Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces

Abstract

The purpose of this paper is to introduce and study a general split variational inclusion problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split variational inclusion problem. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces.

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

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

(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, computer tomography and radiation therapy treatment planning [3–5]. The $SFP$ in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10]. For comprehensive literature, bibliography and a survey on SFP, we refer to [11].

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]:

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

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

On the other hand, let H be a real Hilbert space, and B be a set-valued mapping with domain $D(B):=\{x\in H:B(x)\ne \mathrm{\varnothing }\}$. Recall that B is called monotone, if $〈u-v,x,x-y〉\ge 0$ for any $u\in Bx$ and $v\in By$; B is maximal monotone, if its graph $\{(x,y):x\in D(B),y\in Bx\}$ is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find $x^{\ast }\in H$ such that $0\in B(x^{\ast })$. Here, $x^{\ast }$ is called a zero point of B. A well-known method for approximating a zero point of a maximal monotone mapping defined in a real Hilbert space H is the proximal point algorithm first introduced by Martinet [12] and generated by Rockafellar [13]. This is an iterative procedure, which generates $\{x_n\}$ by $x_1=x\in H$ and

$$x_{n+1}=J_{{\beta }_n}^B{x}_n,n\ge 1,$$

(1.2)

where $\{{\beta }_n\}\subset (0,\mathrm{\infty })$, B is a maximal monotone mapping in a real Hilbert space, and $J_r^B$ is the resolvent mapping of B defined by $J_r^B={(I+rB)}^{-1}$ for each $r>0$. Rockafellar [13] proved that if the solution set $B^{-1}(0)$ is nonempty and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$, then the sequence $\{x_n\}$ in (1.2) converges weakly to an element of $B^{-1}(0)$. In particular, if B is the sub-differential ∂f of a proper convex and lower semicontinuous function $f:H\to \mathbb{R}$, then (1.2) is reduced to

$$x_{n+1}=\underset{y\in H}{argmin}\{f(y)+\frac{1}{2{\beta }_n}{\parallel y-x_n\parallel }^2\},\mathrm{\forall }n\ge 1.$$

(1.3)

In this case, $\{x_n\}$ converges weakly to a minimizer of f. Later, many researchers have studied the convergence problems of the proximal point algorithm in Hilbert spaces (see [14–21] and the references therein).

Motivated by the works in [14–17] and related literature, the purpose of this paper is to introduce and consider the following general split variational inclusion problem.

Let $H_1$ and $H_2$ be two real Hilbert spaces, $B_i:H_1\to H_1$ and $K_i:H_2\to H_2$, $i=1,2,\dots$ be two families of set-valued maximal monotone mappings, $A:H_1\to H_2$ be a linear and bounded operator, and $A^{\ast }$ be the adjoint of A. The so-called general split variational inclusion problem is

$$\text{to find }{x}^{\ast }\in H_1\text{ such that }0\in \bigcap _{i=1}^{\mathrm{\infty }}{B}_i\left(x^{\ast }\right)\text{ and }0\in \bigcap _{i=1}^{\mathrm{\infty }}{K}_i\left(A{x}^{\ast }\right).$$

(1.4)

The following examples are special cases of (GSVIP) (1.4).

Classical split variational inclusion problem. Let $B:H_1\to H_1$ and $K:H_2\to H_2$ be set-valued maximal monotone mappings. The so-called classical split variational inclusion problem (CSVIP) is

$$\text{to find }{x}^{\ast }\in H_1\text{ such that }0\in B\left(x^{\ast }\right)\text{ and }0\in K\left(A{x}^{\ast }\right),$$

(1.5)

which was introduced by Moudafi [17]. It is obvious that problem (1.5) is a special case of (GSVIP) (1.4). In [17], Moudafi proved that the iteration process

$$x_{n+1}=J_{\lambda }^B(x_n+\gamma A^{\ast }(J_{\lambda }^K-I)A{x}_n)$$

converges weakly to a solution of problem (1.5), where λ and γ are given positive numbers.

Split optimization problem. Let $f:H_1\to \mathbb{R}$, $g:H_2\to \mathbb{R}$ be two proper convex and lower semicontinuous functions. The so-called split optimization problem (SOP) is

$$\text{to find }{x}^{\ast }\in H_1\text{ such that }f\left(x^{\ast }\right)=\underset{y\in H_1}{min}f(y)\text{ and }g\left(A{x}^{\ast }\right)=\underset{z\in H_2}{min}g(z).$$

(1.6)

Denote by $B=\partial (f)$ and $K=\partial (g)$, then B and K both are maximal monotone mappings, and problem (1.6) is equivalent to the following classical split variational inclusion problem, i.e.:

$$\text{to find }{x}^{\ast }\in H_1\text{ such that }0\in \partial \left(f\left(x^{\ast }\right)\right)\text{ and }0\in \partial \left(g\left(A{x}^{\ast }\right)\right).$$

(1.7)

Split feasibility problem. As in (1.1), let C and Q be two nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively and A be the same as above. The split feasibility problem is

$$\text{to find }{x}^{\ast }\in C\text{ such }A{x}^{\ast }\in Q.$$

(1.8)

It is well known that this kind of problems was first introduced by Censor and Elfving [1] for modeling inverse problems arising from phase retrievals and in medical image reconstruction [2]. Also it can be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning.

Let $i_C$ ($i_Q$) be the indicator function of C (Q), i.e.,

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

(1.9)

Then $i_C$ and $i_Q$ both are proper convex and lower semicontinuous functions, and its subdifferentials $\partial i_C$ and $\partial i_Q$ are maximal monotone operators. Consequently problem (1.8) is equivalent to the following ‘split optimization problem’ and ‘Moudafi’s classical split variational inclusion problem’, i.e.,

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

(1.10)

For solving (GSVIP) (1.4), in our paper we propose the following iterative algorithms:

$$x_{n+1}={\alpha }_n{x}_n+{\xi }_{n}f(x_n)+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}{J}_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n],\mathrm{\forall }n\ge 0,$$

(1.11)

where $f:H_1\to H_1$ is a contraction mapping with a contractive constant $k\in (0,1)$, $\{{\alpha }_n\}$, $\{{\xi }_n\}$ and $\{{\gamma }_{n,i}\}$ are sequence in $[0,1]$ satisfying some conditions. Under suitable conditions, some strong convergence theorems for the sequence proposed by (1.11) to a solution for (GSVIP) (1.4) in Hilbert spaces are proved. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces. Our results extend and improve the related results of Censor and Elfving [1], Byrne [2], Censor et al. [3–5], Rockafellar [13], Moudafi [14, 17], Eslamian and Latif [15], Eslamian [21], and Chuang [22].

2 Preliminaries

Throughout the paper, we denote by H a real Hilbert space, C be a nonempty closed and convex subset of H. $F(T)$ denote by the set of fixed points of a mapping T. Let $\{x_n\}$ be a sequence in H and $x\in H$. Strong convergence of $\{x_n\}$ to x is denoted by $x_n\to x$, and weak convergence of $\{x_n\}$ to x is denoted by $x_n\rightharpoonup x$. For every point $x\in H$, there exists a unique nearest point in C, denoted by $P_{C}x$. This point satisfies.

$$\parallel x-P_{C}x\parallel \le \parallel x-y\parallel ,\mathrm{\forall }y\in C.$$

The operator $P_C$ is called the metric projection. The metric projection $P_C$ is characterized by the fact that $P_{C}x\in C$ and

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

Recall that a mapping $T:C\to H$ is said to be nonexpansive, if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for every $x,y\in C$. T is said to be quasi-nonexpansive, if $F(T)\ne \mathrm{\varnothing }$ and $\parallel Tx-p\parallel \le \parallel x-p\parallel$ for every $x\in C$ and $p\in F(T)$. It is easy to see that $F(T)$ is a closed convex subset of C if T is a quasi-nonexpansive mapping. 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 (demi-closed principle)

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to H$ be a nonexpansive mapping, and let $\{x_n\}$ be a sequence in C. If $x_n\rightharpoonup w$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$, then $Tw=w$.

Lemma 2.2 [23]

Let H be a (real) Hilbert space. Then for all $x,y\in H$,

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

(2.1)

Lemma 2.3 [24]

Let H be a Hilbert space and let $\{x_n\}$ be a sequence in H. Then, for any given sequence $\{{\lambda }_n\}\subset (0,1)$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n=1$ and for any positive integers i, j with $i<j$,

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

(2.2)

Lemma 2.4 Let $\{a_n\}$ be a sequence of nonnegative real numbers, $\{b_n\}$ be a sequence of real numbers in $(0,1)$ with ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n=\mathrm{\infty }$, $\{u_n\}$ be a sequence of nonnegative real numbers with ${\sum }_{n=1}^{\mathrm{\infty }}{u}_n<\mathrm{\infty }$, $\{t_n\}$ be a real numbers with ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{t}_n\le 0$. If

$$a_{n+1}\le (1-b_n)a_n+b_n{t}_n+u_n,\mathit{\text{for each }}n\ge 1,$$

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

Lemma 2.5 [25]

Let $\{a_n\}$ be a sequence of real numbers such that there exists a subsequence $\{n_i\}$ of $\{n\}$ such that $a_{n_i}<a_{n_i+1}$ for all $i\in \mathbb{N}$. Then there exists a nondecreasing sequence $\{m_k\}\subset \mathbb{N}$ such that $m_k\to \mathrm{\infty }$, $a_{m_k}\le a_{m_k+1}$ and $a_k\le a_{m_k+1}$ are satisfied by all (sufficiently large) numbers $k\in \mathbb{N}$. In fact, $m_k=max\{j\le k:a_j<a_{j+1}\}$.

Lemma 2.6 [22]

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.

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

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

• (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 $x\in H$ and each $w\in B^{-1}(0)$, and each $\beta >0$.

Lemma 2.7 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 $B:H_2\to 2_2^H$ 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 2.6(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 2.7. □

3 Main results

The following lemma will be used in proving our main results.

Lemma 3.1 Let $H_1$ and $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a linear and bounded operator, and $A^{\ast }$ be the adjoint of A. Let $B_i:H_1\to 2^{H_1}$, and $K_i:H_2\to 2^{H_2}$, $i=1,2,\dots$ , be two families of set-valued maximal monotone mappings, and let $\beta >0$ and $\gamma >0$. If $\mathrm{\Omega }\ne \mathrm{\varnothing }$ (the solution set of (GSVIP) (1.4)), then $x^{\ast }\in H_1$ is a solution of (GSVIP) (1.4) if and only if for each $i\ge 1$, for each $\gamma >0$ and for each $\beta >0$

$$x^{\ast }=J_{\beta }^{B_i}(x^{\ast }-\gamma A^{\ast }(I-J_{\beta }^{K_i})A{x}^{\ast }).$$

(3.1)

Proof Indeed, if $x^{\ast }$ is a solution of (GSVIP) (1.4), then for each $i\ge 1$, $\gamma >0$ and $\beta >0$,

$$x^{\ast }\in B_i^{-1}(0)\text{and}A{x}^{\ast }\in K_i^{-1}(0),\mathit{\text{i.e.}},x^{\ast }=J_{\beta }^{B_i}{x}^{\ast }\text{ and }A{x}^{\ast }=J_{\beta }^{K_i}A{x}^{\ast }.$$

This implies that $x^{\ast }=J_{\beta }^{B_i}(x^{\ast }-\gamma A{x}^{\ast }(I-J_{\beta }^{K_i})A{x}^{\ast })$.

Conversely, if $x^{\ast }$ solves (3.1), by Lemma 2.6(v), we have

$$〈x^{\ast }-(x^{\ast }-\gamma A^{\ast }(I-J_{\beta }^{K_i})A{x}^{\ast }),y-x^{\ast }〉\ge 0,\mathrm{\forall }y\in B_i^{-1}(0).$$

Hence we have

$$〈(I-J_{\beta }^{K_i})A{x}^{\ast },Ay-A{x}^{\ast }〉\ge 0,\mathrm{\forall }y\in B_i^{-1}(0).$$

(3.2)

On the other hand, by Lemma 2.6(v) again

$$〈(A{x}^{\ast }-J_{\beta }^{K_i}A{x}^{\ast },J_{\beta }^{K_i}A{x}^{\ast }-v〉\ge 0,\mathrm{\forall }v\in K_i^{-1}(0).$$

(3.3)

Adding up (3.2) and (3.3), we have

$$〈A{x}^{\ast }-J_{\beta }^{K_i}A{x}^{\ast },J_{\beta }^{K_i}A{x}^{\ast }+Ay-A{x}^{\ast }-v〉\ge 0,\mathrm{\forall }y\in B_i^{-1}(0),\text{ and }v\in K_i^{-1}(0).$$

Simplifying it, we have

$${\parallel A{x}^{\ast }-J_{\beta }^{K_i}A{x}^{\ast }\parallel }^2\le 〈A{x}^{\ast }-J_{\beta }^{K_i}A{x}^{\ast },Ay-v〉\ge 0,\mathrm{\forall }y\in B_i^{-1}(0),\text{ and }v\in K_i^{-1}(0).$$

(3.4)

By the assumption that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Taking $w\in \mathrm{\Omega }$, hence for each $i\ge 1$ $w\in B_i^{-1}(0)$ and $Aw\in K_i^{-1}(0)$. In (3.4), taking $y=w$ and $v=Aw$, then we have

$${\parallel A{x}^{\ast }-J_{\beta }^{K_i}A{x}^{\ast }\parallel }^2=0.$$

This implies that $A{x}^{\ast }=J_{\beta }^{K_i}A{x}^{\ast }$, and so $A{x}^{\ast }\in K_i^{-1}(0)$ for each $i\ge 1$. Hence from (3.1), $x^{\ast }=J_{\beta }^{B_i}{x}^{\ast }$, i.e., $x^{\ast }\in B_i^{-1}(0)$. Hence $x^{\ast }$ is a solution of (GSVIP)(1.4).

This completes the proof of Lemma 3.1. □

We are now in a position to prove the following main result.

Theorem 3.2 Let $H_1$, $H_2$, A, $A^{\ast }$, $\{B_i\}$, $\{K_i\}$, Ω be the same as in Lemma  3.1. Let $f:H_1\to H_1$ be a contractive mapping with contractive constant $k\in (0,1)$. Let $\{{\alpha }_n\}$, $\{{\xi }_n\}$, $\{{\gamma }_{n,i}\}$ be the sequences in $(0,1)$ with ${\alpha }_n+{\xi }_n+{\sum }_{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}=1$, for each $n\ge 0$. Let $\{{\beta }_i\}$ be a sequence in $(0,\mathrm{\infty })$, and $\{{\lambda }_{n,i}\}$ be a sequence in $(0,\frac{2}{{\parallel A\parallel }^2})$. Let $\{x_n\}$ be the sequence defined by (1.11). If $\mathrm{\Omega }\ne \mathrm{\varnothing }$ and the following conditions are satisfied:

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

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

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\lambda }_{n,i}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\lambda }_{n,i}<\frac{2}{{\parallel A\parallel }^2}$,

then $x_n\to x^{\ast }\in \mathrm{\Omega }$ where $x^{\ast }=P_{\mathrm{\Omega }}f(x^{\ast })$, where $P_{\mathrm{\Omega }}$ is the metric projection from $H_1$ onto Ω.

Proof (I) First we prove that $\{x_n\}$ is bounded.

In fact, letting $z\in \mathrm{\Omega }$, by Lemma 3.1, for each $i\ge 1$,

$$z=J_{{\beta }_i}^{B_i}[z-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})Az].$$

Hence it follows from Lemma 2.7(iii) that for each $i\ge 1$ and each $n\ge 1$ we have

$$\begin{array}{rl}\parallel x_{n+1}-z\parallel & =\parallel {\alpha }_n{x}_n+{\xi }_{n}f(x_n)+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}{J}_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-z\parallel \\ \le {\alpha }_n\parallel x_n-z\parallel +{\xi }_n\parallel f(x_n)-z\parallel +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-z\parallel \\ \le {\alpha }_n\parallel x_n-z\parallel +{\xi }_n\parallel f(x_n)-z\parallel +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-z\parallel \\ \le {\alpha }_n\parallel x_n-z\parallel +{\xi }_n\parallel f(x_n)-z\parallel +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}\parallel x_n-z\parallel \\ =(1-{\xi }_n)\parallel x_n-z\parallel +{\xi }_n\parallel f(x_n)-z\parallel \\ \le (1-{\xi }_n)\parallel x_n-z\parallel +{\xi }_n\parallel f(x_n)-f(z)\parallel +{\xi }_n\parallel f(z)-z\parallel \\ \le (1-{\xi }_n(1-k))\parallel x_n-z\parallel +\frac{{\xi }_n(1-k)}{1-k}\parallel f(z)-z\parallel \\ \le max\{\parallel x_n-z\parallel ,\frac{1}{1-k}\parallel f(z)-z\parallel \}.\end{array}$$

By induction, we can prove that

$$\parallel x_n-z\parallel \le max\{\parallel x_0-z\parallel ,\frac{1}{1-k}\parallel f(z)-z\parallel \},\mathrm{\forall }n\ge 0.$$

(3.5)

This implies that $\{x_n\}$ is bounded, so is $\{f(x_n)\}$.

(II) Now we prove that for each $j\ge 1$

$$\begin{array}{r}{\alpha }_n{\gamma }_{n,j}{\parallel x_n-J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]\parallel }^2\\ \le {\parallel x_n-z\parallel }^2-{\parallel x_{n+1}-z\parallel }^2+{\xi }_n{\parallel f(x_n)-z\parallel }^2,\text{for each }i\ge 1.\end{array}$$

(3.6)

Indeed, it follows from Lemma 2.3 that for any positive $j\ge 1$

$$\begin{array}{rl}{\parallel x_{n+1}-z\parallel }^2=& {\parallel {\alpha }_n{x}_n+{\xi }_{n}f(x_n)+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}{J}_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-z\parallel }^2\\ \le & {\alpha }_n{\parallel x_n-z\parallel }^2+{\xi }_n{\parallel f(x_n)-z\parallel }^2\\ +\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}{\parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-z\parallel }^2\\ -{\alpha }_n{\gamma }_{n,j}{\parallel x_n-J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]\parallel }^2\\ \le & (1-{\xi }_n){\parallel x_n-z\parallel }^2+{\xi }_n{\parallel f(x_n)-z\parallel }^2\\ -{\alpha }_n{\gamma }_{n,j}{\parallel x_n-J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]\parallel }^2.\end{array}$$

Simplifying it, (3.6) is proved.

By the assumption that $\mathrm{\Omega }\ne \mathrm{\varnothing }$, and it is easy to prove that Ω is closed and convex. This implies that $P_{\mathrm{\Omega }}$ is well defined. Again since $P_{\mathrm{\Omega }}f:H_1\to \mathrm{\Omega }$ is a contraction mapping with contractive constant $k\in (0,1)$, there exists a unique $x^{\ast }\in \mathrm{\Omega }$ such that $x^{\ast }=P_{\mathrm{\Omega }}f{x}^{\ast }$. Since $x^{\ast }\in \mathrm{\Omega }$, it solves (GSVIP) (1.4). By Lemma 3.1,

$$x^{\ast }=J_{{\beta }_j}^{B_j}(x^{\ast }-{\lambda }_{n,j}{A}^{\ast }(I-J_{{\beta }_j}^{K_j})A{x}^{\ast }),\mathrm{\forall }j\ge 1,n\ge 0.$$

(3.7)

• (III) Now we prove that $x_n\to x^{\ast }$.

In order to prove that $x_n\to x^{\ast }$ (as $n\to \mathrm{\infty }$), we consider two cases.

Case 1. Assume that $\{\parallel x_n-x^{\ast }\parallel \}$ is a monotone sequence. In other words, for $n_0$ large enough, ${\{\parallel x_n-x^{\ast }\parallel \}}_{n\ge n_0}$ is either nondecreasing or non-increasing. Since $\{\parallel x_n-x^{\ast }\parallel \}$ is bounded, $\{\parallel x_n-x^{\ast }\parallel \}$ is convergence. Again since ${lim}_{n\to \mathrm{\infty }}{\xi }_n=0$, and $\{f(x_n)\}$ is bounded, from (3.6) we get

$$\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n{\gamma }_{n,j}{\parallel x_n-J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]\parallel }^2=0.$$

By condition (ii), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]\parallel =0.$$

(3.8)

Now we prove that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f\left(x^{\ast }\right)-x^{\ast },x_n-x^{\ast }〉\le 0.$$

(3.9)

To show this inequality, we choose a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $x_{n_k}\rightharpoonup w$, ${\lambda }_{n_k,i}\to {\lambda }_i\in (0,\frac{2}{{\parallel A\parallel }^2})$ for each $i\ge 1$, and

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

(3.10)

It follows from (3.8) that

$$\begin{array}{r}\parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_i{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-x_n\parallel \\ \le \parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_i{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]\parallel \\ +\parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-x_n\parallel \\ \le \parallel [x_n-{\lambda }_i{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]\parallel \\ +\parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-x_n\parallel \\ \le |{\lambda }_i-{\lambda }_{n,i}|\parallel A^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n\parallel \\ +\parallel J_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-x_n\parallel \to 0(\text{as }n\to \mathrm{\infty }).\end{array}$$

For each $i\ge 1$, $J_{{\beta }_i}^{B_i}[I-{\lambda }_i{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A]$ is a nonexpansive mapping. Thus from Lemma 2.1, $w=J_{{\beta }_i}^{B_i}[I-{\lambda }_i{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A]w$. By Lemma 3.1 $w\in \mathrm{\Omega }$, i.e., w is a solution of (GSVIP) (1.4). Consequently we have

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

• (IV) Finally, we prove that $x_n\to P_{\mathrm{\Omega }}f(x^{\ast })$.

In fact, from Lemma 2.2 we have

$$\begin{array}{r}{\parallel x_{n+1}-x^{\ast }\parallel }^2\\ \le {\parallel {\alpha }_n(x_n-x^{\ast })+\sum _{i=1}^{\mathrm{\infty }}{\gamma }_{n,i}{J}_{{\beta }_i}^{B_i}[x_n-{\lambda }_{n,i}{A}^{\ast }(I-J_{{\beta }_i}^{K_i})A{x}_n]-x^{\ast }\parallel }^2\\ +2{\xi }_n〈f(x_n)-x^{\ast },x_{n+1}-x^{\ast }〉\\ \le {(1-{\xi }_n)}^2{\parallel x_n-x^{\ast }\parallel }^2+2{\xi }_n〈f(x_n)-f\left(x^{\ast }\right),x_{n+1}-x^{\ast }〉+2{\xi }_n〈f\left(x^{\ast }\right)-x^{\ast },x_{n+1}-x^{\ast }〉\\ \le {(1-{\xi }_n)}^2{\parallel x_n-x^{\ast }\parallel }^2+2{\xi }_{n}k\parallel x_n-x^{\ast }\parallel \parallel x_{n+1}-x^{\ast }\parallel +2{\xi }_n〈f\left(x^{\ast }\right)-x^{\ast },x_{n+1}-x^{\ast }〉\\ \le {(1-{\xi }_n)}^2{\parallel x_n-x^{\ast }\parallel }^2+{\xi }_{n}k\{{\parallel x_{n+1}-x^{\ast }\parallel }^2+{\parallel x_n-x^{\ast }\parallel }^2\}\\ +2{\xi }_n〈f\left(x^{\ast }\right)-x^{\ast },x_{n+1}-x^{\ast }〉.\end{array}$$

Simplifying it, we have

$$\begin{array}{rl}{\parallel x_{n+1}-x^{\ast }\parallel }^2\le & \frac{{(1-{\xi }_n)}^2+{\xi }_{n}k}{1-{\xi }_{n}k}{\parallel x_n-x^{\ast }\parallel }^2+\frac{2{\xi }_n}{1-{\xi }_{n}k}〈f\left(x^{\ast }\right)-x^{\ast },x_{n+1}-x^{\ast }〉\\ \le & \frac{1-2{\xi }_n+{\xi }_{n}k}{1-{\xi }_{n}k}{\parallel x_n-x^{\ast }\parallel }^2+\frac{{\xi }_n^2}{1-{\xi }_{n}k}{\parallel x_n-x^{\ast }\parallel }^2\\ +\frac{2{\xi }_n}{1-{\xi }_{n}k}〈f\left(x^{\ast }\right)-x^{\ast },x_{n+1}-x^{\ast }〉\\ \le & (1-{\eta }_n){\parallel x_n-x^{\ast }\parallel }^2+{\eta }_n{\delta }_n,\mathrm{\forall }n\ge 0,\end{array}$$

where ${\delta }_n=\frac{{\xi }_{n}M}{2(1-k)}+\frac{1}{1-k}〈f(x^{\ast })-x^{\ast },x_{n+1}-x^{\ast }〉$, $M={sup}_{n\ge 0}{\parallel x_n-x^{\ast }\parallel }^2$, and ${\eta }_n=\frac{2(1-k){\xi }_n}{1-{\xi }_{n}k}$. It is easy to see that ${\eta }_n\to 0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\eta }_n=\mathrm{\infty }$, and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$. Hence by Lemma 2.4, the sequence $\{x_n\}$ converges strongly to $x^{\ast }=P_{\mathrm{\Omega }}f(x^{\ast })$.

Case 2. Assume that $\{\parallel x_n-x^{\ast }\parallel \}$ is not a monotone sequence. Then, by Lemma 2.3, we can define a sequence of positive integers: $\{\tau (n)\}$, $n\ge n_0$ (where $n_0$ large enough) by

$$\tau (n)=max\{k\le n:\parallel x_k-x^{\ast }\parallel \le \parallel x_{k+1}-x^{\ast }\parallel \}.$$

(3.11)

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

$$\parallel x_{\tau (n)}-x^{\ast }\parallel \le \parallel x_{\tau (n)+1}-x^{\ast }\parallel .$$

(3.12)

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

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

This implies that $x_n\to x^{\ast }$ and $x^{\ast }=P_{\mathrm{\Omega }}f(x^{\ast })$ is a solution of (GSVIP) (1.4).

This completes the proof of Theorem 3.2. □

In Theorem 3.2, if $B_i=B$ and $K_i=K$, for each $i\ge 1$, where $B:H_1\to 2^{H_1}$ and $K:H_2\to 2^{H_2}$ are two set-valued maximal monotone mappings, then from Theorem 3.2 we have the following.

Theorem 3.3 Let $H_1$, $H_2$, A, $A^{\ast }$, B, K, Ω, f be the same as in Theorem  3.2. Let $\{{\alpha }_n\}$, $\{{\xi }_n\}$, $\{{\gamma }_n\}$ be the sequence in $(0,1)$ with ${\alpha }_n+{\xi }_n+{\gamma }_n=1$ for each $n\ge 0$. Let $\beta >0$ be any given positive number, and $\{{\lambda }_n\}$ be a sequence in $(0,\frac{2}{{\parallel A\parallel }^2})$. Let $\{x_n\}$ be the sequence defined by

$$x_{n+1}={\alpha }_n{x}_n+{\xi }_{n}f(x_n)+{\gamma }_n{J}_{\beta }^B[x_n-{\lambda }_n{A}^{\ast }(I-J_{\beta }^K)A{x}_n],\mathrm{\forall }n\ge 0.$$

(3.13)

If $\mathrm{\Omega }\ne \mathrm{\varnothing }$ and the following conditions are satisfied:

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

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

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\lambda }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\lambda }_n<\frac{2}{{\parallel A\parallel }^2}$,

then $x_n\to x^{\ast }\in \mathrm{\Omega }$ where $x^{\ast }=P_{\mathrm{\Omega }}f(x^{\ast })$.

4 Applications

In this section we shall utilize the results presented in Theorem 3.2 and Theorem 3.3 to study some problems.

4.1 Application to split optimization problem

Let $H_1$ and $H_2$ be two real Hilbert spaces. Let $h:H_1\to \mathbb{R}$ and $g:H_2\to \mathbb{R}$ be two proper, convex and lower semicontinuous functions, and $A:H_1\to H_2$ be a linear and bounded operators. The so-called split optimization problem (SOP) is

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

(4.1)

Denote by $\partial h=B$ and $\partial g=K$. It is know that $B:H_1\to 2^{H_1}$ (resp. $K:H_2\to 2^{H_2}$) is a maximal monotone mapping, so we can define the resolvent $J_{\beta }^B={(I+\beta B)}^{-1}$ and $J_{\beta }^K={(I+\beta K)}^{-1}$, where $\beta >0$. Since $x^{\ast }$ and $A{x}^{\ast }$ is a minimum of h on $H_1$ and g on $H_2$, respectively, for any given $\beta >0$, we have

$$x^{\ast }\in B^{-1}(0)=F\left(J_{\beta }^B\right),\text{and}A{x}^{\ast }\in K^{-1}(0)=F\left(J_{\beta }^K\right).$$

(4.2)

This implies that the (SOP) (4.1) is equivalent to the split variational inclusion problem (SVIP) (4.2). From Theorem 3.3 we have the following.

Theorem 4.1 Let $H_1$, $H_2$, A, B, K, h, g be the same as above. Let f, $\{{\alpha }_n\}$, $\{{\xi }_n\}$, $\{{\gamma }_n\}$ be the same as in Theorem  3.3. Let $\beta >0$ be any given positive number, and $\{{\lambda }_n\}$ be a sequence in $(0,\frac{2}{{\parallel A\parallel }^2})$. Let $\{x_n\}$ be a sequence generated by $x_0\in H_1$

$$\{\begin{array}{c}{y}_n={argmin}_{z\in H_2}\{g(z)+\frac{1}{2\beta }{\parallel z-A{x}_n\parallel }^2\},\hfill \\ z_n=x_n-{\lambda }_n{A}^{\ast }(A{x}_n-y_n),\hfill \\ w_n={argmin}_{y\in H_1}\{h(y)+\frac{1}{2\beta }{\parallel y-z_n\parallel }^2\},\hfill \\ x_{n+1}={\alpha }_n{x}_n+{\xi }_{n}f(x_n)+{\gamma }_n{w}_n,n\ge 0.\hfill \end{array}$$

(4.3)

If ${\mathrm{\Omega }}_1\ne \mathrm{\varnothing }$, the solution set of the split optimization problem (4.1), and the following conditions are satisfied:

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

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

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\lambda }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\lambda }_n<\frac{2}{{\parallel A\parallel }^2}$,

then $x_n\to x^{\ast }\in {\mathrm{\Omega }}_1$ where $x^{\ast }=P_{{\mathrm{\Omega }}_1}f(x^{\ast })$.

Proof Since $\partial h=B$, $\partial g:=K$, and $y_n={argmin}_{z\in H_2}\{g(z)+\frac{1}{2\beta }{\parallel z-A{x}_n\parallel }^2\}$, we have

$$0\in {[K(z)+\frac{1}{\beta }(z-A{x}_n)]}_{z=y_n},\mathit{\text{i.e.}},A{x}_n\in (\beta K+I)(y_n).$$

This implies that

$$y_n=J_{\beta }^K(A{x}_n).$$

(4.4)

Similarly, from (4.3), we have

$$w_n=J_{\beta }^B(z_n).$$

(4.5)

From (4.3)-(4.5), we have

$$w_n=J_{\beta }^B(x_n-{\lambda }_n{A}^{\ast }(I-J_{\beta }^K)A{x}_n).$$

(4.6)

Therefore (4.3) can be rewritten as

$$x_{n+1}={\alpha }_n{x}_n+{\xi }_{n}f(x_n)+{\gamma }_n{J}_{\beta }^B(x_n-{\lambda }_n{A}^{\ast }(I-J_{\beta }^K)A{x}_n),n\ge 0.$$

(4.7)

The conclusion of Theorem 4.1 can be obtained from Theorem 3.3 immediately. □

4.2 Application to split feasibility problem

Let $C\subset H_1$ and $Q\subset H_2$ be two nonempty closed convex subsets and $A:H_1\to H_2$ be a bounded linear operator. Now we consider the following split feasibility problem, i.e.: to find

$$x^{\ast }\in C\text{ such that }A{x}^{\ast }\in Q.$$

(4.8)

Let $i_C$ and $i_Q$ be the indicator functions of C and Q defined by (1.9). Let $N_C(u)$ be the normal cone at $u\in H_1$ defined by

$$N_C(u)=\{z\in H_1:〈z,v-u〉\le 0,\mathrm{\forall }v\in C\}.$$

Since $i_C$ and $i_Q$ both are proper convex and lower semicontinuous functions on $H_1$ and $H_2$, respectively, and the subdifferential $\partial i_C$ of $i_C$ (resp. $\partial i_Q$ of $i_Q$) is a maximal monotone operator, we can define the resolvents $J_{\beta }^{\partial i_C}$ of $\partial i_C$ and $J_{\beta }^{\partial i_Q}$ of $\partial i_Q$ by

$$\begin{array}{c}{J}_{\beta }^{\partial i_C}(x)={(I+\beta \partial i_C)}^{-1}(x),\mathrm{\forall }x\in H_1,\hfill \\ J_{\beta }^{\partial i_Q}(x)={(I+\beta \partial i_Q)}^{-1}(x),\mathrm{\forall }x\in H_2,\hfill \end{array}$$

where $\beta >0$. By definition, we know that

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

Hence, for each $\beta >0$, 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 $J_{\beta }^{\partial i_C}=P_C$. Similarly $J_{\beta }^{\partial i_Q}=P_Q$. Taking $h(x)=i_C(x)$ and $g(x)=i_Q(x)$ in (4.1), then the (SFP) (4.8) is equivalent to the following split optimization problem:

$$\text{to find }{x}^{\ast }\in H_1\text{ such that }{i}_C\left(x^{\ast }\right)=\underset{y\in H_1}{min}{i}_C(y)\text{ and }{i}_Q\left(A{x}^{\ast }\right)=\underset{z\in H_2}{min}{i}_Q(z).$$

(4.9)

Hence, the following result can be obtained from Theorem 4.1 immediately.

Theorem 4.2 Let $H_1$, $H_2$, A, $A^{\ast }$, $i_C$, $i_Q$ be the same as above. Let f, $\{{\alpha }_n\}$, $\{{\xi }_n\}$, $\{{\gamma }_n\}$ be the same as in Theorem  4.1. Let $\{{\lambda }_n\}$ be a sequence in $(0,\frac{2}{{\parallel A\parallel }^2})$. Let $\{x_n\}$ be the sequence defined by

$$x_{n+1}={\alpha }_n{x}_n+{\xi }_{n}f(x_n)+{\gamma }_n{P}_C[x_n-{\lambda }_n{A}^{\ast }(I-P_Q)A{x}_n],\mathrm{\forall }n\ge 0.$$

(4.10)

If the solution set of the split optimization problem (4.4) ${\mathrm{\Omega }}_2\ne \mathrm{\varnothing }$, and the following conditions are satisfied:

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

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

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\lambda }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\lambda }_n<\frac{2}{{\parallel A\parallel }^2}$,

then $x_n\to x^{\ast }\in {\mathrm{\Omega }}_2$ where $x^{\ast }=P_{{\mathrm{\Omega }}_2}f(x^{\ast })$.

Remark 4.3 Theorem 4.2 extends and improves the main results in Censor and Elfving [1] and Byrne [2].