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 [1] 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 [1–5] and related literature.

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

$$(\text{SFP})\text{Find }\overline{x}\in H_1\text{ such that }\overline{x}\in C\text{ and }A\overline{x}\in Q,$$

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 [6]:

$$(\text{CLIP})\text{Find }\overline{x}\in C\text{ such that }A\overline{x}=b,\text{where }b\in H_2.$$

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 [1] 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, 7–16] and related literature.

In 2002, Byrne [2] first introduced the following recursive procedure:

$$x_{n+1}=P_C(x_n-{\rho }_n{A}^{\ast }(I-P_Q)A{x}_n),$$

(1.1)

where the stepsize ${\tau }_n$ is chosen in the interval $(0,2/{\parallel A\parallel }^2)$, 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 [11] 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 [14] considered (SFP) under the following conditions:

$$C:=\{x\in {\mathbb{R}}^n:c(x)\le 0\}\text{and}Q:=\{x\in {\mathbb{R}}^m:q(x)\le 0\},$$

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 [14] studied the following problem, and we call this problem the relaxed split feasibility problem:

$$(\text{RSFP})\text{Find }\overline{x}\in {\mathbb{R}}^n\text{ such that }c(\overline{x})\le 0\text{ and }q(A\overline{x})\le 0.$$

In 2010, Xu [13] 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 $\mathcal{D}(B):=\{x\in H:B(x)\ne \mathrm{\varnothing }\}$. 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 $\{(x,y):x\in \mathcal{D}(B),y\in Bx\}$ is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find $\overline{x}\in H$ 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 [17] and generated by Rockafellar [18]. This is an iterative procedure, which generates $\{x_n\}$ by $x_1=x\in H$ and

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

(1.2)

where $\{{\beta }_n\}\subseteq (0,\mathrm{\infty })$, 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 $r>0$. In 1976, Rockafellar [18] proved the following in the Hilbert space setting: 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 subdifferential ∂f of a proper lower semicontinuous and convex function $f:H\to \mathbb{R}$, then (1.2) is reduced to

$$x_{n+1}=arg\underset{y\in H}{min}\{f(y)+\frac{1}{2{\beta }_n}{\parallel y-x_n\parallel }^2\},n\in \mathbb{N}.$$

(1.3)

In this case, $\{x_n\}$ 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 [19–24] 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:

$$(\text{SFVIP})\text{Find }\overline{x}\in H_1\text{ such that }0\in B_1(\overline{x})\text{ and }0\in B_2(A\overline{x}).$$

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 [25], and Moudafi [25] gave a weak convergence theorem for problem (SFVIP). The following is an iteration process given by Moudafi [25]:

$$x_{n+1}:=J_{\lambda }^{B_1}(x_n+\gamma A^{\ast }(J_{\lambda }^{B_2}-I)A{x}_n).$$

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 $f(\overline{x})={min}_{y\in H_1}f(y)$ and $g(A\overline{x})={min}_{y\in H_2}g(z)$, 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 ℕ be the set of positive integers and let ℝ 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 $\{x_n\}$ to $x\in H$ by $x_n\to x$ and $x_n\rightharpoonup x$, respectively. From [26], for each $x,y\in H$ and $\lambda \in [0,1]$, we have

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

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 $\alpha ,\beta ,\gamma \in [0,1]$ with $\alpha +\beta +\gamma =1$ [27].

Lemma 2.1 [28]

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 $Fix(T):=\{x\in C:Tx=x\}$. 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 $Fix(T)\ne \mathrm{\varnothing }$ and $\parallel Tx-y\parallel \le \parallel x-y\parallel$ for every $x\in C$ and $y\in Fix(T)$. It is easy to see that $Fix(T)$ 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, ${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel (I-T)x-(I-T)y\parallel }^2$ for every $x,y\in C$.

Lemma 2.2 [29]

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 $Tw=w$.

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 [30]

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 [31].

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.

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

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

• (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 $x\in H$;

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

• (v) Suppose that $B^{-1}(0)\ne \mathrm{\varnothing }$. 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 $\overline{x}\in B^{-1}(0)$, and each $\beta >0$.

• (vi) 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 2.5 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 $Tx:=A^{\ast }(I-J_{\beta }^B)Ax$ for each $x\in H_1$. Then

• (i) ${\parallel (I-J_{\beta }^B)Ax-(I-J_{\beta }^B)Ay\parallel }^2\le 〈Tx-Ty,x-y〉$ for all $x,y\in H_1$;

• (ii) ${\parallel A^{\ast }(I-J_{\beta }^B)Ax-A^{\ast }(I-J_{\beta }^B)Ay\parallel }^2\le {\parallel A\parallel }^2\cdot 〈Tx-Ty,x-y〉$ for all $x,y\in H_1$.

Proof (i) By Lemma 2.4,

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

for all $x,y\in H_1$. (ii) Further, we have

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

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 $\rho \in (0,\frac{2}{{\parallel A\parallel }^2})$. 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

$$\begin{array}{c}{\parallel [x-\rho A^{\ast }(I-J_{\beta }^{B_2})Ax]-[y-\rho A^{\ast }(I-J_{\beta }^{B_2})Ay]\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2-(2\rho -{\rho }^2{\parallel A\parallel }^2){\parallel (I-J_{\beta }^{B_2})Ax-(I-J_{\beta }^{B_2})Ay\parallel }^2\hfill \end{array}$$

for all $x,y\in H_1$. Furthermore, $I-\rho A^{\ast }(I-J_{\beta }^{B_2})A$ is a nonexpansive mapping.

Proof For all $x,y\in H_1$, we have

$$\begin{array}{c}{\parallel [x-\rho A^{\ast }(I-J_{\beta }^{B_2})Ax]-[y-\rho A^{\ast }(I-J_{\beta }^{B_2})Ay]\parallel }^2\hfill \\ ={\parallel x-y\parallel }^2-2\rho 〈x-y,A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay〉\hfill \\ +{\rho }^2{\parallel A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay\parallel }^2\hfill \\ ={\parallel x-y\parallel }^2-2\rho 〈Ax-Ay,(I-J_{\beta }^{B_2})Ax-(I-J_{\beta }^{B_2})Ay〉\hfill \\ +{\rho }^2{\parallel A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay\parallel }^2.\hfill \end{array}$$

(2.1)

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

$$\begin{array}{c}{\parallel [x-\rho A^{\ast }(I-J_{\beta }^{B_2})Ax]-[y-\rho A^{\ast }(I-J_{\beta }^{B_2})Ay]\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2-2\rho {\parallel (I-J_{\beta }^{B_2})Ax-(I-J_{\beta }^{B_2})Ay\parallel }^2\hfill \\ +{\rho }^2{\parallel A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2-(2\rho -{\rho }^2{\parallel A\parallel }^2){\parallel (I-J_{\beta }^{B_2})Ax-(I-J_{\beta }^{B_2})Ay\parallel }^2\hfill \end{array}$$

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 [34]

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 $\{m_k\}\subseteq \mathbb{N}$ 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, $m_k=max\{j\le k:a_j<a_{j+1}\}$.

Lemma 2.8 [35]

Let ${\{a_n\}}_{n\in \mathbb{N}}$ be a sequence of nonnegative real numbers, $\{{\alpha }_n\}$ be a sequence of real numbers in $[0,1]$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, $\{u_n\}$ be a sequence of nonnegative real numbers with ${\sum }_{n=1}^{\mathrm{\infty }}{u}_n<\mathrm{\infty }$, $\{t_n\}$ be a sequence of real numbers with $lim\hspace{0.17em}sup{t}_n\le 0$. Suppose that $a_{n+1}\le (1-{\alpha }_n)a_n+{\alpha }_n{t}_n+u_n$ 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 $\overline{x}\in H_1$.

• (i) If $\overline{x}$ is a solution of (SFVIP), then $J_{\beta }^{B_1}(\overline{x}-\gamma A^{\ast }(I-J_{\beta }^{B_2})A\overline{x})=\overline{x}$.

• (ii) Suppose that $J_{\beta }^{B_1}(\overline{x}-\gamma A^{\ast }(I-J_{{\beta }_n}^{B_2})A\overline{x})=\overline{x}$ 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 $\overline{x}\in B_1^{-1}(0)$ and $A\overline{x}\in B_2^{-1}(0)$. By Lemma 2.4, it is easy to see that

$$J_{\beta }^{B_1}(\overline{x}-\gamma A^{\ast }(I-J_{\beta }^{B_2})A\overline{x})=J_{\beta }^{B_1}(\overline{x}-\gamma A^{\ast }(A\overline{x}-J_{\beta }^{B_2}A\overline{x}))=J_{\beta }^{B_1}(\overline{x})=\overline{x}.$$

(ii) Suppose that $\overline{w}$ is a solution of (SFVIP) and $J_{\beta }^{B_1}(\overline{x}-\gamma A^{\ast }(I-J_{\beta }^{B_2})A\overline{x})=\overline{x}$. By Lemma 2.4,

$$〈(\overline{x}-\gamma A^{\ast }(I-J_{\beta }^{B_2})A\overline{x})-\overline{x},\overline{x}-w〉\ge 0\text{for each }w\in B_1^{-1}(0).$$

That is,

$$〈A^{\ast }(I-J_{\beta }^{B_2})A\overline{x},\overline{x}-w〉\le 0\text{for each }w\in B_1^{-1}(0).$$

(3.1)

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

$$〈A\overline{x}-J_{\beta }^{B_2}A\overline{x},A\overline{x}-Aw〉\le 0\text{for each }w\in B_1^{-1}(0).$$

(3.2)

On the other hand, by Lemma 2.4 again,

$$〈A\overline{x}-J_{\beta }^{B_2}A\overline{x},v-J_{\beta }^{B_2}A\overline{x}〉\le 0\text{for each }v\in B_2^{-1}(0).$$

(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 $w\in B_1^{-1}(0)$ and each $v\in B_2^{-1}(0)$. 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 $w\in B_1^{-1}(0)$ and each $v\in B_2^{-1}(0)$. Since $\overline{w}$ is a solution of (SFVIP), $\overline{w}\in B_1^{-1}(0)$ and $A\overline{w}\in B_2^{-1}(0)$. So, it follows from (3.5) that $A\overline{x}=J_{\beta }^{B_2}A\overline{x}$. So, $A\overline{x}\in Fix(J_{\beta }^{B_2})=B_2^{-1}(0)$. Further,

$$\overline{x}=J_{\beta }^{B_1}(\overline{x}-\gamma A^{\ast }(I-J_{\beta }^{B_2})A\overline{x})=J_{\beta }^{B_1}(\overline{x}).$$

Then $\overline{x}\in Fix(J_{\beta }^{B_1})=B_1^{-1}(0)$. 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 $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, and $\{d_n\}$ be sequences of real numbers in $[0,1]$ with $a_n+b_n+c_n+d_n=1$ and $0<a_n<1$ for each $n\in \mathbb{N}$. Let $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$. Let $\{v_n\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\{{\rho }_n\}\subseteq (0,\frac{2}{{\parallel A\parallel }^2+1})$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$x_{n+1}:=a_{n}u+b_n{x}_n+c_n{J}_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]+d_n{v}_n$$

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

• (i) ${lim}_{n\to \mathrm{\infty }}{a}_n={lim}_{n\to \mathrm{\infty }}\frac{d_n}{a_n}=0$; ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}{d}_n<\mathrm{\infty }$;

• (ii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_n{\rho }_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{b}_n{c}_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$.

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

$$\begin{array}{c}\parallel x_{n+1}-\overline{x}\parallel \hfill \\ \le a_n\parallel u-\overline{x}\parallel +b_n\parallel x_n-\overline{x}\parallel +d_n\parallel v_n-\overline{x}\parallel \hfill \\ +c_n\parallel J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-\overline{x}\parallel \hfill \\ \le a_n\parallel u-\overline{x}\parallel +b_n\parallel x_n-\overline{x}\parallel +d_n\parallel v_n-\overline{x}\parallel +c_n\parallel [x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]\hfill \\ -[\overline{x}-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A\overline{x}]\parallel \hfill \\ \le a_n\parallel u-\overline{x}\parallel +(b_n+c_n)\parallel x_n-\overline{x}\parallel +d_n\parallel v_n-\overline{x}\parallel .\hfill \end{array}$$

This implies that $\{x_n\}$ is a bounded sequence. Besides, by Lemmas 2.4 and 2.6, we have

$$\begin{array}{c}{\parallel J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-\overline{x}\parallel }^2\hfill \\ \le \parallel [x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-[\overline{x}-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A\overline{x}]\parallel \hfill \\ \le {\parallel x_n-\overline{x}\parallel }^2-(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel (I-J_{{\beta }_n}^{B_2})A{x}_n-(I-J_{{\beta }_n}^{B_2})A\overline{x}\parallel }^2\hfill \\ ={\parallel x_n-\overline{x}\parallel }^2-(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel (I-J_{{\beta }_n}^{B_2})A{x}_n\parallel }^2.\hfill \end{array}$$

(3.6)

Hence, it follows from Lemma 2.1 that

$$\begin{array}{c}{\parallel x_{n+1}-\overline{x}\parallel }^2\hfill \\ ={\parallel a_{n}u+b_n{x}_n+c_n{J}_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]+d_n{v}_n-\overline{x}\parallel }^2\hfill \\ \le {\parallel b_n(x_n-\overline{x})+c_n(J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-\overline{x})+d_n(v_n-\overline{x})\parallel }^2\hfill \\ +2a_n〈u-\overline{x},x_{n+1}-\overline{x}〉\hfill \\ ={(1-a_n)}^2{\parallel b_n^{\prime }(x_n-\overline{x})+c_n^{\prime }(J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-\overline{x})+d_n^{\prime }(v_n-\overline{x})\parallel }^2\hfill \\ +2a_n〈u-\overline{x},x_{n+1}-\overline{x}〉,\hfill \end{array}$$

(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

$$\begin{array}{c}{\parallel x_{n+1}-\overline{x}\parallel }^2\hfill \\ \le b_n{\parallel x_n-\overline{x}\parallel }^2+c_n{\parallel J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Ax]-\overline{x}\parallel }^2+d_n{\parallel v_n-\overline{x}\parallel }^2\hfill \\ +2a_n〈u-\overline{x},x_{n+1}-v〉-b_n{c}_n{\parallel x_n-J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Ax]\parallel }^2\hfill \\ \le b_n{\parallel x_n-\overline{x}\parallel }^2+c_n({\parallel x_n-\overline{x}\parallel }^2-(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel }^2)\hfill \\ +d_n{\parallel v_n-\overline{x}\parallel }^2+2a_n〈u-\overline{x},x_{n+1}-v〉-b_n{c}_n{\parallel x_n-J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Ax]\parallel }^2\hfill \\ =(b_n+c_n){\parallel x_n-\overline{x}\parallel }^2+d_n{\parallel v_n-\overline{x}\parallel }^2+2a_n〈u-\overline{x},x_{n+1}-\overline{x}〉\hfill \\ -c_n(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel }^2\hfill \\ -b_n{c}_n{\parallel x_n-J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Ax]\parallel }^2.\hfill \end{array}$$

(3.8)

Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$, 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

$$\underset{n\to \mathrm{\infty }}{lim}{c}_n(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel }^2=0.$$

Clearly, $c_n(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2)\ge \frac{c_n{\rho }_n}{{\parallel A\parallel }^2+1}$. Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_n{\rho }_n>0$, 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

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]\parallel =0.$$

(3.11)

Further, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $x_{n_k}\rightharpoonup z$ for some $z\in C$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-\overline{x},x_{n+1}-\overline{x}〉=\underset{k\to \mathrm{\infty }}{lim}〈u-\overline{x},x_{n_k}-\overline{x}〉=〈u-\overline{x},z-\overline{x}〉.$$

(3.12)

Clearly, $A{x}_{n_k}\rightharpoonup Az$. By (3.10), Lemmas 2.2 and 2.4, we know that $Az\in B_2^{-1}(0)$. Besides, it follows from Lemma 2.4 that

$$\parallel J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-J_{{\beta }_n}^{B_1}{x}_n\parallel \le {\rho }_n\parallel A\parallel \cdot \parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel .$$

(3.13)

By (3.9) and (3.13),

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-J_{{\beta }_n}^{B_1}{x}_n\parallel =0.$$

(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 $z\in B_1^{-1}(0)$. So, z is a solution of (SFVIP). By (3.12) and Lemma 2.3,

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-\overline{x},x_{n+1}-\overline{x}〉\le 0.$$

(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 $\{n_i\}$ of $\{n\}$ 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 $\{m_k\}$ in ℕ such that $m_k\to \mathrm{\infty }$,

$$\parallel x_{m_k}-\overline{x}\parallel \le \parallel x_{m_k+1}-\overline{x}\parallel \text{and}\parallel x_k-\overline{x}\parallel \le \parallel x_{m_k+1}-\overline{x}\parallel$$

(3.18)

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

$$\begin{array}{c}\parallel x_{m_k}-\overline{x}\parallel \hfill \\ \le {\parallel x_{m_k+1}-\overline{x}\parallel }^2\hfill \\ \le (b_{m_k}+c_{m_k}){\parallel x_{m_k}-\overline{x}\parallel }^2+d_{m_k}{\parallel v_{m_k}-\overline{x}\parallel }^2+2a_{m_k}〈u-\overline{x},x_{m_k+1}-\overline{x}〉\hfill \\ -c_{m_k}(2{\rho }_{m_k}-{\rho }_{m_k}^2{\parallel A\parallel }^2){\parallel A{x}_{m_k}-J_{{\beta }_{m_k}}^{B_2}A{x}_{m_k}\parallel }^2\hfill \\ -b_{m_k}{c}_{m_k}{\parallel x_{m_k}-J_{{\beta }_{m_k}}^{B_1}[x_{m_k}-{\rho }_{m_k}{A}^{\ast }(I-J_{{\beta }_{m_k}}^{B_2})A{x}_{m_k}]\parallel }^2.\hfill \end{array}$$

(3.19)

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

$$\underset{k\to \mathrm{\infty }}{lim}\parallel x_{m_k}-J_{{\beta }_{m_k}}^{B_1}[x_{m_k}-{\rho }_{m_k}{A}^{\ast }(I-J_{{\beta }_{m_k}}^{B_2})A{x}_{m_k}]\parallel =0,$$

(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

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u-\overline{x},x_{m_k+1}-\overline{x}〉\le 0.$$

(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

$$\begin{array}{c}\parallel x_{m_k+1}-x_{m_k}\parallel \hfill \\ \le a_{m_k}\parallel u-x_{m_k}\parallel +c_{m_k}\parallel x_{m_k}-J_{{\beta }_{m_k}}^{B_1}[x_{m_k}-{\rho }_{m_k}{A}^{\ast }(I-J_{{\beta }_{m_k}}^{B_2})A{x}_{m_k}]\parallel \hfill \\ +d_{m_k}\parallel v_{m_k}-x_{m_k}\parallel .\hfill \end{array}$$

(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 $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be sequences of real numbers in $[0,1]$ with $a_n+b_n+c_n=1$ and $0<a_n<1$ for each $n\in \mathbb{N}$. Let $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$. Let $u\in H$ be fixed. Let $\{{\rho }_n\}\subseteq (0,\frac{2}{{\parallel A\parallel }^2+1})$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$x_{n+1}:=a_{n}u+b_n{x}_n+c_n{J}_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]$$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_n{\rho }_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{b}_n{c}_n>0$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$. 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 $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be sequences of real numbers in $[0,1]$ with $a_n+b_n+c_n=1$ and $0<a_n<1$ for each $n\in \mathbb{N}$. Let $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$. Let $\{v_n\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\{{\rho }_n\}\subseteq (0,\frac{2}{{\parallel A\parallel }^2+1})$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{v_n\}$ be a bounded sequence. Let $\{x_n\}$ be defined by

$$x_{n+1}:=a_{n}u+b_n{x}_n+c_n{J}_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]+v_n$$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}\parallel v_n\parallel <\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_n{\rho }_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{b}_n{c}_n>0$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$. Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{\mathrm{\Omega }}u$.

Proof Let $\{y_n\}$ be defined by

$$y_{n+1}:=a_{n}u+b_n{y}_n+c_n{J}_{{\beta }_n}^{B_1}[y_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{y}_n].$$

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

$$\begin{array}{c}\parallel x_{n+1}-y_{n+1}\parallel \hfill \\ \le c_n\parallel J_{{\beta }_n}^{B_1}[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-J_{{\beta }_n}^{B_1}[y_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{y}_n]\parallel \hfill \\ +b_n\parallel x_n-y_n\parallel +\parallel v_n\parallel \hfill \\ \le (b_n+c_n)\parallel x_n-y_n\parallel +\parallel v_n\parallel \hfill \\ =(1-a_n)\parallel x_n-y_n\parallel +\parallel v_n\parallel .\hfill \end{array}$$

(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 $\beta >0$, $a\in (0,1)$, and $\rho \in (0,2/({\parallel A\parallel }^2+2))$. Then

$$\parallel J_{\beta }^{B_1}[(1-a\rho )x-\rho A^{\ast }(I-J_{\beta }^{B_2})Ax]-J_{\beta }^{B_1}[(1-a\rho )y-\rho A^{\ast }(I-J_{\beta }^{B_2})Ay]\parallel \le (1-a\rho )\parallel x-y\parallel$$

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

$$\begin{array}{c}{\parallel J_{\beta }^{B_1}((1-a\rho )x-\rho A^{\ast }(I-J_{\beta }^{B_2})Ax)-J_{\beta }^{B_1}((1-a\rho )y-\rho A^{\ast }(I-J_{\beta }^{B_2})Ay)\parallel }^2\hfill \\ \le {\parallel (1-a\rho )(x-y)-\rho (A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay)\parallel }^2\hfill \\ ={(1-a\rho )}^2{\parallel x-y\parallel }^2-2(1-a\rho )\rho 〈x-y,A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay〉\hfill \\ +{\rho }^2{\parallel A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay\parallel }^2\hfill \\ \le (1-a\rho ){\parallel x-y\parallel }^2-2(1-{\alpha }_n\rho )\rho \frac{1}{{\parallel A\parallel }^2}{\parallel A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay\parallel }^2\hfill \\ +{\rho }^2{\parallel A^{\ast }(I-J_{\beta }^{B_2})Ax-A^{\ast }(I-J_{\beta }^{B_2})Ay)\parallel }^2.\hfill \end{array}$$

If $\rho \in (0,2/{\parallel A\parallel }^2+2)$, then $2(1-a\rho )\rho (1/{\parallel A\parallel }^2)\ge {\rho }^2$. 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 $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$, $\{a_n\}\subseteq (0,1)$, and $\{{\rho }_n\}\subseteq (0,2/({\parallel A\parallel }^2+2))$. Let Ω be the solution set of (SFVIP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$x_{n+1}:=J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]$$

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

$$\underset{n\to \mathrm{\infty }}{lim}{a}_n=0,\sum _{n=1}^{\mathrm{\infty }}{a}_n{\rho }_n=\mathrm{\infty },\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\rho }_n>0\mathit{\text{and}}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\beta }_n>0.$$

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

$$\begin{array}{c}\parallel x_{n+1}-w\parallel \hfill \\ =\parallel J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-w\parallel \hfill \\ =\parallel J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-J_{{\beta }_n}^{B_1}[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]\parallel \hfill \\ \le \parallel J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]\parallel \hfill \\ +\parallel J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]-J_{{\beta }_n}^{B_1}[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]\parallel \hfill \\ \le (1-a_n{\rho }_n)\parallel x_n-w\parallel +a_n{\rho }_n\parallel w\parallel \hfill \end{array}$$

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

$$\begin{array}{c}{\parallel x_{n+1}-w\parallel }^2\hfill \\ ={\parallel J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-J_{{\beta }_n}^{B_1}[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]\parallel }^2\hfill \\ \le {\parallel [(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]\parallel }^2\hfill \\ ={\parallel [x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]-a_n{\rho }_n{x}_n\parallel }^2\hfill \\ ={\parallel [x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]\parallel }^2\hfill \\ -2a_n{\rho }_n〈[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw],x_n〉\hfill \\ +a_n{\rho }_n{\parallel x_n\parallel }^2\hfill \end{array}$$

(4.1)

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

$$\begin{array}{c}{\parallel x_{n+1}-w\parallel }^2\hfill \\ \le {\parallel x_n-w\parallel }^2-(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel (I-J_{{\beta }_n}^{B_2})A{x}_n-(I-J_{{\beta }_n}^{B_2})Aw\parallel }^2\hfill \\ -2a_n{\rho }_n〈[x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw],x_n〉\hfill \\ +a_n{\rho }_n{\parallel x_n\parallel }^2\hfill \\ \le {\parallel x_n-w\parallel }^2-(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel }^2+a_n{\rho }_n{\parallel x_n\parallel }^2\hfill \\ +2a_n{\rho }_n\parallel [x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-[w-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw]\parallel \cdot \parallel x_n\parallel \hfill \\ \le {\parallel x_n-w\parallel }^2-(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel }^2+a_n{\rho }_n{\parallel x_n\parallel }^2\hfill \\ +2a_n{\rho }_n\parallel x_n-w\parallel \cdot \parallel x_n\parallel \hfill \end{array}$$

(4.2)

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

$$\begin{array}{c}{\parallel (1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-x_{n+1}\parallel }^2+{\parallel x_{n+1}-w\parallel }^2\hfill \\ \le {\parallel (1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-w\parallel }^2\hfill \\ ={\parallel (1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-w+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})Aw\parallel }^2\hfill \\ \le {\parallel x_n-w\parallel }^2+2a_n{\rho }_n\parallel x_n-w\parallel \cdot \parallel x_n\parallel +a_n{\rho }_n{\parallel x_n\parallel }^2\hfill \end{array}$$

(4.3)

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

$$\begin{array}{c}{\parallel (1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-x_{n+1}\parallel }^2\hfill \\ ={\parallel x_n-x_{n+1}\parallel }^2+{\parallel a_n{\rho }_n{x}_n+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n\parallel }^2\hfill \\ -2〈x_n-x_{n+1},a_n{\rho }_n{x}_n+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n〉\hfill \end{array}$$

(4.4)

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

$$\begin{array}{c}\parallel x_{n+1}-J_{{\beta }_n}^{B_1}{x}_n\parallel \hfill \\ =\parallel J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-J_{{\beta }_n}^{B_1}{x}_n\parallel \hfill \\ \le \parallel [(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-x_n\parallel \hfill \\ \le a_n{\rho }_n\parallel x_n\parallel +{\rho }_n\parallel A^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n\parallel \hfill \\ \le a_n{\rho }_n\parallel x_n\parallel +{\rho }_n\parallel A\parallel \cdot \parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel \hfill \end{array}$$

(4.5)

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

$$\begin{array}{c}{\parallel x_{n+1}-\overline{x}\parallel }^2\hfill \\ ={\parallel J_{{\beta }_n}^{B_1}[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n]-\overline{x}\parallel }^2\hfill \\ \le 〈(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-\overline{x}+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A\overline{x},x_{n+1}-\overline{x}〉\hfill \\ =〈(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-(1-a_n{\rho }_n)\overline{x}+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A\overline{x},x_{n+1}-\overline{x}〉\hfill \\ -a_n{\rho }_n〈\overline{x},x_{n+1}-\overline{x}〉\hfill \end{array}$$

(4.6)

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

$$\begin{array}{c}{\parallel x_{n+1}-\overline{x}\parallel }^2\hfill \\ \le \parallel (1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-(1-a_n{\rho }_n)\overline{x}+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A\overline{x}\parallel \cdot \parallel x_{n+1}-\overline{x}\parallel \hfill \\ +a_n{\rho }_n〈-\overline{x},x_{n+1}-\overline{x}〉\hfill \\ \le (1-a_n{\rho }_n)\parallel x_n-\overline{x}\parallel \cdot \parallel x_{n+1}-\overline{x}\parallel +a_n{\rho }_n〈-\overline{x},x_{n+1}-\overline{x}〉\hfill \\ \le \frac{{(1-a_n{\rho }_n)}^2}{2}{\parallel x_n-\overline{x}\parallel }^2+\frac{1}{2}{\parallel x_{n+1}-\overline{x}\parallel }^2+a_n{\rho }_n〈-\overline{x},x_{n+1}-\overline{x}〉\hfill \\ \le \left(\frac{1-a_n{\rho }_n}{2}\right){\parallel x_n-\overline{x}\parallel }^2+\frac{1}{2}{\parallel x_{n+1}-\overline{x}\parallel }^2+a_n{\rho }_n〈-\overline{x},x_{n+1}-\overline{x}〉\hfill \end{array}$$

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

$${\parallel x_{n+1}-\overline{x}\parallel }^2\le (1-a_n{\rho }_n){\parallel x_n-\overline{x}\parallel }^2+2a_n{\rho }_n〈-\overline{x},x_{n+1}-\overline{x}〉$$

(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

$$\underset{n\to \mathrm{\infty }}{lim}(2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2){\parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel }^2=0.$$

(4.8)

Clearly,

$$2{\rho }_n-{\rho }_n^2{\parallel A\parallel }^2={\rho }_n(2-{\rho }_n{\parallel A\parallel }^2)\ge {\rho }_n(2-\frac{2{\parallel A\parallel }^2}{{\parallel A\parallel }^2+2})=\frac{4{\rho }_n}{{\parallel A\parallel }^2+2}.$$

(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, $\{x_n\}$ is a bounded sequence, and (4.3), we know that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel (1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n-x_{n+1}\parallel =0.$$

(4.13)

Clearly,

$$\parallel a_n{\rho }_n{x}_n+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n\parallel \le a_n{\rho }_n\parallel x_n\parallel +{\rho }_n\parallel A\parallel \cdot \parallel A{x}_n-J_{{\beta }_n}^{B_2}A{x}_n\parallel$$

(4.14)

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

$$\underset{n\to \mathrm{\infty }}{lim}\parallel a_n{\rho }_n{x}_n+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n\parallel =0.$$

(4.15)

By (4.15),

$$\underset{n\to \mathrm{\infty }}{lim}〈x_n-x_{n+1},a_n{\rho }_n{x}_n+{\rho }_n{A}^{\ast }(I-J_{{\beta }_n}^{B_2})A{x}_n〉=0.$$

(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 $\{x_n\}$ is a bounded sequence, there exists a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ such that $x_{n_j}\rightharpoonup z$ for some $z\in H_1$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈-\overline{x},x_{n+1}-\overline{x}〉=\underset{n\to \mathrm{\infty }}{lim}〈-\overline{x},x_{n_j}-\overline{x}〉=〈-\overline{x},z-\overline{x}〉.$$

Then $A{x}_{n_j}\rightharpoonup Az\in H_2$. By (4.11), (4.18), Lemma 2.2, and Lemma 2.4, we know that $z\in B_1^{-1}(0)$ and $Az\in B_2^{-1}(0)$. That is, $z\in \mathrm{\Omega }$. By Lemma 2.3,

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈-\overline{x},x_{n+1}-\overline{x}〉=〈-\overline{x},z-\overline{x}〉\le 0.$$

(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 $\{n_i\}$ of $\{n\}$ 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 $\{m_k\}$ in ℕ such that $m_k\to \mathrm{\infty }$,

$$\parallel x_{m_k}-\overline{x}\parallel \le \parallel x_{m_k+1}-\overline{x}\parallel \text{and}\parallel x_k-\overline{x}\parallel \le \parallel x_{m_k+1}-\overline{x}\parallel$$

(4.20)

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

$$\begin{array}{c}{\parallel x_{m_k+1}-\overline{x}\parallel }^2\hfill \\ \le {\parallel x_{m_k}-\overline{x}\parallel }^2-(2{\rho }_{m_k}-{\rho }_{m_k}^2{\parallel A\parallel }^2){\parallel A{x}_{m_k}-J_{{\beta }_{m_k}}^{B_2}A{x}_{m_k}\parallel }^2\hfill \\ +a_{m_k}{\rho }_{m_k}{\parallel x_{m_k}\parallel }^2+2a_{m_k}{\rho }_{m_k}\parallel x_{m_k}-\overline{x}\parallel \cdot \parallel x_{m_k}\parallel \hfill \end{array}$$

(4.21)

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

$$\begin{array}{c}(2{\rho }_{m_k}-{\rho }_{m_k}^2{\parallel A\parallel }^2){\parallel A{x}_{m_k}-J_{{\beta }_{m_k}}^{B_2}A{x}_{m_k}\parallel }^2\hfill \\ \le {\parallel x_{m_k}-\overline{x}\parallel }^2-{\parallel x_{m_k+1}-\overline{x}\parallel }^2+a_{m_k}{\rho }_{m_k}{\parallel x_{m_k}\parallel }^2+2a_{m_k}{\rho }_{m_k}\parallel x_{m_k}-\overline{x}\parallel \cdot \parallel x_{m_k}\parallel \hfill \\ \le a_{m_k}{\rho }_{m_k}{\parallel x_{m_k}\parallel }^2+2a_{m_k}{\rho }_{m_k}\parallel x_{m_k}-\overline{x}\parallel \cdot \parallel x_{m_k}\parallel \hfill \end{array}$$

(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),

$$\begin{array}{c}{\parallel (1-a_{m_k}{\rho }_{m_k})x_{m_k}-{\rho }_{m_k}{A}^{\ast }(I-J_{{\beta }_{m_k}}^{B_2})A{x}_{m_k}-x_{m_k+1}\parallel }^2\hfill \\ \le {\parallel x_{m_k}-\overline{x}\parallel }^2-{\parallel x_{m_k+1}-\overline{x}\parallel }^2+2a_{m_k}{\rho }_{m_k}\parallel x_{m_k}-\overline{x}\parallel \cdot \parallel x_{m_k}\parallel +a_{m_k}{\rho }_{m_k}{\parallel x_{m_k}\parallel }^2\hfill \\ \le 2a_{m_k}{\rho }_{m_k}\parallel x_{m_k}-\overline{x}\parallel \cdot \parallel x_{m_k}\parallel +a_{m_k}{\rho }_{m_k}{\parallel x_{m_k}\parallel }^2\hfill \end{array}$$

(4.26)

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

$$\underset{k\to \mathrm{\infty }}{lim}{\parallel (1-a_{m_k}{\rho }_{m_k})x_{m_k}-{\rho }_{m_k}{A}^{\ast }(I-J_{{\beta }_{m_k}}^{B_2})A{x}_{m_k}-x_{m_k+1}\parallel }^2=0.$$

(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

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈-\overline{x},x_{m_k+1}-\overline{x}〉=〈-\overline{x},z-\overline{x}〉\le 0.$$

(4.29)

By (4.7) and (4.20),

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

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 $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, and $\{d_n\}$ be sequences of real numbers in $[0,1]$ with $a_n+b_n+c_n+d_n=1$ and $0<a_n<1$ for each $n\in \mathbb{N}$. Let $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$. Let $\{v_n\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\{{\rho }_n\}\subseteq (0,\frac{2}{{\parallel A\parallel }^2+1})$. Let Ω be the solution set of (SFOP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$\{\begin{array}{l}{y}_n=arg{min}_{z\in H_2}\{g(z)+\frac{1}{2{\beta }_n}{\parallel z-A{x}_n\parallel }^2\},\\ z_n=x_n-{\rho }_n{A}^{\ast }(A{x}_n-y_n),\\ w_n=arg{min}_{y\in H_1}\{f(y)+\frac{1}{2{\beta }_n}{\parallel y-z_n\parallel }^2\},\\ x_{n+1}:=a_{n}u+b_n{x}_n+c_n{w}_n+d_n{v}_n,n\in \mathbb{N}.\end{array}$$

Assume that:

• (i) ${lim}_{n\to \mathrm{\infty }}{a}_n={lim}_{n\to \mathrm{\infty }}\frac{d_n}{a_n}=0$; ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}{d}_n<\mathrm{\infty }$;

• (ii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_n{\rho }_n>0$; ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{b}_n{c}_n>0$; ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$.

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 $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be sequences of real numbers in $[0,1]$ with $a_n+b_n+c_n=1$ and $0<a_n<1$ for each $n\in \mathbb{N}$. Let $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$. Let $\{v_n\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\{{\rho }_n\}\subseteq (0,\frac{2}{{\parallel A\parallel }^2+1})$. Let Ω be the solution set of (SFOP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$\{\begin{array}{l}{y}_n=arg{min}_{z\in H_2}\{g(z)+\frac{1}{2{\beta }_n}{\parallel z-A{x}_n\parallel }^2\},\\ z_n=x_n-{\rho }_n{A}^{\ast }(A{x}_n-y_n),\\ w_n=arg{min}_{y\in H_1}\{f(y)+\frac{1}{2{\beta }_n}{\parallel y-z_n\parallel }^2\},\\ x_{n+1}:=a_{n}u+b_n{x}_n+c_n{w}_n+v_n,n\in \mathbb{N}.\end{array}$$

Assume that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}\parallel v_n\parallel <\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_n{\rho }_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{b}_n{c}_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$. 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 $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$, $\{a_n\}\subseteq (0,1)$, and $\{{\rho }_n\}\subseteq (0,2/({\parallel A\parallel }^2+2))$. Let Ω be the solution set of (SFOP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$\{\begin{array}{l}{y}_n=arg{min}_{z\in H_2}\{g(z)+\frac{1}{2{\beta }_n}{\parallel z-A{x}_n\parallel }^2\},\\ z_n=(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(A{x}_n-y_n),\\ x_{n+1}=arg{min}_{y\in H_1}\{g(y)+\frac{1}{2{\beta }_n}{\parallel y-z_n\parallel }^2\},n\in \mathbb{N}.\end{array}$$

Assume that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n{\rho }_n=\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\rho }_n>0$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$. 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 $(-\mathrm{\infty },\mathrm{\infty })$. Then the subdifferential ∂g of g is defined as follows:

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

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.,

$$i_{C}x=\{\begin{array}{ll}0& \text{if }x\in C,\\ \mathrm{\infty }& \text{if }x\notin C.\end{array}$$

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

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

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.,

$$J_{\lambda }^{\partial i_C}x={(I+\lambda \partial i_C)}^{-1}x$$

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

$$\begin{array}{rcl}\partial i_{C}x& =& \{z\in H:i_{C}x+〈z,y-x〉\le i_{C}y,\mathrm{\forall }y\in H\}\\ =& \{z\in H:〈z,y-x〉\le 0,\mathrm{\forall }y\in C\}\\ =& N_{C}x\end{array}$$

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

$$\begin{array}{rcl}u=J_{\beta }^{\partial i_C}x& \iff & x\in u+\beta \partial i_{C}u\iff x-u\in \beta N_{C}u\\ \iff & 〈x-u,y-u〉\le 0,\mathrm{\forall }y\in C\\ \iff & u=P_{C}x.\end{array}$$

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 $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be sequences of real numbers in $[0,1]$ with $a_n+b_n+c_n=1$ and $0<a_n<1$ for each $n\in \mathbb{N}$. Let $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$. Let $\{v_n\}$ be a bounded sequence in H. Let $u\in H$ be fixed. Let $\{{\rho }_n\}\subseteq (0,\frac{2}{{\parallel A\parallel }^2+1})$. Let Ω be the solution set of (SFP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$x_{n+1}:=a_{n}u+b_n{x}_n+c_n{P}_C[x_n-{\rho }_n{A}^{\ast }(I-P_Q)A{x}_n]+v_n$$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_n{\rho }_n>0$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{b}_n{c}_n>0$. 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 $\{{\beta }_n\}$ be a sequence in $(0,\mathrm{\infty })$, $\{a_n\}\subseteq (0,1)$, and $\{{\rho }_n\}\subseteq (0,2/({\parallel A\parallel }^2+2))$. Let Ω be the solution set of (SFP) and suppose that $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by

$$x_{n+1}:=P_C[(1-a_n{\rho }_n)x_n-{\rho }_n{A}^{\ast }(I-P_Q)A{x}_n]$$

for each $n\in \mathbb{N}$. Assume that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n{\rho }_n=\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\rho }_n>0$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n>0$. 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 [[13], Theorem 5.5].