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

## Abstract

In this paper, we first consider a split variational inclusion problem and give several strong convergence theorems in Hilbert spaces, like the Halpern-Mann type iteration method and the regularized iteration method. As applications, we consider the algorithms for a split feasibility problem and a split optimization problem and give strong convergence theorems for these problems in Hilbert spaces. Our results for the split feasibility problem improve the related results in the literature.

MSC:47H10, 49J40, 54H25.

## 1 Introduction

In 1994, the split feasibility problem in finite dimensional Hilbert spaces was first introduced by Censor and Elfving  for modeling inverse problems which arise from medical image reconstruction. Since then, the split feasibility problem has received much attention due to its applications in signal processing, image reconstruction, approximation theory, control theory, biomedical engineering, communications, and geophysics. For examples, one can refer to  and related literature.

We know that the split feasibility problem can be formulated as the following problem:

where C and Q are nonempty closed convex subsets of Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively, and $A:{H}_{1}\to {H}_{2}$ is an operator. It is worth noting that a special case of problem (SFP) is the convexly constrained linear inverse problem in the finite dimensional Hilbert space :

Originally, problem (SFP) was considered in Euclidean spaces. (Note that if ${H}_{1}$ and ${H}_{2}$ are two Euclidean spaces, then A is a matrix.) In 1994, problem (SFP) in finite dimensional Hilbert spaces was first introduced by Censor and Elfving  for modeling inverse problems which arise from medical image reconstruction. Since then, many researchers have studied (SFP) in finite dimensional or infinite dimensional Hilbert spaces. For example, one can see [2, 716] and related literature.

In 2002, Byrne  first introduced the following recursive procedure:

${x}_{n+1}={P}_{C}\left({x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{P}_{Q}\right)A{x}_{n}\right),$
(1.1)

where the stepsize ${\tau }_{n}$ is chosen in the interval $\left(0,2/{\parallel A\parallel }^{2}\right)$, and ${P}_{C}$ and ${P}_{Q}$ are the metric projections onto $C\subseteq {\mathbb{R}}^{n}$ and $Q\subseteq {\mathbb{R}}^{m}$, respectively. This algorithm is called CQ algorithm. Note that A may be not invertible. In 2010, Wang and Xu  modified Byrne’s CQ algorithm and gave a weak convergence theorem in infinite dimensional Hilbert spaces.

In 2004, motivated by the works on CQ algorithm (1.1), Yang  considered (SFP) under the following conditions:

$C:=\left\{x\in {\mathbb{R}}^{n}:c\left(x\right)\le 0\right\}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Q:=\left\{x\in {\mathbb{R}}^{m}:q\left(x\right)\le 0\right\},$

where $c:{\mathbb{R}}^{n}\to \mathbb{R}$ and $q:{\mathbb{R}}^{m}\to \mathbb{R}$ are convex and lower semicontinuous functions. In fact, Yang  studied the following problem, and we call this problem the relaxed split feasibility problem:

In 2010, Xu  modified and extended Yang’s algorithm and gave a weak convergence theorem in infinite dimensional Hilbert spaces.

On the other hand, let H be a real Hilbert space, and B be a set-valued mapping with domain $\mathcal{D}\left(B\right):=\left\{x\in H:B\left(x\right)\ne \mathrm{\varnothing }\right\}$. Recall that B is called monotone if $〈u-v,x-y〉\ge 0$ for any $u\in Bx$ and $v\in By$; B is maximal monotone if its graph $\left\{\left(x,y\right):x\in \mathcal{D}\left(B\right),y\in Bx\right\}$ is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find $\overline{x}\in H$ such that $0\in B\overline{x}$. Here, $\overline{x}$ 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 is the proximal point algorithm first introduced by Martinet  and generated by Rockafellar . This is an iterative procedure, which generates $\left\{{x}_{n}\right\}$ by ${x}_{1}=x\in H$ and

${x}_{n+1}={J}_{{\beta }_{n}}^{B}{x}_{n},\phantom{\rule{1em}{0ex}}n\in \mathbb{N},$
(1.2)

where $\left\{{\beta }_{n}\right\}\subseteq \left(0,\mathrm{\infty }\right)$, B is a maximal monotone mapping in a real Hilbert space, and ${J}_{r}^{B}$ is the resolvent mapping of B defined by ${J}_{r}^{B}={\left(I+rB\right)}^{-1}$ for each $r>0$. In 1976, Rockafellar  proved the following in the Hilbert space setting: If the solution set ${B}^{-1}\left(0\right)$ is nonempty and ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$, then the sequence $\left\{{x}_{n}\right\}$ in (1.2) converges weakly to an element of ${B}^{-1}\left(0\right)$. In particular, if B is the subdifferential ∂f of a proper lower semicontinuous and convex function $f:H\to \mathbb{R}$, then (1.2) is reduced to

${x}_{n+1}=arg\underset{y\in H}{min}\left\{f\left(y\right)+\frac{1}{2{\beta }_{n}}{\parallel y-{x}_{n}\parallel }^{2}\right\},\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.$
(1.3)

In this case, $\left\{{x}_{n}\right\}$ converges weakly to a minimizer of f. Later, many researchers have studied the convergence theorems of the proximal point algorithm in Hilbert spaces. For examples, one can refer to  and references therein.

Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, ${B}_{1}:{H}_{1}⊸{H}_{1}$ and ${B}_{2}:{H}_{2}⊸{H}_{2}$ be two 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. In this paper, motivated by the works in [13, 14] and related literature, we consider the following split variational inclusion problem:

Clearly, we know that the following split variational inclusion problem (SFVIP) is a generalization of variational inclusion problem. Further, we observed that problem (SFVIP) was introduced by Moudafi , and Moudafi  gave a weak convergence theorem for problem (SFVIP). The following is an iteration process given by Moudafi :

${x}_{n+1}:={J}_{\lambda }^{{B}_{1}}\left({x}_{n}+\gamma {A}^{\ast }\left({J}_{\lambda }^{{B}_{2}}-I\right)A{x}_{n}\right).$

It is worth noting that λ and γ are fixed numbers. Hence, it is important to establish generalized iteration processes and the related strong convergence theorems for problem (SFVIP).

Besides, we know that the following problems are special cases of problem (SFVIP).

(SFOP) Find $\overline{x}\in {H}_{1}$ such that $f\left(\overline{x}\right)={min}_{y\in {H}_{1}}f\left(y\right)$ and $g\left(A\overline{x}\right)={min}_{y\in {H}_{2}}g\left(z\right)$, where $f:{H}_{1}\to \mathbb{R}$ and $g:{H}_{2}\to \mathbb{R}$ are two proper, lower semicontinuous, and convex functions.

(SFP) Find $\overline{x}\in {H}_{1}$ such that $\overline{x}\in C$ and $A\overline{x}\in Q$, where C and Q are two nonempty closed convex subsets of real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively.

In this paper, we first consider a split variational inclusion problem and give several strong convergence theorems in Hilbert spaces, like the Halpern-Mann type iteration method, the regularized iteration method. As applications, we consider algorithms for a split feasibility problem and a split optimization problem and give strong convergence theorems for these problems in Hilbert spaces. Our results for the split feasibility problem improve the related results in the literature.

## 2 Preliminaries

Throughout this paper, let be the set of positive integers and let be the set of real numbers. Let H be a (real) Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$, respectively. We denote the strong convergence and the weak convergence of $\left\{{x}_{n}\right\}$ to $x\in H$ by ${x}_{n}\to x$ and ${x}_{n}⇀x$, respectively. From , for each $x,y\in H$ and $\lambda \in \left[0,1\right]$, we have

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

Hence, we also have

$2〈x-y,u-v〉={\parallel x-v\parallel }^{2}+{\parallel y-u\parallel }^{2}-{\parallel x-u\parallel }^{2}-{\parallel y-v\parallel }^{2}$

for all $x,y,u,v\in H$. Furthermore, we know that

${\parallel \alpha x+\beta y+\gamma z\parallel }^{2}=\alpha {\parallel x\parallel }^{2}+\beta {\parallel y\parallel }^{2}+\gamma {\parallel z\parallel }^{2}-\alpha \beta {\parallel x-y\parallel }^{2}-\alpha \gamma {\parallel x-z\parallel }^{2}-\beta \gamma {\parallel y-z\parallel }^{2}$

for each $x,y,z\in H$ and $\alpha ,\beta ,\gamma \in \left[0,1\right]$ with $\alpha +\beta +\gamma =1$ .

Lemma 2.1 

Let H be a (real) Hilbert space, and let $x,y\in H$. Then ${\parallel x+y\parallel }^{2}\le {\parallel x\parallel }^{2}+2〈y,x+y〉$.

Let C be a nonempty closed convex subset of a real Hilbert space H, and let $T:C\to H$ be a mapping. Let $Fix\left(T\right):=\left\{x\in C:Tx=x\right\}$. Then T is said to be a nonexpansive mapping if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for every $x,y\in C$. T is said to be a quasi-nonexpansive mapping if $Fix\left(T\right)\ne \mathrm{\varnothing }$ and $\parallel Tx-y\parallel \le \parallel x-y\parallel$ for every $x\in C$ and $y\in Fix\left(T\right)$. It is easy to see that $Fix\left(T\right)$ is a closed convex subset of C if T is a quasi-nonexpansive mapping. Besides, T is said to be a firmly nonexpansive mapping if ${\parallel Tx-Ty\parallel }^{2}\le 〈x-y,Tx-Ty〉$ for every $x,y\in C$, that is, ${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}-{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2}$ for every $x,y\in C$.

Lemma 2.2 

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 $\left\{{x}_{n}\right\}$ be a sequence in C. If ${x}_{n}⇀w$ and ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-T{x}_{n}\parallel =0$, then $Tw=w$.

Let C be a nonempty closed convex subset of a real Hilbert space H. Then, for each $x\in H$, there is a unique element $\overline{x}\in C$ such that $\parallel x-\overline{x}\parallel ={min}_{y\in C}\parallel x-y\parallel$. Here, we set ${P}_{C}x=\overline{x}$ and ${P}_{C}$ is said to be the metric projection from H onto C.

Lemma 2.3 

Let C be a nonempty closed convex subset of a Hilbert space H. Let ${P}_{C}$ be the metric projection from H onto C. Then, for each $x\in H$ and $z\in C$, we know that $z={P}_{C}x$ if and only if $〈x-z,z-y〉\ge 0$ for all $y\in C$.

The following result is an important tool in this paper. For similar results, one can see .

Lemma 2.4 Let H be a real Hilbert space. Let $B:H⊸H$ be a set-valued maximal monotone mapping, $\beta >0$, and let ${J}_{\beta }^{B}$ be a resolvent mapping of B.

1. (i)

For each $\beta >0$, ${J}_{\beta }^{B}$ is a single-valued and firmly nonexpansive mapping;

2. (ii)

$\mathcal{D}\left({J}_{\beta }^{B}\right)=H$ and $Fix\left({J}_{\beta }^{B}\right)=\left\{x\in \mathcal{D}\left(B\right):0\in Bx\right\}$;

3. (iii)

$\parallel x-{J}_{\beta }^{B}x\parallel \le \parallel x-{J}_{\gamma }^{B}x\parallel$ for all $0<\beta \le \gamma$ and for all $x\in H$;

4. (iv)

$\left(I-{J}_{\beta }^{B}\right)$ is a firmly nonexpansive mapping for each $\beta >0$;

5. (v)

Suppose that ${B}^{-1}\left(0\right)\ne \mathrm{\varnothing }$. Then ${\parallel x-{J}_{\beta }^{B}x\parallel }^{2}+{\parallel {J}_{\beta }^{B}x-\overline{x}\parallel }^{2}\le {\parallel x-\overline{x}\parallel }^{2}$ for each $x\in H$, each $\overline{x}\in {B}^{-1}\left(0\right)$, and each $\beta >0$.

6. (vi)

Suppose that ${B}^{-1}\left(0\right)\ne \mathrm{\varnothing }$. Then $〈x-{J}_{\beta }^{B}x,{J}_{\beta }^{B}x-w〉\ge 0$ for each $x\in H$ and each $w\in {B}^{-1}\left(0\right)$, and each $\beta >0$.

Lemma 2.5 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear operator, and ${A}^{\ast }$ be the adjoint of A, and let $\beta >0$ be fixed. Let $B:{H}_{2}⊸{H}_{2}$ be a set-valued maximal monotone mapping, and let ${J}_{\beta }^{B}$ be a resolvent mapping of B. Let $T:{H}_{1}\to {H}_{1}$ be defined by $Tx:={A}^{\ast }\left(I-{J}_{\beta }^{B}\right)Ax$ for each $x\in {H}_{1}$. Then

1. (i)

${\parallel \left(I-{J}_{\beta }^{B}\right)Ax-\left(I-{J}_{\beta }^{B}\right)Ay\parallel }^{2}\le 〈Tx-Ty,x-y〉$ for all $x,y\in {H}_{1}$;

2. (ii)

${\parallel {A}^{\ast }\left(I-{J}_{\beta }^{B}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{B}\right)Ay\parallel }^{2}\le {\parallel A\parallel }^{2}\cdot 〈Tx-Ty,x-y〉$ for all $x,y\in {H}_{1}$.

Proof (i) By Lemma 2.4,

$\begin{array}{rl}〈Tx-Ty,x-y〉& =〈{A}^{\ast }\left(I-{J}_{\beta }^{B}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{B}\right)Ay,x-y〉\\ =〈\left(I-{J}_{\beta }^{B}\right)Ax-\left(I-{J}_{\beta }^{B}\right)Ay,Ax-Ay〉\\ \ge {\parallel \left(I-{J}_{\beta }^{B}\right)Ax-\left(I-{J}_{\beta }^{B}\right)Ay\parallel }^{2}\end{array}$

for all $x,y\in {H}_{1}$. (ii) Further, we have

$\begin{array}{rcl}{\parallel {A}^{\ast }\left(I-{J}_{\beta }^{B}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{B}\right)Ay\parallel }^{2}& \le & {\parallel A\parallel }^{2}\cdot {\parallel \left(I-{J}_{\beta }^{B}\right)Ax-\left(I-{J}_{\beta }^{B}\right)Ay\parallel }^{2}\\ \le & {\parallel A\parallel }^{2}\cdot 〈Tx-Ty,x-y〉\end{array}$

for all $x,y\in {H}_{1}$. Therefore, the proof is completed. □

Lemma 2.6 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear operator, and ${A}^{\ast }$ be the adjoint of A, and let $\beta >0$ be fixed, and let $\rho \in \left(0,\frac{2}{{\parallel A\parallel }^{2}}\right)$. Let ${B}_{2}:{H}_{2}⊸{H}_{2}$ be a set-valued maximal monotone mapping, and let ${J}_{\beta }^{{B}_{2}}$ be a resolvent mapping of ${B}_{2}$. Then

$\begin{array}{c}{\parallel \left[x-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax\right]-\left[y-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\right]\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}-\left(2\rho -{\rho }^{2}{\parallel A\parallel }^{2}\right){\parallel \left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}\hfill \end{array}$

for all $x,y\in {H}_{1}$. Furthermore, $I-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)A$ is a nonexpansive mapping.

Proof For all $x,y\in {H}_{1}$, we have

$\begin{array}{c}{\parallel \left[x-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax\right]-\left[y-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\right]\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel x-y\parallel }^{2}-2\rho 〈x-y,{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay〉\hfill \\ \phantom{\rule{2em}{0ex}}+{\rho }^{2}{\parallel {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel x-y\parallel }^{2}-2\rho 〈Ax-Ay,\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay〉\hfill \\ \phantom{\rule{2em}{0ex}}+{\rho }^{2}{\parallel {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}.\hfill \end{array}$
(2.1)

Hence, it follows from (2.1) and Lemma 2.4 that

$\begin{array}{c}{\parallel \left[x-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax\right]-\left[y-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\right]\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}-2\rho {\parallel \left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\rho }^{2}{\parallel {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}\hfill \\ \le {\parallel x-y\parallel }^{2}-\left(2\rho -{\rho }^{2}{\parallel A\parallel }^{2}\right){\parallel \left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}\hfill \end{array}$

for all $x,y\in {H}_{1}$. Therefore, the proof is completed. □

The following is a very important result for various strong convergence theorems. Recently, many researchers have studied Halpern’s type strong convergence theorems by using the following lemma and got many generalized results. For examples, one can see [32, 33]. In this paper, we also use this result to get our strong convergence theorems, and our results for the split feasibility problem improve the results in the literature.

Lemma 2.7 

Let $\left\{{a}_{n}\right\}$ be a sequence of real numbers such that there exists a subsequence $\left\{{n}_{i}\right\}$ of $\left\{n\right\}$ such that ${a}_{{n}_{i}}<{a}_{{n}_{i}+1}$ for all $i\in \mathbb{N}$. Then there exists a nondecreasing sequence $\left\{{m}_{k}\right\}\subseteq \mathbb{N}$ 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, ${m}_{k}=max\left\{j\le k:{a}_{j}<{a}_{j+1}\right\}$.

Lemma 2.8 

Let ${\left\{{a}_{n}\right\}}_{n\in \mathbb{N}}$ be a sequence of nonnegative real numbers, $\left\{{\alpha }_{n}\right\}$ be a sequence of real numbers in $\left[0,1\right]$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$, $\left\{{u}_{n}\right\}$ be a sequence of nonnegative real numbers with ${\sum }_{n=1}^{\mathrm{\infty }}{u}_{n}<\mathrm{\infty }$, $\left\{{t}_{n}\right\}$ be a sequence of real numbers with $lim sup{t}_{n}\le 0$. Suppose that ${a}_{n+1}\le \left(1-{\alpha }_{n}\right){a}_{n}+{\alpha }_{n}{t}_{n}+{u}_{n}$ for each $n\in \mathbb{N}$. Then ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$.

## 3 Halpern-Mann type algorithm with perturbations

In this section, we first give the following result.

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 let ${A}^{\ast }$ denote the adjoint of A. Let ${B}_{1}:{H}_{1}⊸{H}_{1}$, and ${B}_{2}:{H}_{2}⊸{H}_{2}$ be two set-valued maximal monotone mappings, and let $\beta >0$ and $\gamma >0$. Given any $\overline{x}\in {H}_{1}$.

1. (i)

If $\overline{x}$ is a solution of (SFVIP), then ${J}_{\beta }^{{B}_{1}}\left(\overline{x}-\gamma {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)A\overline{x}\right)=\overline{x}$.

2. (ii)

Suppose that ${J}_{\beta }^{{B}_{1}}\left(\overline{x}-\gamma {A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A\overline{x}\right)=\overline{x}$ and the solution set of (SFVIP) is nonempty. Then $\overline{x}$ is a solution of (SFVIP).

Proof (i) Suppose that $\overline{x}\in {H}_{1}$ is a solution of (SFVIP). Then $\overline{x}\in {B}_{1}^{-1}\left(0\right)$ and $A\overline{x}\in {B}_{2}^{-1}\left(0\right)$. By Lemma 2.4, it is easy to see that

${J}_{\beta }^{{B}_{1}}\left(\overline{x}-\gamma {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)A\overline{x}\right)={J}_{\beta }^{{B}_{1}}\left(\overline{x}-\gamma {A}^{\ast }\left(A\overline{x}-{J}_{\beta }^{{B}_{2}}A\overline{x}\right)\right)={J}_{\beta }^{{B}_{1}}\left(\overline{x}\right)=\overline{x}.$

(ii) Suppose that $\overline{w}$ is a solution of (SFVIP) and ${J}_{\beta }^{{B}_{1}}\left(\overline{x}-\gamma {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)A\overline{x}\right)=\overline{x}$. By Lemma 2.4,

That is,

(3.1)

By (3.1) and ${A}^{\ast }$ is the adjoint of A,

(3.2)

On the other hand, by Lemma 2.4 again,

(3.3)

By (3.2) and (3.3),

$〈A\overline{x}-{J}_{\beta }^{{B}_{2}}A\overline{x},v-{J}_{\beta }^{{B}_{2}}A\overline{x}+A\overline{x}-Aw〉\le 0$
(3.4)

for each $w\in {B}_{1}^{-1}\left(0\right)$ and each $v\in {B}_{2}^{-1}\left(0\right)$. That is,

${\parallel A\overline{x}-{J}_{\beta }^{{B}_{2}}A\overline{x}\parallel }^{2}\le 〈A\overline{x}-{J}_{\beta }^{{B}_{2}}A\overline{x},Aw-v〉$
(3.5)

for each $w\in {B}_{1}^{-1}\left(0\right)$ and each $v\in {B}_{2}^{-1}\left(0\right)$. Since $\overline{w}$ is a solution of (SFVIP), $\overline{w}\in {B}_{1}^{-1}\left(0\right)$ and $A\overline{w}\in {B}_{2}^{-1}\left(0\right)$. So, it follows from (3.5) that $A\overline{x}={J}_{\beta }^{{B}_{2}}A\overline{x}$. So, $A\overline{x}\in Fix\left({J}_{\beta }^{{B}_{2}}\right)={B}_{2}^{-1}\left(0\right)$. Further,

$\overline{x}={J}_{\beta }^{{B}_{1}}\left(\overline{x}-\gamma {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)A\overline{x}\right)={J}_{\beta }^{{B}_{1}}\left(\overline{x}\right).$

Then $\overline{x}\in Fix\left({J}_{\beta }^{{B}_{1}}\right)={B}_{1}^{-1}\left(0\right)$. Therefore, $\overline{x}$ is a solution of (SFVIP). □

Theorem 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 let ${A}^{\ast }$ denote the adjoint of A. Let ${B}_{1}:{H}_{1}⊸{H}_{1}$ and ${B}_{2}:{H}_{2}⊸{H}_{2}$ be two set-valued maximal monotone mappings. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, and $\left\{{d}_{n}\right\}$ be sequences of real numbers in $\left[0,1\right]$ with ${a}_{n}+{b}_{n}+{c}_{n}+{d}_{n}=1$ and $0<{a}_{n}<1$ for each $n\in \mathbb{N}$. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$. Let $\left\{{v}_{n}\right\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\left\{{\rho }_{n}\right\}\subseteq \left(0,\frac{2}{{\parallel A\parallel }^{2}+1}\right)$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

${x}_{n+1}:={a}_{n}u+{b}_{n}{x}_{n}+{c}_{n}{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]+{d}_{n}{v}_{n}$

for each $n\in \mathbb{N}$. Assume that:

1. (i)

${lim}_{n\to \mathrm{\infty }}{a}_{n}={lim}_{n\to \mathrm{\infty }}\frac{{d}_{n}}{{a}_{n}}=0$; ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}=\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}{d}_{n}<\mathrm{\infty }$;

2. (ii)

${lim inf}_{n\to \mathrm{\infty }}{c}_{n}{\rho }_{n}>0$, ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}{c}_{n}>0$, ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$.

Proof Let $\overline{x}={P}_{\mathrm{\Omega }}u$, where ${P}_{\mathrm{\Omega }}$ is the metric projection from ${H}_{1}$ onto Ω. Then, for each $n\in \mathbb{N}$, it follows from Lemma 2.6 that

$\begin{array}{c}\parallel {x}_{n+1}-\overline{x}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {a}_{n}\parallel u-\overline{x}\parallel +{b}_{n}\parallel {x}_{n}-\overline{x}\parallel +{d}_{n}\parallel {v}_{n}-\overline{x}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+{c}_{n}\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\overline{x}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {a}_{n}\parallel u-\overline{x}\parallel +{b}_{n}\parallel {x}_{n}-\overline{x}\parallel +{d}_{n}\parallel {v}_{n}-\overline{x}\parallel +{c}_{n}\parallel \left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]\hfill \\ \phantom{\rule{2em}{0ex}}-\left[\overline{x}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A\overline{x}\right]\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {a}_{n}\parallel u-\overline{x}\parallel +\left({b}_{n}+{c}_{n}\right)\parallel {x}_{n}-\overline{x}\parallel +{d}_{n}\parallel {v}_{n}-\overline{x}\parallel .\hfill \end{array}$

This implies that $\left\{{x}_{n}\right\}$ is a bounded sequence. Besides, by Lemmas 2.4 and 2.6, we have

$\begin{array}{c}{\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \parallel \left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\left[\overline{x}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A\overline{x}\right]\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-\overline{x}\parallel }^{2}-\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel \left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {x}_{n}-\overline{x}\parallel }^{2}-\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel \left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\parallel }^{2}.\hfill \end{array}$
(3.6)

Hence, it follows from Lemma 2.1 that

$\begin{array}{c}{\parallel {x}_{n+1}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {a}_{n}u+{b}_{n}{x}_{n}+{c}_{n}{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]+{d}_{n}{v}_{n}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {b}_{n}\left({x}_{n}-\overline{x}\right)+{c}_{n}\left({J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\overline{x}\right)+{d}_{n}\left({v}_{n}-\overline{x}\right)\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{a}_{n}〈u-\overline{x},{x}_{n+1}-\overline{x}〉\hfill \\ \phantom{\rule{1em}{0ex}}={\left(1-{a}_{n}\right)}^{2}{\parallel {b}_{n}^{\prime }\left({x}_{n}-\overline{x}\right)+{c}_{n}^{\prime }\left({J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\overline{x}\right)+{d}_{n}^{\prime }\left({v}_{n}-\overline{x}\right)\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{a}_{n}〈u-\overline{x},{x}_{n+1}-\overline{x}〉,\hfill \end{array}$
(3.7)

where ${b}_{n}^{\prime }:=\frac{{b}_{n}}{{b}_{n}+{c}_{n}+{d}_{n}}$, ${c}_{n}^{\prime }:=\frac{{c}_{n}}{{b}_{n}+{c}_{n}+{d}_{n}}$, ${d}_{n}^{\prime }:=\frac{{d}_{n}}{{b}_{n}+{c}_{n}+{d}_{n}}$. Further, by (3.6) and (3.7), we have

$\begin{array}{c}{\parallel {x}_{n+1}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {b}_{n}{\parallel {x}_{n}-\overline{x}\parallel }^{2}+{c}_{n}{\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Ax\right]-\overline{x}\parallel }^{2}+{d}_{n}{\parallel {v}_{n}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{a}_{n}〈u-\overline{x},{x}_{n+1}-v〉-{b}_{n}{c}_{n}{\parallel {x}_{n}-{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Ax\right]\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {b}_{n}{\parallel {x}_{n}-\overline{x}\parallel }^{2}+{c}_{n}\left({\parallel {x}_{n}-\overline{x}\parallel }^{2}-\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel }^{2}\right)\hfill \\ \phantom{\rule{2em}{0ex}}+{d}_{n}{\parallel {v}_{n}-\overline{x}\parallel }^{2}+2{a}_{n}〈u-\overline{x},{x}_{n+1}-v〉-{b}_{n}{c}_{n}{\parallel {x}_{n}-{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Ax\right]\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=\left({b}_{n}+{c}_{n}\right){\parallel {x}_{n}-\overline{x}\parallel }^{2}+{d}_{n}{\parallel {v}_{n}-\overline{x}\parallel }^{2}+2{a}_{n}〈u-\overline{x},{x}_{n+1}-\overline{x}〉\hfill \\ \phantom{\rule{2em}{0ex}}-{c}_{n}\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{b}_{n}{c}_{n}{\parallel {x}_{n}-{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Ax\right]\parallel }^{2}.\hfill \end{array}$
(3.8)

Since ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$, we may assume that ${\beta }_{n}>\beta >0$ for each $n\in \mathbb{N}$. Next, we consider two cases.

Case 1: There exists a natural number N such that $\parallel {x}_{n+1}-\overline{x}\parallel \le \parallel {x}_{n}-\overline{x}\parallel$ for each $n\ge N$. So, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-\overline{x}\parallel$ exists. Hence, it follows from (3.8) and (i) that

$\underset{n\to \mathrm{\infty }}{lim}{c}_{n}\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel }^{2}=0.$

Clearly, ${c}_{n}\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right)\ge \frac{{c}_{n}{\rho }_{n}}{{\parallel A\parallel }^{2}+1}$. Since ${lim inf}_{n\to \mathrm{\infty }}{c}_{n}{\rho }_{n}>0$, we have

$\underset{n\to \mathrm{\infty }}{lim}\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel =0.$
(3.9)

By (3.9) and Lemma 2.4,

$\underset{n\to \mathrm{\infty }}{lim}\parallel A{x}_{n}-{J}_{\beta }^{{B}_{2}}A{x}_{n}\parallel =0.$
(3.10)

Similarly, we know that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]\parallel =0.$
(3.11)

Further, there exists a subsequence $\left\{{x}_{{n}_{k}}\right\}$ of $\left\{{x}_{n}\right\}$ such that ${x}_{{n}_{k}}⇀z$ for some $z\in C$ and

$\underset{n\to \mathrm{\infty }}{lim sup}〈u-\overline{x},{x}_{n+1}-\overline{x}〉=\underset{k\to \mathrm{\infty }}{lim}〈u-\overline{x},{x}_{{n}_{k}}-\overline{x}〉=〈u-\overline{x},z-\overline{x}〉.$
(3.12)

Clearly, $A{x}_{{n}_{k}}⇀Az$. By (3.10), Lemmas 2.2 and 2.4, we know that $Az\in {B}_{2}^{-1}\left(0\right)$. Besides, it follows from Lemma 2.4 that

$\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{J}_{{\beta }_{n}}^{{B}_{1}}{x}_{n}\parallel \le {\rho }_{n}\parallel A\parallel \cdot \parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel .$
(3.13)

By (3.9) and (3.13),

$\underset{n\to \mathrm{\infty }}{lim}\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{J}_{{\beta }_{n}}^{{B}_{1}}{x}_{n}\parallel =0.$
(3.14)

By (3.11) and (3.14),

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{J}_{{\beta }_{n}}^{{B}_{1}}{x}_{n}\parallel =0.$
(3.15)

By (3.15) and Lemma 2.4,

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{J}_{\beta }^{{B}_{1}}{x}_{n}\parallel =0.$
(3.16)

Then it follows from (3.16) and Lemma 2.2 that $z\in {B}_{1}^{-1}\left(0\right)$. So, z is a solution of (SFVIP). By (3.12) and Lemma 2.3,

$\underset{n\to \mathrm{\infty }}{lim sup}〈u-\overline{x},{x}_{n+1}-\overline{x}〉\le 0.$
(3.17)

By assumptions, (3.8), (3.17), and Lemma 2.8, we know that ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$.

Case 2: Suppose that there exists $\left\{{n}_{i}\right\}$ of $\left\{n\right\}$ such that $\parallel {x}_{{n}_{i}}-\overline{x}\parallel \le \parallel {x}_{{n}_{i+1}}-\overline{x}\parallel$ for all $i\in \mathbb{N}$. By Lemma 2.7, there exists a nondecreasing sequence $\left\{{m}_{k}\right\}$ in such that ${m}_{k}\to \mathrm{\infty }$,

$\parallel {x}_{{m}_{k}}-\overline{x}\parallel \le \parallel {x}_{{m}_{k}+1}-\overline{x}\parallel \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\parallel {x}_{k}-\overline{x}\parallel \le \parallel {x}_{{m}_{k}+1}-\overline{x}\parallel$
(3.18)

for all $k\in \mathbb{N}$. By (3.8) and (3.18), we have

$\begin{array}{c}\parallel {x}_{{m}_{k}}-\overline{x}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{{m}_{k}+1}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \left({b}_{{m}_{k}}+{c}_{{m}_{k}}\right){\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}+{d}_{{m}_{k}}{\parallel {v}_{{m}_{k}}-\overline{x}\parallel }^{2}+2{a}_{{m}_{k}}〈u-\overline{x},{x}_{{m}_{k}+1}-\overline{x}〉\hfill \\ \phantom{\rule{2em}{0ex}}-{c}_{{m}_{k}}\left(2{\rho }_{{m}_{k}}-{\rho }_{{m}_{k}}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}A{x}_{{m}_{k}}\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{b}_{{m}_{k}}{c}_{{m}_{k}}{\parallel {x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{1}}\left[{x}_{{m}_{k}}-{\rho }_{{m}_{k}}{A}^{\ast }\left(I-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}\right)A{x}_{{m}_{k}}\right]\parallel }^{2}.\hfill \end{array}$
(3.19)

Following a similar argument as the proof of Case 1, we have

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{1}}\left[{x}_{{m}_{k}}-{\rho }_{{m}_{k}}{A}^{\ast }\left(I-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}\right)A{x}_{{m}_{k}}\right]\parallel =0,$
(3.20)
$\underset{k\to \mathrm{\infty }}{lim}\parallel A{x}_{{m}_{k}}-{J}_{\beta }^{{B}_{2}}A{x}_{{m}_{k}}\parallel =\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}}-{J}_{\beta }^{{B}_{1}}{x}_{{m}_{k}}\parallel =0$
(3.21)

and

$\underset{k\to \mathrm{\infty }}{lim sup}〈u-\overline{x},{x}_{{m}_{k}+1}-\overline{x}〉\le 0.$
(3.22)

By (3.19),

${\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}\le \frac{{d}_{{m}_{k}}}{{a}_{{m}_{k}}}{\parallel {v}_{{m}_{k}}-\overline{x}\parallel }^{2}+2〈u-\overline{x},{x}_{{m}_{k}+1}-\overline{x}〉.$
(3.23)

By assumption, (3.22), and (3.23),

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}}-\overline{x}\parallel =0.$
(3.24)

Besides, we have

$\begin{array}{c}\parallel {x}_{{m}_{k}+1}-{x}_{{m}_{k}}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {a}_{{m}_{k}}\parallel u-{x}_{{m}_{k}}\parallel +{c}_{{m}_{k}}\parallel {x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{1}}\left[{x}_{{m}_{k}}-{\rho }_{{m}_{k}}{A}^{\ast }\left(I-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}\right)A{x}_{{m}_{k}}\right]\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+{d}_{{m}_{k}}\parallel {v}_{{m}_{k}}-{x}_{{m}_{k}}\parallel .\hfill \end{array}$
(3.25)

By assumptions, (3.20), and (3.25),

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}+1}-{x}_{{m}_{k}}\parallel =0.$
(3.26)

By (3.24) and (3.26),

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}+1}-\overline{x}\parallel =0.$
(3.27)

By (3.18) and (3.27),

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{k}-\overline{x}\parallel =0.$

Therefore, the proof is completed. □

In Theorem 3.1, if we set ${v}_{n}=0$ and ${d}_{n}=0$ for each $n\in \mathbb{N}$, then we get the following result.

Corollary 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 let ${A}^{\ast }$ denote the adjoint of A. Let ${B}_{1}:{H}_{1}⊸{H}_{1}$ and ${B}_{2}:{H}_{2}⊸{H}_{2}$ be two set-valued maximal monotone mappings. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, and $\left\{{c}_{n}\right\}$ be sequences of real numbers in $\left[0,1\right]$ with ${a}_{n}+{b}_{n}+{c}_{n}=1$ and $0<{a}_{n}<1$ for each $n\in \mathbb{N}$. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$. Let $u\in H$ be fixed. Let $\left\{{\rho }_{n}\right\}\subseteq \left(0,\frac{2}{{\parallel A\parallel }^{2}+1}\right)$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

${x}_{n+1}:={a}_{n}u+{b}_{n}{x}_{n}+{c}_{n}{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}=\mathrm{\infty }$, ${lim inf}_{n\to \mathrm{\infty }}{c}_{n}{\rho }_{n}>0$, ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}{c}_{n}>0$, and ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$. Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$.

Further, we can get the following result by Corollary 3.1 and Lemma 2.8. In fact, Corollary 3.1 and Theorem 3.2 are equivalent.

Theorem 3.2 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let ${B}_{1}:{H}_{1}⊸{H}_{1}$ and ${B}_{2}:{H}_{2}⊸{H}_{2}$ be two set-valued maximal monotone mappings. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, and $\left\{{c}_{n}\right\}$ be sequences of real numbers in $\left[0,1\right]$ with ${a}_{n}+{b}_{n}+{c}_{n}=1$ and $0<{a}_{n}<1$ for each $n\in \mathbb{N}$. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$. Let $\left\{{v}_{n}\right\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\left\{{\rho }_{n}\right\}\subseteq \left(0,\frac{2}{{\parallel A\parallel }^{2}+1}\right)$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{v}_{n}\right\}$ be a bounded sequence. Let $\left\{{x}_{n}\right\}$ be defined by

${x}_{n+1}:={a}_{n}u+{b}_{n}{x}_{n}+{c}_{n}{J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]+{v}_{n}$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}\parallel {v}_{n}\parallel <\mathrm{\infty }$, ${lim inf}_{n\to \mathrm{\infty }}{c}_{n}{\rho }_{n}>0$, ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}{c}_{n}>0$, and ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$. Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$.

Proof Let $\left\{{y}_{n}\right\}$ be defined by

${y}_{n+1}:={a}_{n}u+{b}_{n}{y}_{n}+{c}_{n}{J}_{{\beta }_{n}}^{{B}_{1}}\left[{y}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{y}_{n}\right].$

By Corollary 3.1, ${lim}_{n\to \mathrm{\infty }}{y}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$. Besides, we know that

$\begin{array}{c}\parallel {x}_{n+1}-{y}_{n+1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {c}_{n}\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{J}_{{\beta }_{n}}^{{B}_{1}}\left[{y}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{y}_{n}\right]\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+{b}_{n}\parallel {x}_{n}-{y}_{n}\parallel +\parallel {v}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \left({b}_{n}+{c}_{n}\right)\parallel {x}_{n}-{y}_{n}\parallel +\parallel {v}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\left(1-{a}_{n}\right)\parallel {x}_{n}-{y}_{n}\parallel +\parallel {v}_{n}\parallel .\hfill \end{array}$
(3.28)

By (3.28) and Lemma 2.8, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{y}_{n}\parallel =0$. So, ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$. Therefore, the proof is completed. □

## 4 Regularized method for (SFVIP)

Lemma 4.1 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let ${B}_{1}:{H}_{1}⊸{H}_{1}$ and ${B}_{2}:{H}_{2}⊸{H}_{2}$ be two set-valued maximal monotone mappings. Let $\beta >0$, $a\in \left(0,1\right)$, and $\rho \in \left(0,2/\left({\parallel A\parallel }^{2}+2\right)\right)$. Then

$\parallel {J}_{\beta }^{{B}_{1}}\left[\left(1-a\rho \right)x-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax\right]-{J}_{\beta }^{{B}_{1}}\left[\left(1-a\rho \right)y-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\right]\parallel \le \left(1-a\rho \right)\parallel x-y\parallel$

for all $x,y\in {H}_{1}$.

Proof For each $x,y\in {H}_{1}$, it follows from Lemma 2.4 and Lemma 2.5 that

$\begin{array}{c}{\parallel {J}_{\beta }^{{B}_{1}}\left(\left(1-a\rho \right)x-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax\right)-{J}_{\beta }^{{B}_{1}}\left(\left(1-a\rho \right)y-\rho {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel \left(1-a\rho \right)\left(x-y\right)-\rho \left({A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\left(1-a\rho \right)}^{2}{\parallel x-y\parallel }^{2}-2\left(1-a\rho \right)\rho 〈x-y,{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay〉\hfill \\ \phantom{\rule{2em}{0ex}}+{\rho }^{2}{\parallel {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(1-a\rho \right){\parallel x-y\parallel }^{2}-2\left(1-{\alpha }_{n}\rho \right)\rho \frac{1}{{\parallel A\parallel }^{2}}{\parallel {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\rho }^{2}{\parallel {A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ax-{A}^{\ast }\left(I-{J}_{\beta }^{{B}_{2}}\right)Ay\right)\parallel }^{2}.\hfill \end{array}$

If $\rho \in \left(0,2/{\parallel A\parallel }^{2}+2\right)$, then $2\left(1-a\rho \right)\rho \left(1/{\parallel A\parallel }^{2}\right)\ge {\rho }^{2}$. This implies that the conclusion of Lemma 4.1 holds. □

Theorem 4.1 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let ${B}_{1}:{H}_{1}⊸{H}_{1}$ and ${B}_{2}:{H}_{2}⊸{H}_{2}$ be two set-valued maximal monotone mappings. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$, $\left\{{a}_{n}\right\}\subseteq \left(0,1\right)$, and $\left\{{\rho }_{n}\right\}\subseteq \left(0,2/\left({\parallel A\parallel }^{2}+2\right)\right)$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

${x}_{n+1}:={J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]$

for each $n\in \mathbb{N}$. Assume that:

$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=0,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}{a}_{n}{\rho }_{n}=\mathrm{\infty },\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim inf}{\rho }_{n}>0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty }}{lim inf}{\beta }_{n}>0.$

Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}0$, i.e., $\overline{x}$ is the minimal norm solution of (SFVIP).

Proof Let $\overline{x}={P}_{\mathrm{\Omega }}0$. Take any $w\in \mathrm{\Omega }$ and let w be fixed. Then we know that

$\begin{array}{c}\parallel {x}_{n+1}-w\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-w\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{J}_{{\beta }_{n}}^{{B}_{1}}\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right)w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right)w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]-{J}_{{\beta }_{n}}^{{B}_{1}}\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \left(1-{a}_{n}{\rho }_{n}\right)\parallel {x}_{n}-w\parallel +{a}_{n}{\rho }_{n}\parallel w\parallel \hfill \end{array}$

for each $n\in \mathbb{N}$. Then $\left\{{x}_{n}\right\}$ is a bounded sequence. Further, we have

$\begin{array}{c}{\parallel {x}_{n+1}-w\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{J}_{{\beta }_{n}}^{{B}_{1}}\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel \left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]-{a}_{n}{\rho }_{n}{x}_{n}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-2{a}_{n}{\rho }_{n}〈\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right],{x}_{n}〉\hfill \\ \phantom{\rule{2em}{0ex}}+{a}_{n}{\rho }_{n}{\parallel {x}_{n}\parallel }^{2}\hfill \end{array}$
(4.1)

for each $n\in \mathbb{N}$. By (4.1) and Lemma 2.6,

$\begin{array}{c}{\parallel {x}_{n+1}-w\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-w\parallel }^{2}-\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel \left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-2{a}_{n}{\rho }_{n}〈\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right],{x}_{n}〉\hfill \\ \phantom{\rule{2em}{0ex}}+{a}_{n}{\rho }_{n}{\parallel {x}_{n}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-w\parallel }^{2}-\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel }^{2}+{a}_{n}{\rho }_{n}{\parallel {x}_{n}\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{a}_{n}{\rho }_{n}\parallel \left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\left[w-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\right]\parallel \cdot \parallel {x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-w\parallel }^{2}-\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel }^{2}+{a}_{n}{\rho }_{n}{\parallel {x}_{n}\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{a}_{n}{\rho }_{n}\parallel {x}_{n}-w\parallel \cdot \parallel {x}_{n}\parallel \hfill \end{array}$
(4.2)

for each $n\in \mathbb{N}$. By (4.1)-(4.2), Lemma 2.4, we know that

$\begin{array}{c}{\parallel \left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-{x}_{n+1}\parallel }^{2}+{\parallel {x}_{n+1}-w\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel \left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-w\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-w+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)Aw\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-w\parallel }^{2}+2{a}_{n}{\rho }_{n}\parallel {x}_{n}-w\parallel \cdot \parallel {x}_{n}\parallel +{a}_{n}{\rho }_{n}{\parallel {x}_{n}\parallel }^{2}\hfill \end{array}$
(4.3)

for each $n\in \mathbb{N}$. Next, we know that

$\begin{array}{c}{\parallel \left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-{x}_{n+1}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {x}_{n}-{x}_{n+1}\parallel }^{2}+{\parallel {a}_{n}{\rho }_{n}{x}_{n}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-2〈{x}_{n}-{x}_{n+1},{a}_{n}{\rho }_{n}{x}_{n}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}〉\hfill \end{array}$
(4.4)

for each $n\in \mathbb{N}$, and

$\begin{array}{c}\parallel {x}_{n+1}-{J}_{{\beta }_{n}}^{{B}_{1}}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{J}_{{\beta }_{n}}^{{B}_{1}}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \parallel \left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {a}_{n}{\rho }_{n}\parallel {x}_{n}\parallel +{\rho }_{n}\parallel {A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {a}_{n}{\rho }_{n}\parallel {x}_{n}\parallel +{\rho }_{n}\parallel A\parallel \cdot \parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel \hfill \end{array}$
(4.5)

for each $n\in \mathbb{N}$. Further, we have

$\begin{array}{c}{\parallel {x}_{n+1}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {J}_{{\beta }_{n}}^{{B}_{1}}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\right]-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le 〈\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-\overline{x}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A\overline{x},{x}_{n+1}-\overline{x}〉\hfill \\ \phantom{\rule{1em}{0ex}}=〈\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-\left(1-{a}_{n}{\rho }_{n}\right)\overline{x}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A\overline{x},{x}_{n+1}-\overline{x}〉\hfill \\ \phantom{\rule{2em}{0ex}}-{a}_{n}{\rho }_{n}〈\overline{x},{x}_{n+1}-\overline{x}〉\hfill \end{array}$
(4.6)

for each $n\in \mathbb{N}$. Hence,

$\begin{array}{c}{\parallel {x}_{n+1}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \parallel \left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-\left(1-{a}_{n}{\rho }_{n}\right)\overline{x}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A\overline{x}\parallel \cdot \parallel {x}_{n+1}-\overline{x}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+{a}_{n}{\rho }_{n}〈-\overline{x},{x}_{n+1}-\overline{x}〉\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(1-{a}_{n}{\rho }_{n}\right)\parallel {x}_{n}-\overline{x}\parallel \cdot \parallel {x}_{n+1}-\overline{x}\parallel +{a}_{n}{\rho }_{n}〈-\overline{x},{x}_{n+1}-\overline{x}〉\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(1-{a}_{n}{\rho }_{n}\right)}^{2}}{2}{\parallel {x}_{n}-\overline{x}\parallel }^{2}+\frac{1}{2}{\parallel {x}_{n+1}-\overline{x}\parallel }^{2}+{a}_{n}{\rho }_{n}〈-\overline{x},{x}_{n+1}-\overline{x}〉\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(\frac{1-{a}_{n}{\rho }_{n}}{2}\right){\parallel {x}_{n}-\overline{x}\parallel }^{2}+\frac{1}{2}{\parallel {x}_{n+1}-\overline{x}\parallel }^{2}+{a}_{n}{\rho }_{n}〈-\overline{x},{x}_{n+1}-\overline{x}〉\hfill \end{array}$

for each $n\in \mathbb{N}$. This implies that

${\parallel {x}_{n+1}-\overline{x}\parallel }^{2}\le \left(1-{a}_{n}{\rho }_{n}\right){\parallel {x}_{n}-\overline{x}\parallel }^{2}+2{a}_{n}{\rho }_{n}〈-\overline{x},{x}_{n+1}-\overline{x}〉$
(4.7)

for each $n\in \mathbb{N}$.

Case 1: There exists a natural number N such that $\parallel {x}_{n+1}-\overline{x}\parallel \le \parallel {x}_{n}-\overline{x}\parallel$ for each $n\ge N$. So, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-\overline{x}\parallel$ exists.

Hence, it follows from ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-\overline{x}\parallel$ exists and (4.2) that

$\underset{n\to \mathrm{\infty }}{lim}\left(2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel }^{2}=0.$
(4.8)

Clearly,

$2{\rho }_{n}-{\rho }_{n}^{2}{\parallel A\parallel }^{2}={\rho }_{n}\left(2-{\rho }_{n}{\parallel A\parallel }^{2}\right)\ge {\rho }_{n}\left(2-\frac{2{\parallel A\parallel }^{2}}{{\parallel A\parallel }^{2}+2}\right)=\frac{4{\rho }_{n}}{{\parallel A\parallel }^{2}+2}.$
(4.9)

By assumption, (4.8), and (4.9),

$\underset{n\to \mathrm{\infty }}{lim}\parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel =0.$
(4.10)

Without loss of generality, we may assume that ${\beta }_{n}\ge \beta >0$ for each $n\in \mathbb{N}$. By (4.10) and Lemma 2.4,

$\underset{n\to \mathrm{\infty }}{lim}\parallel A{x}_{n}-{J}_{\beta }^{{B}_{2}}A{x}_{n}\parallel =0.$
(4.11)

By assumption, (4.5), and (4.10),

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-{J}_{{\beta }_{n}}^{{B}_{1}}{x}_{n}\parallel =0.$
(4.12)

By assumption, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-\overline{x}\parallel$ exists, $\left\{{x}_{n}\right\}$ is a bounded sequence, and (4.3), we know that

$\underset{n\to \mathrm{\infty }}{lim}\parallel \left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}-{x}_{n+1}\parallel =0.$
(4.13)

Clearly,

$\parallel {a}_{n}{\rho }_{n}{x}_{n}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\parallel \le {a}_{n}{\rho }_{n}\parallel {x}_{n}\parallel +{\rho }_{n}\parallel A\parallel \cdot \parallel A{x}_{n}-{J}_{{\beta }_{n}}^{{B}_{2}}A{x}_{n}\parallel$
(4.14)

for each $n\in \mathbb{N}$. By assumption, (4.10), and (4.14),

$\underset{n\to \mathrm{\infty }}{lim}\parallel {a}_{n}{\rho }_{n}{x}_{n}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}\parallel =0.$
(4.15)

By (4.15),

$\underset{n\to \mathrm{\infty }}{lim}〈{x}_{n}-{x}_{n+1},{a}_{n}{\rho }_{n}{x}_{n}+{\rho }_{n}{A}^{\ast }\left(I-{J}_{{\beta }_{n}}^{{B}_{2}}\right)A{x}_{n}〉=0.$
(4.16)

By (4.4), (4.13), (4.15), and (4.16), we know that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-{x}_{n}\parallel =0.$
(4.17)

By (4.12) and (4.17),

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{J}_{{\beta }_{n}}^{{B}_{1}}{x}_{n}\parallel =0.$
(4.18)

Since $\left\{{x}_{n}\right\}$ is a bounded sequence, there exists a subsequence $\left\{{x}_{{n}_{j}}\right\}$ of $\left\{{x}_{n}\right\}$ such that ${x}_{{n}_{j}}⇀z$ for some $z\in {H}_{1}$ and

$\underset{n\to \mathrm{\infty }}{lim sup}〈-\overline{x},{x}_{n+1}-\overline{x}〉=\underset{n\to \mathrm{\infty }}{lim}〈-\overline{x},{x}_{{n}_{j}}-\overline{x}〉=〈-\overline{x},z-\overline{x}〉.$

Then $A{x}_{{n}_{j}}⇀Az\in {H}_{2}$. By (4.11), (4.18), Lemma 2.2, and Lemma 2.4, we know that $z\in {B}_{1}^{-1}\left(0\right)$ and $Az\in {B}_{2}^{-1}\left(0\right)$. That is, $z\in \mathrm{\Omega }$. By Lemma 2.3,

$\underset{n\to \mathrm{\infty }}{lim sup}〈-\overline{x},{x}_{n+1}-\overline{x}〉=〈-\overline{x},z-\overline{x}〉\le 0.$
(4.19)

By (4.7), (4.19), and Lemma 2.8, we know that ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}0$.

Case 2: Suppose that there exists $\left\{{n}_{i}\right\}$ of $\left\{n\right\}$ such that $\parallel {x}_{{n}_{i}}-\overline{x}\parallel \le \parallel {x}_{{n}_{i}+1}-\overline{x}\parallel$ for all $i\in \mathbb{N}$. By Lemma 2.7, there exists a nondecreasing sequence $\left\{{m}_{k}\right\}$ in such that ${m}_{k}\to \mathrm{\infty }$,

$\parallel {x}_{{m}_{k}}-\overline{x}\parallel \le \parallel {x}_{{m}_{k}+1}-\overline{x}\parallel \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\parallel {x}_{k}-\overline{x}\parallel \le \parallel {x}_{{m}_{k}+1}-\overline{x}\parallel$
(4.20)

for each $k\in \mathbb{N}$. By (4.2), we have

$\begin{array}{c}{\parallel {x}_{{m}_{k}+1}-\overline{x}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}-\left(2{\rho }_{{m}_{k}}-{\rho }_{{m}_{k}}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}A{x}_{{m}_{k}}\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{a}_{{m}_{k}}{\rho }_{{m}_{k}}{\parallel {x}_{{m}_{k}}\parallel }^{2}+2{a}_{{m}_{k}}{\rho }_{{m}_{k}}\parallel {x}_{{m}_{k}}-\overline{x}\parallel \cdot \parallel {x}_{{m}_{k}}\parallel \hfill \end{array}$
(4.21)

for each $k\in \mathbb{N}$. By (4.20) and (4.21),

$\begin{array}{c}\left(2{\rho }_{{m}_{k}}-{\rho }_{{m}_{k}}^{2}{\parallel A\parallel }^{2}\right){\parallel A{x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}A{x}_{{m}_{k}}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}-{\parallel {x}_{{m}_{k}+1}-\overline{x}\parallel }^{2}+{a}_{{m}_{k}}{\rho }_{{m}_{k}}{\parallel {x}_{{m}_{k}}\parallel }^{2}+2{a}_{{m}_{k}}{\rho }_{{m}_{k}}\parallel {x}_{{m}_{k}}-\overline{x}\parallel \cdot \parallel {x}_{{m}_{k}}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {a}_{{m}_{k}}{\rho }_{{m}_{k}}{\parallel {x}_{{m}_{k}}\parallel }^{2}+2{a}_{{m}_{k}}{\rho }_{{m}_{k}}\parallel {x}_{{m}_{k}}-\overline{x}\parallel \cdot \parallel {x}_{{m}_{k}}\parallel \hfill \end{array}$
(4.22)

for each $k\in \mathbb{N}$. Then following the same argument as the above, we know that

$\underset{k\to \mathrm{\infty }}{lim}\parallel A{x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}A{x}_{{m}_{k}}\parallel =0,$
(4.23)
$\underset{k\to \mathrm{\infty }}{lim}\parallel A{x}_{{m}_{k}}-{J}_{\beta }^{{B}_{2}}A{x}_{{m}_{k}}\parallel =0,$
(4.24)
$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}+1}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{1}}A{x}_{{m}_{k}}\parallel =0.$
(4.25)

By (4.3),

$\begin{array}{c}{\parallel \left(1-{a}_{{m}_{k}}{\rho }_{{m}_{k}}\right){x}_{{m}_{k}}-{\rho }_{{m}_{k}}{A}^{\ast }\left(I-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}\right)A{x}_{{m}_{k}}-{x}_{{m}_{k}+1}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}-{\parallel {x}_{{m}_{k}+1}-\overline{x}\parallel }^{2}+2{a}_{{m}_{k}}{\rho }_{{m}_{k}}\parallel {x}_{{m}_{k}}-\overline{x}\parallel \cdot \parallel {x}_{{m}_{k}}\parallel +{a}_{{m}_{k}}{\rho }_{{m}_{k}}{\parallel {x}_{{m}_{k}}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le 2{a}_{{m}_{k}}{\rho }_{{m}_{k}}\parallel {x}_{{m}_{k}}-\overline{x}\parallel \cdot \parallel {x}_{{m}_{k}}\parallel +{a}_{{m}_{k}}{\rho }_{{m}_{k}}{\parallel {x}_{{m}_{k}}\parallel }^{2}\hfill \end{array}$
(4.26)

for each $k\in \mathbb{N}$. This implies that

$\underset{k\to \mathrm{\infty }}{lim}{\parallel \left(1-{a}_{{m}_{k}}{\rho }_{{m}_{k}}\right){x}_{{m}_{k}}-{\rho }_{{m}_{k}}{A}^{\ast }\left(I-{J}_{{\beta }_{{m}_{k}}}^{{B}_{2}}\right)A{x}_{{m}_{k}}-{x}_{{m}_{k}+1}\parallel }^{2}=0.$
(4.27)

Following the same argument as the above, we know that

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}+1}-{x}_{{m}_{k}}\parallel =\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}}-{J}_{{\beta }_{{m}_{k}}}^{{B}_{1}}{x}_{{m}_{k}}\parallel =0$
(4.28)

and

$\underset{k\to \mathrm{\infty }}{lim sup}〈-\overline{x},{x}_{{m}_{k}+1}-\overline{x}〉=〈-\overline{x},z-\overline{x}〉\le 0.$
(4.29)

By (4.7) and (4.20),

$\begin{array}{rcl}{a}_{{m}_{k}}{\rho }_{{m}_{k}}{\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}& \le & {\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}-{\parallel {x}_{{m}_{k}+1}-\overline{x}\parallel }^{2}+2{a}_{{m}_{k}}{\rho }_{{m}_{k}}〈-\overline{x},{x}_{{m}_{k}+1}-\overline{x}〉\\ \le & 2{a}_{{m}_{k}}{\rho }_{{m}_{k}}〈-\overline{x},{x}_{{m}_{k}+1}-\overline{x}〉\end{array}$

for each $k\in \mathbb{N}$. This implies that

${\parallel {x}_{{m}_{k}}-\overline{x}\parallel }^{2}\le 2〈-\overline{x},{x}_{{m}_{k}+1}-\overline{x}〉$
(4.30)

for each $k\in \mathbb{N}$. By (4.29) and (4.30),

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}}-\overline{x}\parallel =0.$
(4.31)

By (4.28) and (4.31),

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}+1}-\overline{x}\parallel \le \underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}}-{x}_{{m}_{k}+1}\parallel +\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{{m}_{k}}-\overline{x}\parallel =0.$
(4.32)

By (4.20) and (4.32),

$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{k}-\overline{x}\parallel =0.$
(4.33)

Therefore, the proof is completed. □

## 5 Applications: (SFOP) and (SFP)

We get the following results by Theorems 3.1 and 3.2, respectively.

Theorem 5.1 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let $f:{H}_{1}\to \mathbb{R}$ and $g:{H}_{2}\to \mathbb{R}$ be two proper lower semicontinuous and convex functions. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, and $\left\{{d}_{n}\right\}$ be sequences of real numbers in $\left[0,1\right]$ with ${a}_{n}+{b}_{n}+{c}_{n}+{d}_{n}=1$ and $0<{a}_{n}<1$ for each $n\in \mathbb{N}$. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$. Let $\left\{{v}_{n}\right\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\left\{{\rho }_{n}\right\}\subseteq \left(0,\frac{2}{{\parallel A\parallel }^{2}+1}\right)$. Let Ω be the solution set of (SFOP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

$\left\{\begin{array}{l}{y}_{n}=arg{min}_{z\in {H}_{2}}\left\{g\left(z\right)+\frac{1}{2{\beta }_{n}}{\parallel z-A{x}_{n}\parallel }^{2}\right\},\\ {z}_{n}={x}_{n}-{\rho }_{n}{A}^{\ast }\left(A{x}_{n}-{y}_{n}\right),\\ {w}_{n}=arg{min}_{y\in {H}_{1}}\left\{f\left(y\right)+\frac{1}{2{\beta }_{n}}{\parallel y-{z}_{n}\parallel }^{2}\right\},\\ {x}_{n+1}:={a}_{n}u+{b}_{n}{x}_{n}+{c}_{n}{w}_{n}+{d}_{n}{v}_{n},\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.\end{array}$

Assume that:

1. (i)

${lim}_{n\to \mathrm{\infty }}{a}_{n}={lim}_{n\to \mathrm{\infty }}\frac{{d}_{n}}{{a}_{n}}=0$; ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}=\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}{d}_{n}<\mathrm{\infty }$;

2. (ii)

${lim inf}_{n\to \mathrm{\infty }}{c}_{n}{\rho }_{n}>0$; ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}{c}_{n}>0$; ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$.

Theorem 5.2 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let $f:{H}_{1}\to \mathbb{R}$ and $g:{H}_{2}\to \mathbb{R}$ be two proper lower semicontinuous and convex functions. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, and $\left\{{c}_{n}\right\}$ be sequences of real numbers in $\left[0,1\right]$ with ${a}_{n}+{b}_{n}+{c}_{n}=1$ and $0<{a}_{n}<1$ for each $n\in \mathbb{N}$. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$. Let $\left\{{v}_{n}\right\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\left\{{\rho }_{n}\right\}\subseteq \left(0,\frac{2}{{\parallel A\parallel }^{2}+1}\right)$. Let Ω be the solution set of (SFOP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

$\left\{\begin{array}{l}{y}_{n}=arg{min}_{z\in {H}_{2}}\left\{g\left(z\right)+\frac{1}{2{\beta }_{n}}{\parallel z-A{x}_{n}\parallel }^{2}\right\},\\ {z}_{n}={x}_{n}-{\rho }_{n}{A}^{\ast }\left(A{x}_{n}-{y}_{n}\right),\\ {w}_{n}=arg{min}_{y\in {H}_{1}}\left\{f\left(y\right)+\frac{1}{2{\beta }_{n}}{\parallel y-{z}_{n}\parallel }^{2}\right\},\\ {x}_{n+1}:={a}_{n}u+{b}_{n}{x}_{n}+{c}_{n}{w}_{n}+{v}_{n},\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.\end{array}$

Assume that ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}\parallel {v}_{n}\parallel <\mathrm{\infty }$, ${lim inf}_{n\to \mathrm{\infty }}{c}_{n}{\rho }_{n}>0$, ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}{c}_{n}>0$, ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$. Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$.

By Theorem 4.1, we get the following result.

Theorem 5.3 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let $f:{H}_{1}\to \mathbb{R}$ and $g:{H}_{2}\to \mathbb{R}$ be two proper lower semicontinuous and convex functions. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$, $\left\{{a}_{n}\right\}\subseteq \left(0,1\right)$, and $\left\{{\rho }_{n}\right\}\subseteq \left(0,2/\left({\parallel A\parallel }^{2}+2\right)\right)$. Let Ω be the solution set of (SFOP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

$\left\{\begin{array}{l}{y}_{n}=arg{min}_{z\in {H}_{2}}\left\{g\left(z\right)+\frac{1}{2{\beta }_{n}}{\parallel z-A{x}_{n}\parallel }^{2}\right\},\\ {z}_{n}=\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(A{x}_{n}-{y}_{n}\right),\\ {x}_{n+1}=arg{min}_{y\in {H}_{1}}\left\{g\left(y\right)+\frac{1}{2{\beta }_{n}}{\parallel y-{z}_{n}\parallel }^{2}\right\},\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.\end{array}$

Assume that ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}{\rho }_{n}=\mathrm{\infty }$, ${lim inf}_{n\to \mathrm{\infty }}{\rho }_{n}>0$, and ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$. Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}0$, i.e., $\overline{x}$ is the minimal norm solution of (SFOP).

Let H be a Hilbert space and let g be a proper lower semicontinuous convex function of H into $\left(-\mathrm{\infty },\mathrm{\infty }\right)$. Then the subdifferential ∂g of g is defined as follows:

$\partial g\left(x\right)=\left\{z\in H:g\left(x\right)+〈z,y-x〉\le g\left(y\right),\mathrm{\forall }y\in H\right\}$

for all $x\in H$. Let C be a nonempty closed convex subset of a real Hilbert space H, and ${i}_{C}$ be the indicator function of C, i.e.,

Further, we also define the normal cone ${N}_{C}u$ of C at u as follows:

${N}_{C}u=\left\{z\in H:〈z,v-u〉\le 0,\mathrm{\forall }v\in C\right\}.$

Then ${i}_{C}$ is a proper lower semicontinuous convex function on H, and the subdifferential $\partial {i}_{C}$ of ${i}_{C}$ is a maximal monotone operator. So, we can define the resolvent ${J}_{\lambda }^{\partial {i}_{C}}$ of $\partial {i}_{C}$ for $\lambda >0$, i.e.,

${J}_{\lambda }^{\partial {i}_{C}}x={\left(I+\lambda \partial {i}_{C}\right)}^{-1}x$

for all $x\in H$. By definitions, we know that

$\begin{array}{rcl}\partial {i}_{C}x& =& \left\{z\in H:{i}_{C}x+〈z,y-x〉\le {i}_{C}y,\mathrm{\forall }y\in H\right\}\\ =& \left\{z\in H:〈z,y-x〉\le 0,\mathrm{\forall }y\in C\right\}\\ =& {N}_{C}x\end{array}$

for all $x\in C$. Hence, for each $\beta >0$, we have that

$\begin{array}{rcl}u={J}_{\beta }^{\partial {i}_{C}}x\phantom{\rule{1em}{0ex}}& ⇔& \phantom{\rule{1em}{0ex}}x\in u+\beta \partial {i}_{C}u\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}x-u\in \beta {N}_{C}u\\ ⇔& \phantom{\rule{1em}{0ex}}〈x-u,y-u〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C\\ ⇔& \phantom{\rule{1em}{0ex}}u={P}_{C}x.\end{array}$

Hence, we have the following result by Theorem 3.2.

Theorem 5.4 Let C and Q be two nonempty closed convex subsets of ${H}_{1}$ and ${H}_{2}$, respectively. Let $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, and $\left\{{c}_{n}\right\}$ be sequences of real numbers in $\left[0,1\right]$ with ${a}_{n}+{b}_{n}+{c}_{n}=1$ and $0<{a}_{n}<1$ for each $n\in \mathbb{N}$. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$. Let $\left\{{v}_{n}\right\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\left\{{\rho }_{n}\right\}\subseteq \left(0,\frac{2}{{\parallel A\parallel }^{2}+1}\right)$. Let Ω be the solution set of (SFP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

${x}_{n+1}:={a}_{n}u+{b}_{n}{x}_{n}+{c}_{n}{P}_{C}\left[{x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{P}_{Q}\right)A{x}_{n}\right]+{v}_{n}$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}=\mathrm{\infty }$, ${lim inf}_{n\to \mathrm{\infty }}{c}_{n}{\rho }_{n}>0$, and ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}{c}_{n}>0$. Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}u$.

By Theorem 4.1, we get the following result.

Theorem 5.5 Let C and Q be two nonempty closed convex subsets of ${H}_{1}$ and ${H}_{2}$, respectively. Let $A:{H}_{1}\to {H}_{2}$ be a linear and bounded operator, and let ${A}^{\ast }$ denote the adjoint of A. Let $\left\{{\beta }_{n}\right\}$ be a sequence in $\left(0,\mathrm{\infty }\right)$, $\left\{{a}_{n}\right\}\subseteq \left(0,1\right)$, and $\left\{{\rho }_{n}\right\}\subseteq \left(0,2/\left({\parallel A\parallel }^{2}+2\right)\right)$. Let Ω be the solution set of (SFP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\left\{{x}_{n}\right\}$ be defined by

${x}_{n+1}:={P}_{C}\left[\left(1-{a}_{n}{\rho }_{n}\right){x}_{n}-{\rho }_{n}{A}^{\ast }\left(I-{P}_{Q}\right)A{x}_{n}\right]$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{n}{\rho }_{n}=\mathrm{\infty }$, ${lim inf}_{n\to \mathrm{\infty }}{\rho }_{n}>0$, and ${lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}>0$. Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$, where $\overline{x}={P}_{\mathrm{\Omega }}0$, i.e., $\overline{x}$ is the minimal norm solution of (SFP).

Remark 5.1 Theorem 5.5 improves some conditions of [, Theorem 5.5].

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