Moudafi’s open question and simultaneous iterative algorithm for general split equality variational inclusion problems and general split equality optimization problems

Abstract

The purpose of this paper is first to introduce and study the general split equality variational inclusion problems and the general split equality optimization problems in the setting of infinite-dimensional Hilbert spaces and then propose a new simultaneous iterative algorithm. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm converging strongly to a solution for these two kinds of problems are proved. As special cases, we shall utilize our results to study the split feasibility problems, the split equality equilibrium problems, and the split optimization problems. The results presented in the paper not only extend and improve the corresponding recent results announced by many authors, but they also provide an affirmative answer to an open question raised by Moudafi in his recent work.

1 Introduction

Let C and Q be nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. The split feasibility problem (SFP) is formulated as

$$\text{to finding }{x}^{\ast }\in C\text{ and }A{x}^{\ast }\in Q,$$

(1.1)

where $A:H_1\to H_2$ is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the (SFP) in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been found that the (SFP) can also be used in various disciplines such as image restoration, and computer tomograph and radiation therapy treatment planning [3–5]. The (SFP) in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10].

Assuming that the (SFP) is consistent, it is not hard to see that $x^{\ast }\in C$ solves (SFP) if and only if it solves the fixed-point equation

$$x^{\ast }=P_C(I-\gamma A^{\ast }(I-P_Q)A)x^{\ast },$$

where $P_C$ and $P_Q$ are the metric projection from $H_1$ onto C and from $H_2$ onto Q, respectively, $\gamma >0$ is a positive constant and $A^{\ast }$ is the adjoint of A.

A popular algorithm to be used to solves the $SFP$ (1.1) is due to Byrne’s CQ-algorithm [2]:

$$x_{k+1}=P_C(I-\gamma A^{\ast }(I-P_Q)A)x_k,k\ge 1,$$

(1.2)

where $\gamma \in (0,2/\lambda )$ with λ being the spectral radius of the operator $A^{\ast }A$.

Recently, Moudafi [11, 12] introduced the following split equality feasibility problem (SEFP):

$$\text{to find }x\in C,y\in Q\text{ such that }Ax=By,$$

(1.3)

where $A:H_1\to H_3$ and $B:H_2\to H_3$ are two bounded linear operators. Obviously, if $B=I$ (identity mapping on $H_2$) and $H_3=H_2$, then (1.3) reduces to (1.1). The kind of split equality problems (1.3) allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, such as decomposition methods for PDEs, and applications in game theory and intensity-modulated radiation therapy.

In order to solve the split equality feasibility problem (1.3), Moudafi [11] introduced the following simultaneous iterative method:

$$\{\begin{array}{l}{x}_{k+1}=P_C(x_k-\gamma A^{\ast }(A{x}_k-B{y}_k)),\\ y_{k+1}=P_Q(y_k+\beta B^{\ast }(A{x}_{k+1}-B{y}_k)),\end{array}$$

(1.4)

and under suitable conditions he proved the weak convergence of the sequence $\{(x_n,y_n)\}$ to a solution of (1.3) in Hilbert spaces.

At the same time, he raised the following open question.

Moudafi’s Open Question 1.1 Is there any strong convergence theorem of an alternating algorithm for the split equality feasibility problem (1.3) in real Hilbert spaces?

More recently, Eslamian and Latif [13], Chen et al. [14], Chuang [15] and Chang and Wang [16] introduced and studied some kinds of general split feasibility problem, general split equality problem, and split variational inclusion problem in real Hilbert spaces. Under suitable conditions some strong convergence theorems are proved. Also a comprehensive survey and update bibliography on split feasibility problems are given in Ansari and Rehan [17].

Motivated by the above works and related literature, in this paper, we continue to consider the problem (1.3). We obtain some strongly convergent theorems to a solution of the problem (1.3) which provide an affirmative answer to Moudafi’s open question.

For the purpose we first introduce and consider the following more general problems.

(I) General split equality variational inclusion problem:

$$\begin{array}{rl}\text{(GSEVIP)}& \text{to find }{x}^{\ast }\in H_1\text{ and }{y}^{\ast }\in H_2\text{ such that}\\ 0\in \bigcap _{i=1}^{\mathrm{\infty }}{U}_i\left(x^{\ast }\right),0\in \bigcap _{i=1}^{\mathrm{\infty }}{K}_i\left(y^{\ast }\right)\text{and}A{x}^{\ast }=B{y}^{\ast },\end{array}$$

(1.5)

where $H_1$, $H_2$ and $H_3$ are three real Hilbert spaces, $U_i:H_1\to H_1$ and $K_i:H_2\to H_2$, $i=1,2,\dots$ are two families of set-valued maximal monotone mappings, $A:H_1\to H_3$ and $B:H_2\to H_3$ are two linear and bounded operators.

(II) General split equality optimization problem:

$$\begin{array}{rl}\text{(GSEOP)}& \text{to find }{x}^{\ast }\in H_1\text{ and }{y}^{\ast }\in H_2\text{ such that for each }i\ge 1\\ h_i\left(x^{\ast }\right)=\underset{x\in H_1}{min}{h}_i(x),g_i\left(y^{\ast }\right)=\underset{y\in H_2}{min}{g}_i(y)\text{and}A{x}^{\ast }=B{y}^{\ast },\end{array}$$

(1.6)

where $H_1$, $H_2$, and $H_3$ are three real Hilbert spaces, $A:H_1\to H_3$ and $B:H_2\to H_3$ are two linear and bounded operators, $h_i:H_1\to \mathbb{R}$ and $g_i:H_2\to \mathbb{R}$ are two countable families of proper, convex, and lower semicontinuous functions.

The following problems are special cases of Problem I and II.

(III) Split equality feasibility problems.

Let $C\subset H_1$ and $Q\subset H_2$ be two nonempty closed convex subsets and $A:H_1\to H_3$, $B:H_2\to H_3$ be two bounded linear operators. As mentioned above the so-called ‘split equality feasibility problem’ (SEFP) is to find

(1.3**)

Let $i_C$ and $i_Q$ be the indicator functions of C and Q, respectively, i.e.,

$$i_C(x)=\{\begin{array}{ll}0,& \text{if }x\in C,\\ +\mathrm{\infty },& \text{if }x\notin C;\end{array}{i}_Q(y)=\{\begin{array}{ll}0,& \text{if }y\in C,\\ +\mathrm{\infty },& \text{if }y\notin Q.\end{array}$$

(1.7)

Denote by $N_C(x)$ and $N_Q(y)$ the normal cones of C and Q at x and y, respectively:

$$\begin{array}{c}{N}_C(x)=\{z\in H_1:〈z,v-x〉\le 0,\mathrm{\forall }v\in C\},\hfill \\ N_Q(y)=\{z\in H_2:〈z,v-y〉\le 0,\mathrm{\forall }v\in Q\}.\hfill \end{array}$$

It is easy to know that $i_C$ and $i_Q$ both are proper convex and lower semicontinuous functions on $H_1$ and $H_2$, respectively, and the sub-differentials $\partial i_C$ and $\partial i_Q$ both are maximal monotone operators. We define the resolvent operator $J_{\beta }^{\partial i_C}$ of $i_C$ by

$$J_{\beta }^{\partial i_C}(x)={(I+\beta \partial i_C)}^{-1}(x),\beta >0,x\in H_1.$$

Here

$$\begin{array}{rl}\partial i_C(x)& =\{z\in H_1:i_C(x)+〈z,u-x〉\le i_C(u),\mathrm{\forall }u\in H_1\}\\ =\{z\in H_1:〈z,u-x〉\le 0,\mathrm{\forall }u\in C\}=N_C(x),x\in C.\end{array}$$

Hence we have

$$\begin{array}{r}u=J_{\beta }^{\partial i_C}(x)\iff x-u\in \beta N_C(u)\\ \iff 〈x-u,y-u〉\le 0,\mathrm{\forall }y\in C\iff u=P_C(x).\end{array}$$

This implies that $J_{\beta }^{\partial i_C}=P_C$ for any $\beta >0$. Similarly, we also have $\partial i_Q(y)=N_Q(y)$, and $J_{\beta }^{\partial i_Q}=P_Q$ for any $\beta >0$. Therefore the (SEFP) (1.3) is equivalent to the following split equality optimization problem, i.e., to find $x^{\ast }\in H_1$, and $y^{\ast }\in H_2$ such that

$$\begin{array}{r}{i}_C\left(x^{\ast }\right)=\underset{x\in H_1}{min}{i}_C(x),i_Q\left(y^{\ast }\right)=\underset{y\in H_2}{min}{i}_Q(y)\text{and}A{x}^{\ast }=B{y}^{\ast };\\ \iff 0\in \partial i_C\left(x^{\ast }\right),0\in \partial i_Q\left(y^{\ast }\right)\text{and}A{x}^{\ast }=B{y}^{\ast }.\end{array}$$

(IV) Split equality equilibrium problem.

Let D be a nonempty closed and convex subset of a real Hilbert space H. A bifunction $g:D\times D\to (-\mathrm{\infty },+\mathrm{\infty })$ is said to be a equilibrium function, if it satisfies the following conditions:

• (A1) $g(x,x)=0$, for all $x\in D$;

• (A2) g is monotone, i.e., $g(x,y)+g(y,x)\le 0$ for all $x,y\in D$;

• (A3) ${lim\hspace{0.17em}sup}_{t\downarrow 0}g(tz+(1-t)x,y)\le g(x,y)$ for all $x,y,z\in D$;

• (A4) for each $x\in D$, $y\mapsto g(x,y)$ is convex and lower semicontinuous.

The so-called equilibrium problem with respect to the equilibrium function g is

$$\text{to find }{x}^{\ast }\in D\text{ such that }g(x^{\ast },y)\ge 0,\mathrm{\forall }y\in D.$$

(1.8)

Its solution set is denoted by $EP(g)$.

For given $\lambda >0$ and $x\in H$, the resolvent of the equilibrium function g is the operator $R_{\lambda ,g}:H\to D$ defined by

$$R_{\lambda ,g}(x):=\{z\in D:g(z,y)+\frac{1}{\lambda }〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in D\}.$$

(1.9)

Proposition 1.2 [18]

The resolvent operator $R_{\lambda ,g}$ of the equilibrium function g has the following properties:

• (1) $R_{\lambda ,g}$ is single-valued;

• (2) $F(R_{\lambda ,g})=EP(g)$ and $EP(g)$ is a nonempty closed and convex subset of D;

• (3) $R_{\lambda ,g}$ is a firmly nonexpansive mapping.

Let $h,g:D\times D\to (-\mathrm{\infty },+\mathrm{\infty })$ be two equilibrium functions. For given $\lambda >0$, let $R_{\lambda ,h}$ and $R_{\lambda ,g}$ be the resolvent of h and g (defined by (1.9)), respectively.

The so-called split equality equilibrium problem with respective to h, g, and D $(SEEP(h,g,D))$ is to find $x^{\ast }\in D$, $y^{\ast }\in D$ such that

$$h(x^{\ast },u)\ge 0,\mathrm{\forall }u\in D,g(y^{\ast },v)\ge 0,\mathrm{\forall }v\in D\text{and}A{x}^{\ast }=B{y}^{\ast },$$

(1.10)

where $A,B:D\to D$ are two linear and bounded operators.

By Proposition 1.2, the $(SEEP(h,g,D))$ (1.10) is equivalent to find $x^{\ast }\in D$, $y^{\ast }\in D$ such that for each $\lambda >0$

$$\begin{array}{r}{x}^{\ast }\in EP(h,D),y^{\ast }\in EP(g,D)\text{and}A{x}^{\ast }=B{y}^{\ast }\\ \iff x^{\ast }\in F(R_{\lambda h}),y^{\ast }\in F(R_{\lambda g})\text{and}A{x}^{\ast }=B{y}^{\ast }.\end{array}$$

Letting $C=F(R_{\lambda h})$, $Q=F(R_{\lambda g})$, by Proposition 1.2, C and Q both are nonempty closed and convex subset of D. Hence the problem (1.10) is equivalent to the following split equality feasibility problem:

$$\text{to find }{x}^{\ast }\in C,y^{\ast }\in Q\text{ such that }A{x}^{\ast }=B{y}^{\ast }.$$

(1.11)

(V) Split optimization problem.

Let $H_1$ and $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a linear and bounded operators, $h:H_1\to \mathbb{R}$ and $g:H_2\to \mathbb{R}$ be two proper convex and lower semicontinuous functions. The split optimization problem (SOP) is to find $x^{\ast }\in H_1$, $A{x}^{\ast }\in H_2$ such that

$$h\left(x^{\ast }\right)=\underset{x\in H_1}{min}{h}_i(x)\text{and}g\left(A{x}^{\ast }\right)=\underset{z\in H_2}{min}g(z).$$

(1.12)

Denote by $U=\partial h$ and $K=\partial g$, then the (SOP) (1.12) is equivalent to the following split variational inclusion problem (SVIP): to find $x^{\ast }\in H_1$ such that

$$0\in U\left(x^{\ast }\right),0\in K\left(A{x}^{\ast }\right).$$

(1.13)

For solving (GSEVIP) (1.5) and (GSEOP) (1.6), in Sections 3 and 4, we propose a new simultaneous type iterative algorithm. Under suitable conditions some strong convergence theorems for the sequences generated by the algorithm are proved in the setting of infinite-dimensional Hilbert spaces. As special cases, we shall utilize our results to study the split feasibility problem, split equality equilibrium problem and the split optimization problem. By the way, we obtain a strongly convergent iterative sequence to a solution of the problem (1.3), which provides an affirmative answer to the open question raised by Moudafi [11]. The results presented in the paper extend and improve the corresponding results announced by Moudafi et al. [11, 12, 19], Eslamian and Latif [13], Chen et al. [14], Censor et al. [1, 3–5, 20], Chuang [15], Naraghirad [21], Chang and Wang [16], Ansari and Rehan [17], and some others.

2 Preliminaries

We first recall some definitions, notations, and conclusions.

Throughout this paper, we assume that H is a real Hilbert space and C is a nonempty closed convex subset of H. In the sequel, we denote by $F(T)$ the set of fixed points of a mapping T and by $x_n\to x^{\ast }$ and $x_n\rightharpoonup x^{\ast }$, the strong convergence, and weak convergence of a sequence $\{x_n\}$ to a point $x^{\ast }$, respectively.

Recall that a mapping $T:H\to H$ is said to be nonexpansive, if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$, $\mathrm{\forall }x,y\in H$. A typical example of nonexpansive mapping is the metric projection $P_C$ from H onto $C\subseteq H$ defined by $\parallel x-P_{C}x\parallel ={inf}_{y\in C}\parallel x-y\parallel$. The metric projection $P_C$ is firmly nonexpansive, if

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

(2.1)

and it can be characterized by the fact that

$$P_C(x)\in C\text{and}〈y-P_C(x),x-P_C(x)〉\le 0,\mathrm{\forall }x\in H,y\in C.$$

(2.2)

A mapping $T:H\to H$ is said to be quasinonexpansive, if $F(T)\ne \mathrm{\varnothing }$, and

$$\parallel Tx-p\parallel \le \parallel x-p\parallel ,\text{for each }x\in H\text{ and }p\in F(T).$$

It is easy to see that if T is a quasi-nonexpansive mapping, then $F(T)$ is a closed and convex subset of C. Besides, T is said to be a firmly nonexpansive, if

$$\begin{array}{r}{\parallel Tx-Ty\parallel }^2\le 〈x-y,Tx-Ty〉\mathrm{\forall }x,y\in C\\ \iff {\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel (I-T)x-(I-T)y\parallel }^2\mathrm{\forall }x,y\in C.\end{array}$$

Lemma 2.1 [22]

Let H be a real Hilbert space, and $\{x_n\}$ be a sequence in H. Then, for any given sequence $\{{\lambda }_n\}$ of positive numbers with ${\sum }_{i=1}^{\mathrm{\infty }}{\lambda }_n=1$ for any positive integers i, j with $i<j$ the following holds:

$${\parallel \sum _{i=1}^{\mathrm{\infty }}{\lambda }_n{x}_n\parallel }^2\le \sum _{i=1}^{\mathrm{\infty }}{\lambda }_n{\parallel x_n\parallel }^2-{\lambda }_i{\lambda }_j{\parallel x_i-x_j\parallel }^2.$$

Lemma 2.2 [23]

Let H be a real Hilbert space. For any $x,y\in H$, the following inequality holds:

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

Lemma 2.3 [24]

Let $\{t_n\}$ be a sequence of real numbers. If there exists a subsequence $\{n_i\}$ of $\{n\}$ such that $t_{n_i}<t_{n_i+1}$ for all $i\ge 1$, then there exists a nondecreasing sequence $\{\tau (n)\}$ with $\tau (n)\to \mathrm{\infty }$ such that for all (sufficiently large) positive integer number n, the following holds:

$$t_{\tau (n)}\le t_{\tau (n)+1},t_n\le t_{\tau (n)+1}.$$

In fact,

$$\tau (n)=max\{k\le n:t_k\le t_{k+1}\}.$$

Definition 2.4 (Demiclosedness principle)

Let C be a nonempty closed convex subset of a real Hilbert space H, and $T:C\to C$ be a mapping with $F(T)\ne \mathrm{\varnothing }$. Then $I-T$ is said to be demiclosed at zero, if for any sequence $\{x_n\}\subset C$ with $x_n\rightharpoonup x$ and $\parallel x_n-T{x}_n\parallel \to 0$, $x=Tx$.

Remark 2.5 [25]

It is well known that if $T:C\to C$ is a nonexpansive mapping, then $I-T$ is demiclosed at zero.

Lemma 2.6 Let $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ be sequences of positive real numbers satisfying $a_{n+1}\le (1-b_n)a_n+c_n$ for all $n\ge 1$. If the following conditions are satisfied:

• (1) $b_n\in (0,1)$ and ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n=\mathrm{\infty }$,

• (2) ${\sum }_{n=1}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$, or ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{c_n}{b_n}\le 0$,

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

Lemma 2.7 [15]

Let H be a real Hilbert space, $B:H\to 2^H$ be a set-valued maximal monotone mapping, $\beta >0$, and let $J_{\beta }^B$ be the resolvent mapping of B defined by $J_{\beta }^B:={(I+\beta B)}^{-1}$, then

• (i) for each $\beta >0$, $J_{\beta }^B$ is a single-valued and firmly nonexpansive mapping;

• (ii) $D(J_{\beta }^B)=H$ and $F(J_{\beta }^B)=B^{-1}(0)$;

• (iii) $(I-J_{\beta }^B)$ is a firmly nonexpansive mapping for each $\beta >0$;

• (iv) suppose that $B^{-1}(0)\ne \mathrm{\varnothing }$, then for each $x\in H$, each $x^{\ast }\in B^{-1}(0)$ and each $\beta >0$

  $${\parallel x-J_{\beta }^{B}x\parallel }^2+\parallel J_{\beta }^{B}x-x^{\ast }\parallel \le {\parallel x-x^{\ast }\parallel }^2;$$

• (v) suppose that $B^{-1}(0)\ne \mathrm{\varnothing }$. Then $〈x-J_{\beta }^{B}x,J_{\beta }^{B}x-w〉\ge 0$ for each $x\in H$, each $w\in B^{-1}(0)$, and each $\beta >0$.

Lemma 2.8 Let $H_1$, $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a linear bounded operator and $A^{\ast }$ be the adjoint of A. Let $B:H_2\to 2^{H_2}$ be a set-valued maximal monotone mapping, $\beta >0$, and let $J_{\beta }^B$ be the resolvent mapping of B, then

• (i) ${\parallel (I-J_{\beta }^B)Ax-(I-J_{\beta }^B)Ay\parallel }^2\le 〈(I-J_{\beta }^B)Ax-(I-J_{\beta }^B)Ay,Ax-Ay〉$;

• (ii) ${\parallel A^{\ast }(I-J_{\beta }^B)Ax-A^{\ast }(I-J_{\beta }^B)Ay\parallel }^2\le {\parallel A\parallel }^2〈(I-J_{\beta }^B)Ax-(I-J_{\beta }^B)Ay,Ax-Ay〉$;

• (iii) if $\rho \in (0,\frac{2}{{\parallel A\parallel }^2})$, then $(I-\rho A^{\ast }(I-J_{\beta }^B)A)$ is a nonexpansive mapping.

Proof By Lemma 2.7(iii), the mapping $(I-J_{\beta }^B)$ is firmly nonexpansive, hence the conclusions (i) and (ii) are obvious.

Now we prove the conclusion (iii).

In fact, for any $x,y\in H_1$, it follows from the conclusions (i) and (ii) that

$$\begin{array}{r}{\parallel (I-\rho A^{\ast }(I-J_{\beta }^B)A)x-(I-\rho A^{\ast }(I-J_{\beta }^B)A)y\parallel }^2\\ ={\parallel x-y\parallel }^2-2\rho 〈x-y,A^{\ast }(I-J_{\beta }^B)Ax-A^{\ast }(I-J_{\beta }^B)Ay〉\\ +{\rho }^2{\parallel A^{\ast }(I-J_{\beta }^B)Ax-A^{\ast }(I-J_{\beta }^B)Ay\parallel }^2\\ \le {\parallel x-y\parallel }^2-2\rho 〈Ax-Ay,(I-J_{\beta }^B)Ax-(I-J_{\beta }^B)Ay〉\\ +{\rho }^2{\parallel A\parallel }^2{\parallel (I-J_{\beta }^B)Ax-(I-J_{\beta }^B)Ay\parallel }^2\\ \le {\parallel x-y\parallel }^2-\rho (2-\rho {\parallel A\parallel }^2){\parallel (I-J_{\beta }^B)Ax-(I-J_{\beta }^B)Ay\parallel }^2\\ \le {\parallel x-y\parallel }^2(\text{since }\rho (2-\rho {\parallel A\parallel }^2)\ge 0).\end{array}$$

This completes the proof of Lemma 2.8. □

3 General split equality variational inclusion problem and strong convergence theorems

Throughout this section we assume that

• (1) $H_1$, $H_2$, $H_3$ are three real Hilbert spaces;

• (2) ${\{U_i\}}_{i=1}^{\mathrm{\infty }}:H_1\to 2^{H_1}$ and ${\{K_i\}}_{i=1}^{\mathrm{\infty }}:H_2\to 2^{H_2}$ are two families of set-valued maximal monotone mappings, $\beta >0$ and $\gamma >0$ are given positive numbers;

• (3) $A:H_1\to H_3$ and $B:H_2\to H_3$ are two bounded linear operators and $A^{\ast }$, $B^{\ast }$ are the adjoint of A and B, respectively;

• (4) $f=\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]$, where $f_i$, $i=1,2$ is a k-contractive mapping on $H_i$ with $k\in (0,1)$;

• (5) the set of solutions of (GSEVIP) (1.5) $\mathrm{\Omega }\ne \mathrm{\varnothing }$,

  $$J_{{\mu }_i}^{(U_i,K_i)}:=\left[\begin{array}{c}{J}_{{\mu }_i}^{U_i}\\ J_{{\mu }_i}^{K_i}\end{array}\right],G=[A-B],G^{\ast }=\left[\begin{array}{c}{A}^{\ast }\\ -B^{\ast }\end{array}\right],G^{\ast }G=\left[\begin{array}{cc}{A}^{\ast }A& -A^{\ast }B\\ -B^{\ast }A& B^{\ast }B\end{array}\right],$$

• (6) for any given $w_0\in H_1\times H_2$, the iterative sequence $\{w_n\}\subset H_1\times H_2$ is generated by

  $$w_{n+1}={\alpha }_n{w}_n+{\beta }_{n}f(w_n)+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\left(J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n\right),n\ge 0,$$

  (3.1)

or its equivalent form:

((3.1)′)

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_{n,i}\}$ are the sequences of nonnegative numbers satisfying

$${\alpha }_n+{\beta }_n+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}=1,\text{for each }n\ge 0.$$

We are now in a position to give the following results.

Lemma 3.1 Let $H_1$, $H_2$, $H_3$, A, B, $A^{\ast }$, $B^{\ast }$, $\{U_i\}$, $\{K_i\}$, $J_{{\mu }_i}^{(U_i,K_i)}$, G, $G^{\ast }$ be the same as above. If $\mathrm{\Omega }\ne \mathrm{\varnothing }$ (the solution set of (GSEVIP) (1.5)), then $w^{\ast }:=(x^{\ast },y^{\ast })\in H_1\times H_2$ is a solution of (GSEVIP) (1.5) if and only if for each $i\ge 1$, and for any given $\gamma >0$ and $\mu >0$

$$w^{\ast }=J_{\mu }^{(U_i,K_i)}(I-\gamma G^{\ast }G)w^{\ast }.$$

(3.2)

Proof Indeed, if $w^{\ast }=(x^{\ast },y^{\ast })\in H_1\times H_2$ is a solution of (GSEVIP) (1.5), then by Lemma 2.7(ii), for each $i\ge 1$, and for any $\gamma >0$ and $\mu >0$ we have

$$\begin{array}{c}{x}^{\ast }\in U_i^{-1}(0)=F\left(J_{\mu }^{U_i}\right),y^{\ast }\in K_i^{-1}(0)=F\left(J_{\mu }^{K_i}\right)\text{and}A{x}^{\ast }=B{y}^{\ast }\hfill \\ \iff x^{\ast }=J_{\mu }^{U_i}{x}^{\ast },y^{\ast }=J_{\mu }^{K_i}{y}^{\ast }\text{and}A{x}^{\ast }=B{y}^{\ast }.\hfill \end{array}$$

Hence we have $G(w^{\ast })=A{x}^{\ast }-B{y}^{\ast }=0$, and so

$$J_{{\mu }_i}^{(U_i,K_i)}(I-\gamma G^{\ast }G)\left(w^{\ast }\right)=J_{\mu }^{(U_i,K_i)}\left(w^{\ast }\right)=(J_{\mu }^{U_i}{x}^{\ast },J_{\mu }^{K_i}{y}^{\ast })=w^{\ast }.$$

This implies that (3.2) is true.

Conversely, if $w^{\ast }=(x^{\ast },y^{\ast })\in H_1\times H_2$ satisfies (3.2), then we have

$$\{\begin{array}{l}{x}^{\ast }=J_{\mu }^{U_i}[x^{\ast }-\gamma A^{\ast }(A{x}^{\ast }-B{y}^{\ast })],\\ y^{\ast }=J_{\mu }^{K_i}[y^{\ast }+\gamma B^{\ast }(A{x}^{\ast }-B{y}^{\ast })].\end{array}$$

(3.3)

We make the assumption that the solution set Ω of (GSEVIP) (1.5) is nonempty. Hence the sets $U_i^{-1}(0)$ and $K_i^{-1}(0)$ both are nonempty. By Lemma 2.7(v) and (3.3), we have

$$〈x^{\ast }-(x^{\ast }-\gamma A^{\ast }(A{x}^{\ast }-B{y}^{\ast })),x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in U_i^{-1}(0),$$

and so

$$〈A{x}^{\ast }-B{y}^{\ast },Ax-A{x}^{\ast }〉\ge 0,\mathrm{\forall }x\in U_i^{-1}(0).$$

(3.4)

Similarly, by Lemma 2.7(v) and (3.3) again, one gets

$$〈A{x}^{\ast }-B{y}^{\ast },B{y}^{\ast }-By〉\ge 0,\mathrm{\forall }y\in K_i^{-1}(0).$$

(3.5)

Adding up (3.4) and (3.5), we have

$$〈A{x}^{\ast }-B{y}^{\ast },Ax-A{x}^{\ast }+B{y}^{\ast }-By〉\ge 0,\mathrm{\forall }x\in U_i^{-1}(0)\text{ and }y\in K_i^{-1}(0).$$

Simplifying it, we have

$${\parallel A{x}^{\ast }-B{y}^{\ast }\parallel }^2\le 〈A{x}^{\ast }-B{y}^{\ast },Ax-By〉,\mathrm{\forall }x\in U_i^{-1}(0)\text{ and }y\in K_i^{-1}(0).$$

(3.6)

Since $\mathrm{\Omega }\ne \mathrm{\varnothing }$, taking $\overline{w}=(\overline{x},\overline{y})\in \mathrm{\Omega }$, for each $i\ge 1$, we have $\overline{x}\in U_i^{-1}(0)$ and $\overline{y}\in K_i^{-1}(0)$ and $A\overline{x}=B\overline{y}$. In (3.6), taking $x=\overline{x}$ and $y=\overline{y}$, we have

$$\parallel A{x}^{\ast }-B{y}^{\ast }\parallel =0,\mathit{\text{i.e.}},A{x}^{\ast }=B{y}^{\ast }.$$

(3.7)

Hence from (3.3) and (3.7)

$$\{\begin{array}{l}{x}^{\ast }=J_{\mu }^{U_i}(x^{\ast }),\\ y^{\ast }=J_{\mu }^{K_i}(y^{\ast }),\end{array}\iff 0\in U_i\left(x^{\ast }\right),0\in K_i\left(y^{\ast }\right),\mathrm{\forall }i\ge 1.$$

(3.8)

It follows from (3.7) and (3.8) that $w^{\ast }$ is a solution of (GSEVIP) (1.5).

This completes the proof of Lemma 3.1. □

Lemma 3.2 If $\lambda \in (0,\frac{2}{L})$, where $L={\parallel G\parallel }^2$, then $(I-\lambda G^{\ast }G):H_1\times H_2\to H_1\times H_2$ is a nonexpansive mapping.

Proof In fact, for any $w,u\in H_1\times H_2$, we have

$$\begin{array}{r}{\parallel (I-\lambda G^{\ast }G)u-(I-\lambda G^{\ast }G)w\parallel }^2\\ ={\parallel (u-w)-\lambda G^{\ast }G(u-w)\parallel }^2\\ ={\parallel u-w\parallel }^2+{\lambda }^2{\parallel G^{\ast }G(u-w)\parallel }^2-2\lambda 〈u-w,G^{\ast }G(u-w)〉\\ \le {\parallel u-w\parallel }^2+{\lambda }^{2}L{\parallel G(u-w)\parallel }^2-2\lambda 〈G(u-w),G(u-w)〉\\ ={\parallel u-w\parallel }^2+{\lambda }^{2}L{\parallel G(u-w)\parallel }^2-2\lambda {\parallel G(u-w)\parallel }^2\\ ={\parallel u-w\parallel }^2-\lambda (2-\lambda L){\parallel G(u-w)\parallel }^2\\ \le {\parallel u-w\parallel }^2.\end{array}$$

This completes the proof. □

Theorem 3.3 Let $H_1$, $H_2$, $H_3$, A, B, $A^{\ast }$, $B^{\ast }$, $\{U_i\}$, $\{K_i\}$, $J_{{\mu }_i}^{(U_i,K_i)}$, G, $G^{\ast }$, f be the same as above. Let $\{w_n\}$ be the sequence defined by (3.1). If the solution set Ω of (GSEVIP) (1.5) is nonempty and the following conditions are satisfied:

• (i) ${\alpha }_n+{\beta }_n+{\sum }_{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}=1$, for each $n\ge 0$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$, and ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_{n,i}>0$ for each $i\ge 1$;

• (iv) $\{{\lambda }_{n,i}\}\subset (0,\frac{2}{L})$ for each $i\ge 1$, where $L={\parallel G\parallel }^2$,

then the sequence $\{w_n\}$ converges strongly to $w^{\ast }=P_{\mathrm{\Omega }}f(w^{\ast })$, which is a solution of (GSEVIP) (1.5).

Proof (I) First we prove that the sequence $\{w_n\}$ is bounded.

In fact, for any given $z\in \mathrm{\Omega }$, it follows from Lemma 3.1, Lemma 3.2, and condition (iv) that

$$z=J_{\mu }^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)z,\text{for each }i\ge 1,$$

and $(I-{\lambda }_{n,i}{G}^{\ast }G):H_1\times H_2\to H_1\times H_2$ is a nonexpansive mapping. Also by Lemma 2.7(i), for each $i\ge 1$, $J_{{\mu }_i}^{(U_i,K_i)}$ is a firmly nonexpansive mapping. Hence we have

$$\begin{array}{rl}\parallel w_{n+1}-z\parallel =& \parallel ({\alpha }_n{w}_n+{\beta }_{n}f(w_n)+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}{J}_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n)-z\parallel \\ \le & {\alpha }_n\parallel w_n-z\parallel +{\beta }_n\parallel f(w_n)-z\parallel +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n-z\parallel \\ \le & {\alpha }_n\parallel w_n-z\parallel +{\beta }_n\parallel f(w_n)-z\parallel +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel (I-{\lambda }_{n,i}{G}^{\ast }G)w_n-z\parallel \\ \le & {\alpha }_n\parallel w_n-z\parallel +{\beta }_n\parallel f(w_n)-z\parallel \\ +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel (I-{\lambda }_{n,i}{G}^{\ast }G)w_n-(I-{\lambda }_{n,i}{G}^{\ast }G)z\parallel \\ \le & {\alpha }_n\parallel w_n-z\parallel +{\beta }_n\parallel f(w_n)-z\parallel +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel w_n-z\parallel \\ =& (1-{\beta }_n)\parallel w_n-z\parallel +{\beta }_n\parallel f(w_n)-z\parallel \\ \le & (1-{\beta }_n)\parallel w_n-z\parallel +{\beta }_n\parallel f(w_n)-f(z)\parallel +{\beta }_n\parallel f(z)-z\parallel \\ \le & (1-{\beta }_n)\parallel w_n-z\parallel +k{\beta }_n\parallel w_n-z\parallel +{\beta }_n\parallel f(z)-z\parallel \\ =& (1-(1-k){\beta }_n)\parallel w_n-z\parallel +(1-k){\beta }_n\frac{1}{1-k}\parallel f(z)-z\parallel \\ \le & max\{\parallel w_n-z\parallel ,\frac{1}{1-k}\parallel f(z)-z\parallel \}.\end{array}$$

By induction, we can prove that

$$\parallel w_n-z\parallel \le max\{\parallel w_0-z\parallel ,\frac{1}{1-k}\parallel f(z)-z\parallel \},\mathrm{\forall }n\ge 0.$$

This shows that $\{w_n\}$ is bounded, and so is $\{f(w_n)\}$.

(II) Now we prove that the following inequality holds:

$$\begin{array}{r}{\alpha }_n{\gamma }_{n,i}{\parallel w_n-J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n\parallel }^2\\ \le {\parallel w_n-z\parallel }^2-{\parallel w_{n+1}-z\parallel }^2+{\beta }_n{\parallel f(w_n)-z\parallel }^2\text{for each }i\ge 1.\end{array}$$

(3.9)

Indeed, it follows from (3.1) and Lemma 2.1 that for each $i\ge 1$

$$\begin{array}{rl}{\parallel w_{n+1}-z\parallel }^2=& {\parallel {\alpha }_n(w_n-z)+{\beta }_n(f(w_n)-z)+\sum _{j=1}^{\mathrm{\infty }}{\gamma }_{n,j}(J_{{\mu }_j}^{(U_j,K_j)}(I-{\lambda }_{n,j}{G}^{\ast }G)w_n-z)\parallel }^2\\ \le & {\alpha }_n{\parallel w_n-z\parallel }^2+{\beta }_n{\parallel f(w_n)-z\parallel }^2+\sum _{j=1}^{\mathrm{\infty }}{\gamma }_{n,j}{\parallel J_{{\mu }_j}^{(U_j,K_j)}(I-{\lambda }_{n,j}{G}^{\ast }G)w_n-z\parallel }^2\\ -{\alpha }_n{\gamma }_{n,i}{\parallel w_n-J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n\parallel }^2\\ \le & {\alpha }_n{\parallel w_n-z\parallel }^2+{\beta }_n{\parallel f(w_n)-z\parallel }^2+\sum _{j=1}^{\mathrm{\infty }}{\gamma }_{n,j}{\parallel w_n-z\parallel }^2\\ -{\alpha }_n{\gamma }_{n,i}{\parallel w_n-J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n\parallel }^2\\ =& (1-{\beta }_n){\parallel w_n-z\parallel }^2+{\beta }_n{\parallel f(w_n)-z\parallel }^2\\ -{\alpha }_n{\gamma }_{n,i}{\parallel w_n-J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n\parallel }^2.\end{array}$$

This implies that for each $i\ge 1$

$${\alpha }_n{\gamma }_{n,i}{\parallel w_n-J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n\parallel }^2\le {\parallel w_n-z\parallel }^2-{\parallel w_{n+1}-z\parallel }^2+{\beta }_n{\parallel f(w_n)-z\parallel }^2.$$

Inequality (3.3) is proved.

It is easy to see that the solution set Ω of (GSEVIP) (1.5) is a closed and convex subset in $H_1\times H_2$. By the assumption that Ω is nonempty, so it is a nonempty closed and convex subset in $H_1\times H_2$. Hence the metric projection $P_{\mathrm{\Omega }}$ is well defined. In addition, since $P_{\mathrm{\Omega }}f:H_1\times H_2\to \mathrm{\Omega }$ is a contractive mapping, there exists a unique $w^{\ast }\in \mathrm{\Omega }$ such that

$$w^{\ast }=P_{\mathrm{\Omega }}f\left(w^{\ast }\right).$$

(3.10)

(III) Now we prove that $\{w_n\}$ converges strongly to $w^{\ast }$.

For the purpose, we consider two cases.

Case I. Suppose that the sequence $\{\parallel w_n-w^{\ast }\parallel \}$ is monotone. Since $\{\parallel w_n-w^{\ast }\parallel \}$ is bounded, $\{\parallel w_n-w^{\ast }\parallel \}$ is convergent. Since $w^{\ast }\in \mathrm{\Omega }$, in (3.9) taking $z=w^{\ast }$ and letting $n\to \mathrm{\infty }$, in view of conditions (ii) and (iii), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel w_n-J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n\parallel =0,\text{for each }i\ge 1.$$

(3.11)

On the other hand, by Lemma 2.2 and (3.1), we have

$$\begin{array}{rl}{\parallel w_{n+1}-w^{\ast }\parallel }^2=& {\parallel ({\alpha }_n{w}_n+{\beta }_{n}f(w_n)+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}{J}_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n)-w^{\ast }\parallel }^2\\ =& \parallel {\alpha }_n(w_n-w^{\ast })+{\beta }_n(f(w_n)-w^{\ast })\\ +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}(J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n-w^{\ast }){\parallel }^2\\ \le & {\parallel {\alpha }_n(w_n-w^{\ast })+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}(J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)w_n-w^{\ast })\parallel }^2\\ +2{\beta }_n〈f(w_n)-w^{\ast },w_{n+1}-w^{\ast }〉(\text{by Lemma }\text{2.2})\\ \le & {\{{\alpha }_n\parallel w_n-w^{\ast }\parallel +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel w_n-w^{\ast }\parallel \}}^2\\ +2{\beta }_n〈f(w_n)-f\left(w^{\ast }\right),w_{n+1}-w^{\ast }〉+2{\beta }_n〈f\left(w^{\ast }\right)-w^{\ast },w_{n+1}-w^{\ast }〉\\ =& {(1-{\beta }_n)}^2{\parallel w_n-w^{\ast }\parallel }^2+2{\beta }_{n}k\parallel w_n-w^{\ast }\parallel \parallel w_{n+1}-w^{\ast }\parallel \\ +2{\beta }_n〈f\left(w^{\ast }\right)-w^{\ast },w_{n+1}-w^{\ast }〉\\ \le & {(1-{\beta }_n)}^2{\parallel w_n-w^{\ast }\parallel }^2+{\beta }_{n}k\{{\parallel w_n-w^{\ast }\parallel }^2+{\parallel w_{n+1}-w^{\ast }\parallel }^2\}\\ +2{\beta }_n〈f\left(w^{\ast }\right)-w^{\ast },w_{n+1}-w^{\ast }〉.\end{array}$$

Simplifying it we have

$$\begin{array}{rl}{\parallel w_{n+1}-w^{\ast }\parallel }^2\le & \frac{{(1-{\beta }_n)}^2+{\beta }_{n}k}{1-{\beta }_{n}k}{\parallel w_n-w^{\ast }\parallel }^2+\frac{2{\beta }_n}{1-{\beta }_{n}k}〈f\left(w^{\ast }\right)-w^{\ast },w_{n+1}-w^{\ast }〉\\ =& \frac{1-2{\beta }_n+{\beta }_{n}k}{1-{\beta }_{n}k}{\parallel w_n-w^{\ast }\parallel }^2+\frac{{\beta }_n^2}{1-{\beta }_{n}k}{\parallel w_n-w^{\ast }\parallel }^2\\ +\frac{2{\beta }_n}{1-{\beta }_{n}k}〈f\left(w^{\ast }\right)-w^{\ast },w_{n+1}-w^{\ast }〉\\ =& (1-\frac{2(1-k){\beta }_n}{1-{\beta }_{n}k}){\parallel w_n-w^{\ast }\parallel }^2\\ +\frac{2(1-k){\beta }_n}{1-{\beta }_{n}k}\{\frac{{\beta }_{n}M}{2(1-k)}+\frac{1}{1-k}〈f\left(w^{\ast }\right)-w^{\ast },w_{n+1}-w^{\ast }〉\}\\ =& (1-{\eta }_n){\parallel w_n-w^{\ast }\parallel }^2+{\eta }_n{\delta }_n,\end{array}$$

(3.12)

where

$$\begin{array}{r}{\eta }_n=\frac{2(1-k){\beta }_n}{1-{\beta }_{n}k},{\delta }_n=\frac{{\beta }_{n}M}{2(1-k)}+\frac{1}{1-k}〈f\left(w^{\ast }\right)-w^{\ast },w_{n+1}-w^{\ast }〉,\\ M=\underset{n\ge 0}{sup}{\parallel w_n-w^{\ast }\parallel }^2.\end{array}$$

By condition (ii), ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$, and so is ${\sum }_{n=1}^{\mathrm{\infty }}{\eta }_n=\mathrm{\infty }$.

Next we prove that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\delta }_n\le 0.$$

(3.13)

In fact, since $\{w_n\}$ is bounded in $H_1\times H_2$, there exists a subsequence $\{w_{n_k}\}\subset \{w_n\}$ with $w_{n_k}\rightharpoonup v^{\ast }$ (some point in $C\times Q$), and ${\lambda }_{n_k,i}\to {\lambda }_i\in (0,\frac{2}{L})$ such that

$$\underset{n\to \mathrm{\infty }}{lim}〈f\left(w^{\ast }\right)-w^{\ast },w_{n_k}-w^{\ast }〉=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f\left(w^{\ast }\right)-w^{\ast },w_n-w^{\ast }〉.$$

Since

$$\parallel w_{n_k}-J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n_k,i}{G}^{\ast }G)w_{n_k}\parallel \to 0,\text{for each }i\ge 1$$

and $J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n_k,i}{G}^{\ast }G)$ is a nonexpansive mapping, by Remark 2.5, $I-J_{{\mu }_i}^{(B_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)$ is demiclosed at zero, hence we have

$$v^{\ast }=J_{{\mu }_i}^{(U_i,K_i)}(I-{\lambda }_{n,i}{G}^{\ast }G)v^{\ast },\mathrm{\forall }i\ge 1.$$

(3.14)

By Lemma 3.1, this implies that $v^{\ast }\in \mathrm{\Omega }$. In addition, since $w^{\ast }=P_{\mathrm{\Omega }}f(w^{\ast })$, we have

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f\left(w^{\ast }\right)-w^{\ast },w_n-w^{\ast }〉& =\underset{n\to \mathrm{\infty }}{lim}〈f\left(w^{\ast }\right)-w^{\ast },w_{n_k}-w^{\ast }〉\\ =〈f\left(w^{\ast }\right)-w^{\ast },v^{\ast }-w^{\ast }〉\le 0.\end{array}$$

This shows that (3.13) is true. Taking $a_n={\parallel w_n-w^{\ast }\parallel }^2$, $b_n={\eta }_n$, and $c_n={\delta }_n{\eta }_n$ in Lemma 2.6, therefore all conditions in Lemma 2.6 are satisfied. We have $w_n\to w^{\ast }$.

Case II. If the sequence $\{\parallel w_n-w^{\ast }\parallel \}$ is not monotone, by Lemma 2.3, there exists a sequence of positive integers: $\{\tau (n)\}$, $n\ge n_0$ (where $n_0$ large enough) such that

$$\tau (n)=max\{k\le n:\parallel w_k-w^{\ast }\parallel \le \parallel w_{k+1}-w^{\ast }\parallel \}.$$

(3.15)

Clearly $\{\tau (n)\}$ is a nondecreasing, $\tau (n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$, and for all $n\ge n_0$

$$\parallel w_{\tau (n)}-w^{\ast }\parallel \le \parallel w_{\tau (n)+1}-w^{\ast }\parallel ;\parallel w_n-w^{\ast }\parallel \le \parallel w_{\tau (n)+1}-w^{\ast }\parallel .$$

(3.16)

Therefore $\{\parallel w_{\tau (n)}-w^{\ast }\parallel \}$ is a nondecreasing sequence. According to Case I, ${lim}_{n\to \mathrm{\infty }}\parallel w_{\tau (n)}-w^{\ast }\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel w_{\tau (n)+1}-w^{\ast }\parallel =0$. Hence we have

$$0\le \parallel w_n-w^{\ast }\parallel \le max\{\parallel w_n-w^{\ast }\parallel ,\parallel w_{\tau (n)}-w^{\ast }\parallel \}\le \parallel w_{\tau (n)+1}-w^{\ast }\parallel \to 0,\text{as }n\to \mathrm{\infty }.$$

This implies that $w_n\to w^{\ast }$ and $w^{\ast }=P_{\mathrm{\Omega }}f(w^{\ast })$ is a solution of (GSEVIP) (1.5).

This completes the proof of Theorem 3.3. □

Remark 3.4 Theorem 3.3 extends and improves the main results in Moudafi et al. [11, 12, 19], Eslamian and Latif [13], Chen et al. [14], Chuang [15], Naraghirad [21] and Ansari and Rehan [17].

4 General split equality optimization problem and strong convergence theorems

Let $H_1$, $H_2$, and $H_3$ be three real Hilbert spaces. Let $A:H_1\to H_3$ and $B:H_2\to H_3$ be two linear and bounded operators. The so-called general split equality optimization problem (GSEOP) is to find $x^{\ast }\in H_1$, and $y^{\ast }\in H_2$ such that for each $i\ge 1$

$$h_i\left(x^{\ast }\right)=\underset{x\in H_1}{min}{h}_i(x),g_i\left(y^{\ast }\right)=\underset{z\in H_2}{min}{g}_i(z)\text{and}A{x}^{\ast }=B{y}^{\ast },$$

(4.1)

where $h_i:H_1\to \mathbb{R}$ and $g_i:H_2\to \mathbb{R}$ are two families of proper, lower semicontinuous, and convex functions.

For each $i\ge 1$ denote by $\partial h_i=U_i$ and $\partial g_i=K_i$. Then the mappings $U_i:H_1\to 2^{H_1}$ and $K_i:H_2\to 2^{H_2}$, $i=1,2,\dots$ both are set-valued maximal monotone mappings, and

$$\begin{array}{c}{h}_i\left(x^{\ast }\right)=\underset{x\in H_1}{min}{h}_i(x)\iff 0\in \partial h_i\left(x^{\ast }\right)=U_i\left(x^{\ast }\right),\hfill \\ g_i\left(y^{\ast }\right)=\underset{z\in H_2}{min}{g}_i(z)\iff 0\in \partial g_i\left(y^{\ast }\right)=K_i\left(y^{\ast }\right).\hfill \end{array}$$

Therefore (GSEOP) (4.1) is equivalent to the following general split equality variational inclusion problem (GSEVIP): to find $x^{\ast }\in H_1$ and $y^{\ast }\in H_2$ such that

$$0\in \bigcap _{i=1}^{\mathrm{\infty }}{U}_i\left(x^{\ast }\right),0\in \bigcap _{i=1}^{\mathrm{\infty }}{K}_i\left(y^{\ast }\right)\text{and}A{x}^{\ast }=B{y}^{\ast }.$$

(4.2)

Therefore, the following theorem can be obtained from Theorem 3.3 immediately.

Theorem 4.1 Let $H_1$, $H_2$, $H_3$, A, B, $A^{\ast }$, $B^{\ast }$, $\{U_i\}$, $\{K_i\}$ be the same as above. Let $J_{{\mu }_i}^{(U_i,K_i)}$, G, $G^{\ast }$, f be the same as in Theorem  3.3. Let $\{w_n\}$ be the sequence defined by (3.1). If the solution set ${\mathrm{\Omega }}_1$ of (GSEVIP) (4.1) is nonempty and the following conditions are satisfied:

• (i) ${\alpha }_n+{\beta }_n+{\sum }_{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}=1$, for each $n\ge 0$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$, and ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_{n,i}>0$ for each $i\ge 1$;

• (iv) $\{{\lambda }_{n,i}\}\subset (0,\frac{2}{L})$ for each $i\ge 1$, where $L={\parallel G\parallel }^2$,

then the sequence $\{w_n\}$ converges strongly to $w^{\ast }=P_{{\mathrm{\Omega }}_1}f(w^{\ast })$, which is a solution of (GSEOP) (4.1).

By using Theorem 3.3 and Theorem 4.1, now we give some corollaries for the split equality feasibility problem, the split equality equilibrium problem, and the split optimization problem.

Let $H_1$, $H_2$, $H_3$, C, Q, A, B be the same as in the split equality feasibility problem (1.3). Let $i_C$ and $i_Q$ be the indicator function of C and Q, respectively, defined by (1.7). In Theorem 4.1, take $\{U\}=\{\partial i_C\}$, $\{K\}=\{\partial i_Q\}$, and $J_{\mu }^{(U,K)}=P_{C\times Q}:=\left[\begin{array}{c}{P}_C\\ P_Q\end{array}\right]$, therefore we have the following.

Corollary 4.2 Let $H_1$, $H_2$, $H_3$, A, B, $A^{\ast }$, $B^{\ast }$, $P_{C\times Q}$ be the same as above. Let G, $G^{\ast }$, f be the same as in Theorem  4.1. Let $\{w_n\}$ be the sequence generated by $w_0\in H_1\times H_2$

$$w_{n+1}={\alpha }_n{w}_n+{\beta }_{n}f(w_n)+{\gamma }_n\left(P_{C\times Q}(I-{\lambda }_n{G}^{\ast }G)w_n\right),n\ge 0,$$

(4.3)

or its equivalent form

$$\{\begin{array}{l}{x}_{n+1}={\alpha }_n{x}_n+{\beta }_n{f}_1(x_n)+{\gamma }_n(P_C(x_n-{\lambda }_n(A^{\ast }(A{x}_n-B{y}_n)))),\\ y_{n+1}={\alpha }_n{y}_n+{\beta }_n{f}_2(y_n)+{\gamma }_n(P_Q(y_n+{\lambda }_n(B^{\ast }(A{x}_n-B{y}_n)))).\end{array}$$

(4.4)

If the solution set ${\mathrm{\Gamma }}_1$ of (SEFP) (1.3) is nonempty and the following conditions are satisfied:

• (i) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, for each $n\ge 0$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$, and ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

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

• (iv) $\{{\lambda }_n\}\subset (0,\frac{2}{L})$ for each $i\ge 1$, where $L={\parallel G\parallel }^2$,

then the sequence $\{w_n\}$ converges strongly to $w^{\ast }=P_{{\mathrm{\Gamma }}_1}f(w^{\ast })$, which is a solution of (SEFP) (1.3).

Remark 4.3 Since the simultaneous iterative sequence $\{(x_n,y_n)\}$ (4.4) converges strongly to a solution of (SEFP) (1.3). Therefore it provides an affirmative answer to Moudafi’s open question 1.1 [11].

Let $h,g:D\times D\to (-\mathrm{\infty },+\mathrm{\infty })$ be two equilibrium functions. For given $\lambda >0$, let $R_{\lambda ,h}$ and $R_{\lambda ,g}$ be the resolvents of h and g (defined by (1.9)), respectively.

The so-called split equality equilibrium problem with respective to h, g, and D $(SEEP(h,g,D))$ is to find $x^{\ast }\in D$, $y^{\ast }\in D$ such that

$$h(x^{\ast },u)\ge 0,\mathrm{\forall }u\in D,g(y^{\ast },v)\ge 0,\mathrm{\forall }v\in D\text{and}A{x}^{\ast }=B{y}^{\ast },$$

(4.5)

where $A,B:D\to D$ are two linear and bounded operators.

By Proposition 1.2, the $(SEEP(h,g,D))$ (4.5) is equivalent to find $x^{\ast }\in D$, $y^{\ast }\in D$ such that for each $\lambda >0$

$$\begin{array}{r}{x}^{\ast }\in EP(h,D),y^{\ast }\in EP(g,D)\text{and}A{x}^{\ast }=B{y}^{\ast }\\ \iff x^{\ast }\in F(R_{\lambda h}),y^{\ast }\in F(R_{\lambda g})\text{and}A{x}^{\ast }=B{y}^{\ast }.\end{array}$$

(4.6)

Letting $C=F(R_{\lambda h})$, $Q=F(R_{\lambda g})$, by Proposition 1.2, C and Q both are nonempty closed and convex subset of D. Hence the problem (4.5) (and so the problem (4.6)) is equivalent to the following split equality feasibility problem:

$$\text{to find }{x}^{\ast }\in C,y^{\ast }\in Q\text{ such that }A{x}^{\ast }=B{y}^{\ast }.$$

(4.7)

In Corollary 4.2 taking $H_1=H_2=H_3=D$, from Corollary 4.2 we have the following.

Corollary 4.4 Let D, C, Q be the same as above. Let A, B, $A^{\ast }$, $B^{\ast }$, $P_{C\times Q}$, G, $G^{\ast }$, f be the same as in Corollary  4.2. For any given $w_0\in D\times D$, let $\{w_n\}$ be the sequence generated by

$$w_{n+1}={\alpha }_n{w}_n+{\beta }_{n}f(w_n)+{\gamma }_n\left(P_{C\times Q}(I-{\lambda }_n{G}^{\ast }G)w_n\right),n\ge 0.$$

(4.8)

If the solution set ${\mathrm{\Gamma }}_2$ of $(SEEP(h,g,D))$ (4.5) is nonempty and the following conditions are satisfied:

• (i) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, for each $n\ge 0$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$, and ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

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

• (iv) $\{{\lambda }_n\}\subset (0,\frac{2}{L})$ for each $i\ge 1$, where $L={\parallel G\parallel }^2$,

then the sequence $\{w_n\}$ converges strongly to $w^{\ast }=P_{{\mathrm{\Gamma }}_2}f(w^{\ast })$, which is a solution of $(SEEP(h,g,D))$ (4.5).

Let $H_1$ and $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a linear and bounded operators, $h:H_1\to \mathbb{R}$ and $g:H_2\to \mathbb{R}$ be two proper convex and lower semicontinuous functions. The split optimization problem (SOP) is to find $x^{\ast }\in H_1$, $A{x}^{\ast }\in H_2$ such that

$$h\left(x^{\ast }\right)=\underset{x\in H_1}{min}{h}_i(x)\text{and}g\left(A{x}^{\ast }\right)=\underset{z\in H_2}{min}g(z).$$

(4.9)

Denote $U=\partial h$ and $K=\partial g$, then the (SOP) (4.9) is equivalent to the following split variational inclusion problem (SVIP): to find $x^{\ast }\in H_1$ such that

$$0\in U\left(x^{\ast }\right),0\in K\left(A{x}^{\ast }\right).$$

(4.10)

In Theorem 4.1 taking $H_3=H_2$, $B=I$ (the identity mapping on $H_2$) and

$$\tilde{G}=[A-I],{\tilde{G}}^{\ast }=\left[\begin{array}{c}{A}^{\ast }\\ -I\end{array}\right],{\tilde{G}}^{\ast }\tilde{G}=\left[\begin{array}{cc}{A}^{\ast }A& -A^{\ast }\\ -A& I\end{array}\right],$$

then from Theorem 4.1 we have the following.

Corollary 4.5 Let $H_1$, $H_2$, A, I, $\tilde{G}$, ${\tilde{G}}^{\ast }$, U, K, be the same as above. Let $J_{\mu }^{(U,K)}$, f be the same as in Theorem  4.1. For any given $w_0=(x_0,y_0)\in H_1\times H_2$, let $\{w_n=(x_n,y_n)\}$ be the sequence defined by

$$\{\begin{array}{l}{x}_{n+1}={\alpha }_n{x}_n+{\beta }_n{f}_1(x_n)+{\gamma }_n{J}_{\mu }^U(x_n-{\lambda }_n{A}^{\ast }(A{x}_n-y_n)),\\ y_{n+1}={\alpha }_n{y}_n+{\beta }_n{f}_2(y_n)+{\gamma }_n{J}_{\mu }^K(y_n+{\lambda }_n(A{x}_n-y_n)),\end{array}$$

(4.11)

or its equivalent form:

$$w_{n+1}={\alpha }_n{w}_n+{\beta }_{n}f(w_n)+{\gamma }_n\left(J_{\mu }^{(U,K)}(I-{\lambda }_n\widetilde{G^{\ast }}\tilde{G})w_n\right),n\ge 0,$$

(4.12)

If ${\mathrm{\Gamma }}_3:=\{x^{\ast }\in U^{-1}(0)\cap A^{-1}{K}^{-1}(0)\}$, the solution set of (SOP) (4.9) is nonempty, and the following conditions are satisfied:

• (i) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, for each $n\ge 0$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$, and ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

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

• (iv) $\{{\lambda }_n\}\subset (0,\frac{2}{L})$, where $L={\parallel \tilde{G}\parallel }^2$,

then the sequence $\{w_n\}$ converges strongly to $w^{\ast }=P_{{\mathrm{\Gamma }}_3}f(w^{\ast })$, which is a solution of (SOP) (4.9).