})}{2+\gamma (L+{\lambda }_{n-1})}{u}_{n-1}\parallel \hfill \\ +\parallel \frac{[2-\gamma (L+{\lambda }_n)]I}{2+\gamma (L+{\lambda }_n)}{u}_{n-1}-\frac{[2-\gamma (L+{\lambda }_{n-1})]I}{2+\gamma (L+{\lambda }_{n-1})}{u}_{n-1}\parallel \hfill \\ =\parallel \frac{4[2+\gamma (L+{\lambda }_{n-1})]{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})-4[2+\gamma (L+{\lambda }_n)]{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_{n-1}})}{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}{u}_{n-1}\parallel \hfill \\ +\frac{4\gamma |{\lambda }_n-{\lambda }_{n-1}|}{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}\parallel u_{n-1}\parallel \hfill \\ =\parallel \frac{4\gamma ({\lambda }_{n-1}-{\lambda }_n){Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})}{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}{u}_{n-1}\hfill \\ +\frac{4[2+\gamma (L+{\lambda }_n)]({Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})-{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_{n-1}}))}{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}{u}_{n-1}\parallel \hfill \\ +\frac{4\gamma |{\lambda }_n-{\lambda }_{n-1}|}{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}\parallel u_{n-1}\parallel \hfill \\ \le \frac{4\gamma |{\lambda }_n-{\lambda }_{n-1}|\parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})u_{n-1}\parallel }{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}\hfill \\ +\frac{4[2+\gamma (L+{\lambda }_n)]\parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})u_{n-1}-{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_{n-1}})u_{n-1}\parallel }{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}\hfill \\ +\frac{4\gamma |{\lambda }_n-{\lambda }_{n-1}|}{[2+\gamma (L+{\lambda }_n)][2+\gamma (L+{\lambda }_{n-1})]}\parallel u_{n-1}\parallel \hfill \\ \le |{\lambda }_n-{\lambda }_{n-1}|[\gamma \parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})u_{n-1}\parallel +4\gamma \parallel u_{n-1}\parallel +\gamma \parallel u_{n-1}\parallel ]\hfill \\ \le K|{\lambda }_n-{\lambda }_{n-1}|\hfill \end{array}$$

for some appropriate constant $K>0$ such that

$$K\ge \gamma \parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})u_{n-1}\parallel +5\gamma \parallel u_{n-1}\parallel .$$

Thus, we get

$$\begin{array}{c}\parallel x_{n+1}-x_n\parallel \hfill \\ =\parallel [{\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n]-[{\alpha }_{n-1}rV{u}_{n-1}+(I-{\alpha }_{n-1}A)T_{{\lambda }_{n-1}}{u}_{n-1}]\parallel \hfill \\ \le \parallel {\alpha }_{n}rV{u}_n-{\alpha }_{n}rV{u}_{n-1}\parallel +\parallel {\alpha }_{n}rV{u}_{n-1}-{\alpha }_{n-1}rV{u}_{n-1}\parallel \hfill \\ +\parallel (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}{u}_{n-1})\parallel \hfill \\ +\parallel (I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_{n-1}-(I-{\alpha }_{n-1}A)T_{{\lambda }_{n-1}}{u}_{n-1}\parallel \hfill \\ \le {\alpha }_{n}rl\parallel u_n-u_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|r\parallel V{u}_{n-1}\parallel +(1-{\alpha }_n\overline{\gamma })\parallel u_n-u_{n-1}\parallel \hfill \\ +\parallel T_{{\lambda }_n}{u}_{n-1}-T_{{\lambda }_{n-1}}{u}_{n-1}\parallel +\parallel {\alpha }_{n-1}A{T}_{{\lambda }_{n-1}}{u}_{n-1}-{\alpha }_{n}A{T}_{{\lambda }_n}{u}_{n-1}\parallel \hfill \\ \le (1-{\alpha }_n(\overline{\gamma }-rl))\parallel u_n-u_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|r\parallel V{u}_{n-1}\parallel \hfill \\ +\parallel T_{{\lambda }_n}{u}_{n-1}-T_{{\lambda }_{n-1}}{u}_{n-1}\parallel (1+{\alpha }_{n-1}\parallel A\parallel )+|{\alpha }_n-{\alpha }_{n-1}|\parallel A{T}_{{\lambda }_n}{u}_{n-1}\parallel \hfill \\ \le (1-{\alpha }_n(\overline{\gamma }-rl))\parallel u_n-u_{n-1}\parallel \hfill \\ +|{\alpha }_n-{\alpha }_{n-1}|(r\parallel V{u}_{n-1}\parallel +\parallel A{T}_{{\lambda }_n}{u}_{n-1}\parallel )\hfill \\ +|{\lambda }_n-{\lambda }_{n-1}|(K+K\parallel A\parallel ).\hfill \end{array}$$

Since both $\{V{u}_{n-1}\}$ and $\{A{T}_{{\lambda }_n}{u}_{n-1}\}$ are bounded, we can take a constant $E>0$ such that

$$E\ge r\parallel V{u}_{n-1}\parallel +\parallel A{T}_{{\lambda }_n}{u}_{n-1}\parallel ,n\ge 1.$$

Consequently,

$$\begin{array}{rcl}\parallel x_{n+1}-x_n\parallel & \le & (1-{\alpha }_n(\overline{\gamma }-rl))\parallel u_n-u_{n-1}\parallel \\ +E|{\alpha }_n-{\alpha }_{n-1}|+|{\lambda }_n-{\lambda }_{n-1}|(K+\parallel A\parallel K).\end{array}$$

(3.10)

From $u_{n+1}=Q_{{\beta }_{n+1}}{x}_{n+1}$ and $u_n=Q_{{\beta }_n}{x}_n$, we note that

$$\varphi (u_{n+1},y)+\frac{1}{{\beta }_{n+1}}〈y-u_{n+1},u_{n+1}-x_{n+1}〉\ge 0\mathrm{\forall }y\in C$$

(3.11)

and

$$\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C.$$

(3.12)

Putting $y=u_n$ in (3.11) and $y=u_{n+1}$ in (3.12), we have

$$\varphi (u_{n+1},u_n)+\frac{1}{{\beta }_{n+1}}〈u_n-u_{n+1},u_{n+1}-x_{n+1}〉\ge 0$$

and

$$\varphi (u_n,u_{n+1})+\frac{1}{{\beta }_n}〈u_{n+1}-u_n,u_n-x_n〉\ge 0.$$

So, from (A2), we have

$$〈u_{n+1}-u_n,\frac{u_n-x_n}{{\beta }_n}-\frac{u_{n+1}-x_{n+1}}{{\beta }_{n+1}}〉\ge 0$$

and hence

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

Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$, without loss of generality, let us assume that there exists a real number a such that ${\beta }_n>a>0$ for all $n\in \mathbb{N}$. Thus, we have

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

thus,

$$\parallel u_{n+1}-u_n\parallel \le \parallel x_{n+1}-x_n\parallel +\frac{1}{a}|{\beta }_{n+1}-{\beta }_n|M_1,$$

(3.13)

where $M_1=sup\{\parallel u_n-x_n\parallel :n\in \mathbb{N}\}$.

From (3.10) and (3.13), we obtain

$$\begin{array}{rcl}\parallel x_{n+1}-x_n\parallel & \le & (1-{\alpha }_n(\overline{\gamma }-rl))(\parallel x_n-x_{n-1}\parallel +\frac{1}{a}|{\beta }_n-{\beta }_{n-1}|M_1)\\ +E|{\alpha }_n-{\alpha }_{n-1}|+|{\lambda }_n-{\lambda }_{n-1}|(K+\parallel A\parallel K)\\ \le & (1-{\alpha }_n(\overline{\gamma }-rl))\parallel x_n-x_{n-1}\parallel +\frac{M_1}{a}|{\beta }_n-{\beta }_{n-1}|\\ +E|{\alpha }_n-{\alpha }_{n-1}|+|{\lambda }_n-{\lambda }_{n-1}|(K+\parallel A\parallel K)\\ \le & (1-{\alpha }_n(\overline{\gamma }-rl))\parallel x_n-x_{n-1}\parallel \\ +M_2(|{\beta }_n-{\beta }_{n-1}|+|{\alpha }_n-{\alpha }_{n-1}|+|{\lambda }_n-{\lambda }_{n-1}|),\end{array}$$

where $M_2=max\{\frac{M_1}{a},E,K+\parallel A\parallel K\}$. Hence, by Lemma 2.1, we have

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

(3.14)

Then, from (3.13), (3.14) and $|{\beta }_{n+1}-{\beta }_n|\to 0$, we have

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

(3.15)

For any $p\in U\cap EP(\varphi )$, as the same proof of Theorem 3.1, we have

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

(3.16)

Then, from (3.5) and (3.16), by the same argument as in the proof of Theorem 3.1, we derive that

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & ({(1-{\alpha }_n\overline{\gamma })}^2+2(1-{\alpha }_n\overline{\gamma }){\alpha }_{n}rl)({\parallel x_n-p\parallel }^2-{\parallel u_n-x_n\parallel }^2)\\ +{\lambda }_n^2\cdot {(M(p))}^2+2\parallel u_n-p\parallel ({\lambda }_{n}M(p)+{\alpha }_n\parallel rV(p)-A(p)\parallel )\\ +2{\alpha }_n{\lambda }_{n}M(p)(rl\parallel u_n-p\parallel +\parallel rV(p)-A(p)\parallel )\\ +{\alpha }_n^2{(rl\parallel u_n-p\parallel +\parallel rV(p)-A(p)\parallel )}^2.\end{array}$$

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

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

(3.17)

Next,

$$\begin{array}{c}\parallel x_n-T_{{\lambda }_n}{x}_n\parallel \hfill \\ =\parallel x_n-x_{n+1}+x_{n+1}-T_{{\lambda }_n}{u}_n+T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}{x}_n\parallel \hfill \\ \le \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T_{{\lambda }_n}{u}_n\parallel +\parallel T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}{x}_n\parallel \hfill \\ \le \parallel x_n-x_{n+1}\parallel +{\alpha }_n\parallel rV{u}_n-A{T}_{{\lambda }_n}{u}_n\parallel +\parallel u_n-x_n\parallel \hfill \end{array}$$

and then

$$\parallel x_n-T_{{\lambda }_n}{x}_n\parallel \to 0.$$

(3.18)

It follows that $\parallel u_n-T_{{\lambda }_n}{u}_n\parallel \to 0$.

Now, we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈x_n-z,(rV-A)z〉\le 0,$$

where $z\in U\cap EP(\varphi )$ is a unique solution of the variational inequality (3.2).

Indeed, since $\{x_n\}$ is bounded, without loss of generality, we may assume that $x_{n_k}\rightharpoonup \tilde{x}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈x_n-z,(rV-A)z〉=\underset{k\to \mathrm{\infty }}{lim}〈x_{n_k}-z,(rV-A)z〉.$$

(3.19)

By (3.17) and $x_{n_k}\rightharpoonup \tilde{x}$, we derive that $u_{n_k}\rightharpoonup \tilde{x}$.

Note that

$$\begin{array}{rcl}\parallel u_n-T{u}_n\parallel & \le & \parallel u_n-T_{{\lambda }_n}{u}_n\parallel +\parallel T_{{\lambda }_n}{u}_n-T{u}_n\parallel \\ \le & \parallel u_n-T_{{\lambda }_n}{u}_n\parallel +{\lambda }_{n}M(u_n).\end{array}$$

Hence, by $\parallel u_n-T_{{\lambda }_n}{u}_n\parallel \to 0$, we get $\parallel u_n-T{u}_n\parallel \to 0$.

In terms of Lemma 2.2, we get $\tilde{x}\in Fix(T)=U$.

Then, by the same argument as in the proof of Theorem 3.1, we have $\tilde{x}\in U\cap EP(\varphi )$.

Since $z\in U\cap EP(\varphi )$ is the solution of the variational inequality (3.2), we derive from (3.19) that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈x_n-z,(rV-A)z〉\le 0.$$

(3.20)

Finally, we show that $x_n\to z$.

As a matter of fact,

$$\begin{array}{rcl}{x}_{n+1}-z& =& {\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n-z\\ =& (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-z)+{\alpha }_{n}rV{u}_n-{\alpha }_{n}A(z)\\ =& (I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-z)+{\alpha }_{n}r(V{u}_n-Vz)+{\alpha }_n(rV(z)-A(z)).\end{array}$$

So, from (3.5) and (3.9), we derive

$$\begin{array}{rcl}{\parallel x_{n+1}-z\parallel }^2& =& 〈(I-{\alpha }_{n}A)(T_{{\lambda }_n}{u}_n-z),x_{n+1}-z〉\\ +{\alpha }_{n}r〈V{u}_n-Vz,x_{n+1}-z〉+{\alpha }_n〈rV(z)-A(z),x_{n+1}-z〉\\ \le & (1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}{u}_n-z\parallel \parallel x_{n+1}-z\parallel +{\alpha }_{n}rl\parallel u_n-z\parallel \parallel x_{n+1}-z\parallel \\ +{\alpha }_n〈rV(z)-A(z),x_{n+1}-z〉\\ \le & (1-{\alpha }_n\overline{\gamma })(\parallel T_{{\lambda }_n}{u}_n-T_{{\lambda }_n}z\parallel +\parallel T_{{\lambda }_n}z-Tz\parallel )\parallel x_{n+1}-z\parallel \\ +{\alpha }_{n}rl\parallel u_n-z\parallel \parallel x_{n+1}-z\parallel +{\alpha }_n〈rV(z)-A(z),x_{n+1}-z〉\\ \le & (1-{\alpha }_n\overline{\gamma })\parallel u_n-z\parallel \parallel x_{n+1}-z\parallel +(1-{\alpha }_n\overline{\gamma })\parallel T_{{\lambda }_n}z-Tz\parallel \parallel x_{n+1}-z\parallel \\ +{\alpha }_{n}rl\parallel u_n-z\parallel \parallel x_{n+1}-z\parallel +{\alpha }_n〈rV(z)-A(z),x_{n+1}-z〉\\ \le & (1-{\alpha }_n(\overline{\gamma }-rl))\parallel x_n-z\parallel \parallel x_{n+1}-z\parallel \\ +(1-{\alpha }_n\overline{\gamma }){\lambda }_{n}M(z)\parallel x_{n+1}-z\parallel +{\alpha }_n〈rV(z)-A(z),x_{n+1}-z〉\\ \le & (1-{\alpha }_n(\overline{\gamma }-rl))\frac{1}{2}({\parallel x_n-z\parallel }^2+{\parallel x_{n+1}-z\parallel }^2)\\ +{\alpha }_n[〈rV(z)-A(z),x_{n+1}-z〉+\frac{{\lambda }_n}{{\alpha }_n}M(z)\parallel x_{n+1}-z\parallel ].\end{array}$$

It follows that

$$\begin{array}{c}{\parallel x_{n+1}-z\parallel }^2\hfill \\ \le \frac{1-{\alpha }_n(\overline{\gamma }-rl)}{1+{\alpha }_n(\overline{\gamma }-rl)}{\parallel x_n-z\parallel }^2\hfill \\ +\frac{2{\alpha }_n}{1+{\alpha }_n(\overline{\gamma }-rl)}[〈rV(z)-A(z),x_{n+1}-z〉+\frac{{\lambda }_n}{{\alpha }_n}M(z)\parallel x_{n+1}-z\parallel ]\hfill \\ \le (1-{\alpha }_n(\overline{\gamma }-rl)){\parallel x_n-z\parallel }^2\hfill \\ +\frac{2{\alpha }_n}{1+{\alpha }_n(\overline{\gamma }-rl)}[〈rV(z)-A(z),x_{n+1}-z〉+\frac{{\lambda }_n}{{\alpha }_n}M(z)\parallel x_{n+1}-z\parallel ].\hfill \end{array}$$

Since $\{x_n\}$ is bounded, we can take a constant $M_3>0$ such that

$$M_3\ge M(z)\parallel x_{n+1}-z\parallel ,n\ge 1.$$

Then, we obtain that

$${\parallel x_{n+1}-z\parallel }^2\le (1-{\alpha }_n(\overline{\gamma }-rl)){\parallel x_n-z\parallel }^2+{\alpha }_n{\delta }_n,$$

(3.21)

where ${\delta }_n=\frac{2}{1+{\alpha }_n(\overline{\gamma }-rl)}[〈rV(z)-A(z),x_{n+1}-z〉+\frac{{\lambda }_n}{{\alpha }_n}{M}_3]$.

By (3.20) and ${\lambda }_n=o({\alpha }_n)$, we get ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$. Now applying Lemma 2.1 to (3.21) concludes that $x_n\to z$ as $n\to \mathrm{\infty }$. □

4 Application

In this section, we give an application of Theorem 3.2 to the split feasibility problem (SFP for short) which was introduced by Censor and Elfving [25]. Since its inception in 1994, the split feasibility problem (SFP) has received much attention (see [21, 26, 27]) due to its applications in signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy.

The SFP can mathematically be formulated as the problem of finding a point x with the property

$$x\in C\text{and}Bx\in Q,$$

(4.1)

where C and Q are nonempty closed convex subsets of Hilbert spaces $H_1$ and $H_2$, respectively. $B:H_1\to H_2$ is a bounded linear operator.

It is clear that $x^{\ast }$ is a solution to the split feasibility problem (4.1) if and only if $x^{\ast }\in C$ and $B{x}^{\ast }-{Proj}_{Q}B{x}^{\ast }=0$. We define the proximity function f by

$$f(x)=\frac{1}{2}{\parallel Bx-{Proj}_{Q}Bx\parallel }^2$$

and consider the constrained convex minimization problem

$$\underset{x\in C}{min}f(x)=\underset{x\in C}{min}\frac{1}{2}{\parallel Bx-{Proj}_{Q}Bx\parallel }^2.$$

(4.2)

Then $x^{\ast }$ solves the split feasibility problem (4.1) if and only if $x^{\ast }$ solves the minimization problem (4.2) with the minimize equal to 0. Byrne [7] introduced the so-called CQ algorithm to solve the SFP.

$$x_{n+1}={Proj}_C(I-\gamma B^{\ast }(I-{Proj}_Q)B)x_n,n\ge 0,$$

(4.3)

where $0<\gamma <2/{\parallel B\parallel }^2$. He obtained that the sequence $\{x_n\}$ generated by (4.3) converges weakly to a solution of the SFP.

In order to obtain a strong convergence iterative sequence to solve the SFP, we propose the following algorithm:

$$\{\begin{array}{c}\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(4.4)

where $u_n=Q_{{\beta }_n}{x}_n$. Let $\{T_{{\lambda }_n}\}$ satisfy the following conditions:

• (i) ${Proj}_C(I-\gamma (B^{\ast }(I-{Proj}_Q)B+{\lambda }_{n}I))={\theta }_{n}I+(1-{\theta }_n)T_{{\lambda }_n}$ and $\gamma \in (0,2/{\parallel B\parallel }^2)$;

• (ii) ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$ for all n,

where $V:C\to H$ is Lipschitzian with a constant $l>0$ and $A:C\to H$ is a strongly positive bounded linear operator with a constant $\overline{\gamma }>0$. Suppose that $0<r<\overline{\gamma }/l$. We can show that the sequence $\{x_n\}$ generated by (4.4) converges strongly to a solution of SFP (4.1) if the sequence $\{{\alpha }_n\}\subset (0,1)$ and the sequence $\{{\lambda }_n\}$ of parameters satisfy appropriate conditions.

Applying Theorem 3.2, we obtain the following result.

Theorem 4.1 Assume that the split feasibility problem (4.1) is consistent. Let the sequence $\{x_n\}$ be generated by (4.4). Where the sequence $\{{\beta }_n\},\{{\alpha }_n\}\subset (0,1)$ and the sequence $\{{\lambda }_n\}$ satisfy the conditions (C1)-(C3). Then the sequence $\{x_n\}$ converges strongly to a solution of the split feasibility problem (4.1).

Proof By the definition of the proximity function f, we have

$$\mathrm{\nabla }f(x)=B^{\ast }(I-{Proj}_Q)Bx$$

and ∇f is Lipschitz continuous, i.e.,

$$\parallel \mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel \le L\parallel x-y\parallel ,$$

where $L={\parallel B\parallel }^2$.

Set $f_{{\lambda }_n}(x)=f(x)+\frac{{\lambda }_n}{2}{\parallel x\parallel }^2$; consequently,

$$\begin{array}{rcl}\mathrm{\nabla }{f}_{{\lambda }_n}(x)& =& \mathrm{\nabla }f(x)+{\lambda }_{n}I(x)\\ =& B^{\ast }(I-{Proj}_Q)Bx+{\lambda }_{n}x.\end{array}$$

Then the iterative scheme (4.4) is equivalent to

$$\{\begin{array}{c}\varphi (u_n,y)+\frac{1}{{\beta }_n}〈y-u_n,u_n-x_n〉\ge 0\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}rV{u}_n+(I-{\alpha }_{n}A)T_{{\lambda }_n}{u}_n\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(4.5)

where $u_n=Q_{{\beta }_n}{x}_n$. $\{T_{{\lambda }_n}\}$ satisfy the following conditions:

• (i) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})={\theta }_{n}I+(1-{\theta }_n)T_{{\lambda }_n}$ and $\gamma \in (0,2/L)$;

• (ii) ${\theta }_n=\frac{2-\gamma (L+{\lambda }_n)}{4}$ for all n.

Due to Theorem 3.2, we have the conclusion immediately. □

5 Conclusion

Methods for solving the equilibrium problem (EP) and the constrained convex minimization problem have been extensively studied, respectively, in a Hilbert space. In 2012, Tian and Liu proposed an iterative method for finding a common solution of an EP and a constrained convex minimization problem. But, in this paper, it is the first time that we combine the regularized gradient-projection algorithm and the averaged mappings approach to propose implicit and explicit algorithms for finding the common solution of an EP and a constrained convex minimization problem, which also solves a certain variational inequality.