Iterative algorithms for solving the split feasibility problem in Hilbert spaces

Abstract

In this paper, we introduce two iterative algorithms for the split feasibility problem in real Hilbert spaces by reformulating it as a fixed point equation. Under suitable conditions, weak and strong convergence theorems are established. As a consequence, we obtain weak and strong convergence iterative sequences for the split equality problem introduced by Moudafi. The efficiency of the proposed algorithms is illustrated by numerical experiments. Our results improve and extend the corresponding results announced by many others.

1 Introduction

The split feasibility problem (SFP) in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction [2], with particular progress in the intensity-modulated radiation therapy [3,4,5].

In this paper, we work in the framework of infinite-dimensional Hilbert spaces. Let C and Q be nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. The SFP can mathematically be formulated as the problem of finding a point $x^*\in C$ with the property

$$\begin{aligned} Ax^*\in Q, \end{aligned}$$

(1.1)

where $A:H_1\rightarrow H_2$ is a bounded linear operator.

Throughout this paper, we use $\Gamma$ to denote the solution set of the SFP (1.1), i.e.,

$$\begin{aligned} \Gamma = \{x\in C: Ax \in Q\}, \end{aligned}$$

and assume the consistency of (1.1) so that $\Gamma$ is closed, convex and nonempty.

Note that the SFP (1.1) can be formulated as a fixed point equation by using the fact

$$\begin{aligned} P_C(I-\gamma A^*(I-P_Q)A)x^*= x^*, \end{aligned}$$

(1.2)

where $P_C$ and $P_Q$ are the (orthogonal) projections onto C and Q, respectively, $\gamma >0$ is any positive constant and $A^*$ denotes the adjoint of A. That is, $x^*$ solves the SFP (1.1) if and only if $x^*$ solves the fixed point equation (1.2) (see [6] for the details). This implies that we can use fixed point algorithms (see [7,8,9,10,11,12,13,14,15]) to solve SFP. Byrne [2, 16] proposed the so-called CQ algorithm which generates a sequence $\{x_k\}$ by

$$\begin{aligned} x_{k+1}=P_C(x_k-\gamma A^*(I-P_Q)Ax_k), \end{aligned}$$

(1.3)

where $\gamma \in (0,2/ \lambda )$ with $\lambda$ being the spectral radius of the operator $A^*A$. The CQ algorithm only involves the computations of the projections $P_C$ and $P_Q$ onto the sets C and Q, respectively, and is therefore implementable in the case where $P_C$ and $P_Q$ have closed-form expressions.

We can reformulate the SFP (1.1) as an optimization problem. Indeed, $x\in \Gamma$ means that there is an $x\in C$ such that $Ax-q =0$ for some $q\in Q$. This motivates us to introduce the (convex) objective function

$$\begin{aligned} f(x):=\frac{1}{2}\Vert (I-P_Q)Ax\Vert ^2, \end{aligned}$$

(1.4)

and consider the convex minimization problem

$$\begin{aligned} \min _{x\in C}f(x). \end{aligned}$$

(1.5)

The objective function f is continuously differentiable with gradient given by

$$\begin{aligned} \nabla f(x)=A^*(I-P_Q)Ax. \end{aligned}$$

(1.6)

(Here $A^*$ is the adjoint of A.) Due to the fact that $I-P_Q$ is (firmly) nonexpansive, we obtain that $\nabla f$ is Lipschitz continuous with Lipschitz constant $L=\Vert A\Vert ^2$. It is well known that the gradient-projection algorithm (GPA) is one of the powerful methods for solving constrained optimization problems. Recall that the GPA, for the minimization problem (1.5), generates a sequence $\{x_k\}$ via the recursion:

$$\begin{aligned} x_{k+1}=P_C(x_k-\gamma \nabla f(x_k)), \end{aligned}$$

(1.7)

where $\gamma$ is chosen in the interval (0, 2 / L) with L being the Lipschitz constant of $\nabla f$. For solving (1.5), the GPM with gradient $\nabla f$ given as in (1.6) is the CQ algorithm (1.3).

By (1.5), we can see the SFP (1.1) can be written as the following convex separable minimization problem

$$\begin{aligned} \min _{x\in H_1} \iota _C(x)+f(x), \end{aligned}$$

(1.8)

where f(x) is defined by (1.4) and $\iota _C(x)$ is an indicator function of the set C defined by

$$\begin{aligned} \iota _C(x)= {\left\{ \begin{array}{ll} 1, &{} x\in C,\\ +\infty , &{} x\not \in C. \end{array}\right. } \end{aligned}$$

Chen et al. [17] designed and discussed an efficient algorithm for minimizing the sum of two proper lower semi-continuous convex functions, i.e.,

$$\begin{aligned} \min _{x\in R^n}( f_1 \circ B)(x) + f_2(x), \end{aligned}$$

(1.9)

where $f_1\in \Gamma _0(R^m)$, $f_2\in \Gamma _0(R^n)$, $f_2$ is differentiable on $R^n$ with $1/\beta$-Lipschitz continuous gradient for some $\beta \in (0,+\infty )$ and $B: R^n\rightarrow R^m$ a linear transform. When $B = I$ and $m=n$, the problem (1.9) becomes the following problem often considered in the literature

$$\begin{aligned} \min _{x\in R^n} f_1(x) + f_2(x). \end{aligned}$$

(1.10)

For any two positive numbers $\lambda$ and $\gamma$, define $T: R^m \times R^n\rightarrow R^m$ as

$$\begin{aligned} T(v, x) = (I-\mathrm{prox}_{\frac{\gamma }{\lambda }f_1})(B(x -\gamma \nabla f_2(x)) + (I-\lambda BB^T)v) \end{aligned}$$

(1.11)

and $S : R^m \times R^n\rightarrow R^n$ as

$$\begin{aligned} S(v, x) = x-\gamma \nabla f_2(x)-\lambda B^T\circ T(v,x). \end{aligned}$$

(1.12)

Denote $G: R^m\times R^n\rightarrow R^m\times R^n$ as

$$\begin{aligned} G(v, x) = (T(v, x), S(v, x)). \end{aligned}$$

(1.13)

The authors in [17] obtained the following fixed point formulation for the solution of the convex separable problem (1.9).

Theorem 1.1

[17] (Theorem 3.1) Let $\lambda$ and $\gamma$ be two positive numbers. Suppose that $x^*$ is a solution of (1.9). Then there exists $v^*\in R^m$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} v^*= T(v^*, x^*),\\ x^*= S(v^*, x^*).\end{array}\right. } \end{aligned}$$

In other words, $u^*= (v^*, x^*)$ is a fixed point of G. Conversely, if $u^*\in R^m \times R^n$ is a fixed point of G, with $u^*= (v^*, x^*)$, $v^*\in R^m$, $x^*\in R^n$, then $x^*$ is a solution of (1.9).

They proposed the following Picard sequence $\{u_k\}$ of G

$$\begin{aligned} {\left\{ \begin{array}{ll} v_{k+1}= T(v_k, x_k ) = (I-\mathrm{prox}_{\frac{\gamma }{\lambda }f_1})(B(x_k-\gamma \nabla f_2(x_k )) + (I-\lambda BB^\mathrm{T} )v_k ),\\ x_{k+1} = S(v_k, x_k ) = x_k -\gamma \nabla f_2(x_k )-\lambda B^\mathrm{T}\circ T(v_k,x_k) \\ \qquad \qquad \qquad \quad \,\,\,\, = x_k-\gamma \nabla f_2(x_k )-\lambda B^\mathrm{T}(v_{k+1}). \end{array}\right. } \end{aligned}$$

(1.14)

It was showed [17] that under appropriate conditions $\{u_k\}$ converges to a fixed point of G and $\{x_k\}$ converges to a solution of problem (1.9). Since x is the primal variable related to (1.9), it is very natural to ask what role the variable v plays in above algorithm. In [17] they found out that v is actually the dual variable of the primal-dual form related to (1.9).

Motivated the above works, we construct two algorithms for the SFP (1.1). Note that the step-size $\gamma$ in (1.14) is fixed, the aim of this paper is twofold: first to propose a new algorithm with variable step-sizes in infinite-dimensional Hilbert space; second to modify the proposed algorithm so that it has strongly convergence result. As a consequence, we obtain weak and strong convergence sequences for the split equality problem introduced by Moudafi [18]. Finally, some numerical results are presented to show the efficiency of the proposed algorithms.

2 Preliminaries

Throughout this paper, we always assume that H is a real Hilbert space with inner product $\langle \cdot ,\cdot \rangle$ and norm $\Vert \cdot \Vert$. Let I denote the identity operator on H. Let $U:H\rightarrow H$ be a mapping. A point $x\in H$ is said to be a fixed point of U provided $Ux=x$. In this paper, we use F(U) to denote the fixed point set and use $\rightarrow$ to denote the strong convergence. Here and in what follows, for a real Hilbert space H, $\Gamma _0(H)$ denotes the collection of all proper lower semi-continuous convex functions from H to $(-\infty ,+\infty ]$.

Definition 2.1

An operator $U:H\rightarrow H$ is said to be

(i) nonexpansive if

$$\begin{aligned} \Vert Ux-Uy\Vert \le \Vert x-y\Vert ,\ \ \forall x,y\in H; \end{aligned}$$

(ii) firmly nonexpansive if

$$\begin{aligned} \Vert Ux-Uy\Vert ^2\le \Vert x-y\Vert ^2-\Vert (x-y)-(Ux-Uy)\Vert ^2,\ \ \forall x,y\in H. \end{aligned}$$

It is easily to obtain that U is firmly nonexpansive if and only if

$$\begin{aligned} \Vert Ux-Uy\Vert ^2\le \langle Ux-Uy,x-y\rangle ,\ \ \forall \ x,\ y\in H. \end{aligned}$$

More information on firmly nonexpansive mappings can be found on pages 42–44 of the book [19].

Definition 2.2

An operator $U:H\rightarrow H$ is called demi-closed at the origin if, for any sequence $\{x_n\}$ which weakly converges to x, and if the sequence $\{U(x_n)\}$ strongly converges to 0, then $Ux=0$.

Definition 2.3

An operator $h:H\rightarrow H$ is said to be

(i) L-Lipschitzian if there exists a positive constant L such that

$$\begin{aligned} \Vert h(x)-h(y)\Vert \le L\Vert x-y\Vert ,\ \ \forall x,y\in H; \end{aligned}$$

(ii) $\rho$-strongly pseudo-contraction if there exists a constant $\rho \in [0,1)$ such that

$$\begin{aligned} \langle h(x)-h(y), x-y\rangle \le \rho \Vert x-y\Vert ^2,\ \ \forall x,y\in H; \end{aligned}$$

(iii) $\rho$-contraction if there exists a constant $\rho \in [0,1)$ such that

$$\begin{aligned} \Vert h(x)-h(y)\Vert \le \rho \Vert x-y\Vert ,\ \ \forall x,y\in H. \end{aligned}$$

Definition 2.4

An operator $h:C\rightarrow H$ is said to be $\eta$-strongly monotone, if there exists a positive constant $\eta$ such that

$$\begin{aligned} \langle h(x)-h(y),x-y\rangle \ge \eta \Vert x-y\Vert ^2,\ \ \forall x,y\in C. \end{aligned}$$

It is obvious that if h is a $\rho$-strongly pseudo-contraction, then $I-h$ is a $(1-\rho )$-strongly monotone mapping. Recall the variational inequality problem [20] is to find a point $x^*\in C$ such that

$$\begin{aligned} \langle Fx^*, x-x^*\rangle \ge 0, \ \ \ \forall x\in C, \end{aligned}$$

where $F:C\rightarrow H$ is a nonlinear operator. It is well known that [21] if $F:C\rightarrow H$ is a Lipschitzian and strongly monotone operator, then the above variational inequality problem has a unique solution.

Recall that the metric (nearest point) projection from H onto a nonempty closed convex subset C of H, denoted by $P_C$, is defined as follows: for each $x\in H$,

$$\begin{aligned} P_{C}(x)=\mathrm{arg}\min _{y\in C} \{\Vert x-y\Vert \}. \end{aligned}$$

(2.1)

For $f\in \Gamma _0(H)$ and $\rho \in (0,+\infty )$, the proximal operator of f of order $\rho$, denoted by prox$_{\rho f}$, is defined by, for each $x\in H$,

$$\begin{aligned} \mathrm{prox}_{\rho f}(x)=\mathrm{arg}\min _{y\in H} \left\{ f(y)+\frac{1}{2\rho }\Vert x-y\Vert ^2\right\} . \end{aligned}$$

(2.2)

It is well known that

$$\begin{aligned} \mathrm{prox}_{\rho f} = P_C \end{aligned}$$

(2.3)

for $f =\iota _C$ and $\rho >0$. Conversely, if prox$_{\rho f} = P_C$, we can choose $f=\iota _C$. Proximity operators are therefore generalization of projection operators.

Lemma 2.1

(Lemma 2.4 of [22]). Let f be a function in $\Gamma _0(H)$. Then prox$_f$ and $I-\mathrm{prox}_f$ are both firmly nonexpansive operators.

Lemma 2.2

[23, 24] Let E be a uniformly convex Banach space, K be a nonempty closed convex subset of E and $U:K\rightarrow K$ be a nonexpansive operator. Then $I-U$ is demi-closed at origin.

Lemma 2.3

Let K be a nonempty closed convex subset of a Hilbert space H. Let $\{x_k\}$ be a bounded sequence which satisfies the following properties:

• (i) every weak limit point of $\{x_k\}$ lies in K;

• (ii) $\lim _{k\rightarrow \infty }\Vert x_k-x\Vert$ exists for every $x\in K$.

Then $\{x_k\}$ converges weakly to a point in K.

Now, taking $f_1(x)=\iota _C(x)$, $f_2(x)=\frac{1}{2}\Vert (I-P_Q)Ax\Vert ^2$ and $B=I$, it follows from (1.6) and (2.3) that

$$\begin{aligned} \mathrm{prox}_{\frac{\gamma }{\lambda }f_1}=P_C,\ \ \nabla f_2(x)=A^*(I-P_Q)Ax, \end{aligned}$$

for any two positive numbers $\lambda$ and $\gamma$. We can obtain a fixed point formulation for the solution of the SFP (1.1) from Theorem 1.1.

For any two positive numbers $\lambda$ and $\gamma$, define $g:H_1\rightarrow H_1$ as

$$\begin{aligned} g(x)=x-\gamma A^*(I-P_Q)Ax, \end{aligned}$$

(2.4)

$T : H_1 \times H_1\rightarrow H_1$ as

$$\begin{aligned} T(v, x) = (I-P_C)(g(x)+ (1-\lambda )v) \end{aligned}$$

(2.5)

and $S : H_1 \times H_1\rightarrow H_1$ as

$$\begin{aligned} S(v, x) = g(x)-\lambda T(v,x). \end{aligned}$$

(2.6)

Denote $G : H_1\times H_1\rightarrow H_1\times H_1$ as

$$\begin{aligned} G(v, x) = (T(v, x), S(v, x)). \end{aligned}$$

(2.7)

Lemma 2.4

(Theorem 3.1 of [17]) Let $\lambda$ and $\gamma$ be two positive numbers. Suppose that $x^*$ is a solution of the SFP (1.1). Then there exists $v^*\in H_1$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} v^*= T(v^*, x^*),\\ x^*= S(v^*, x^*).\end{array}\right. } \end{aligned}$$

In other words, $u^*= (v^*, x^*)$ is a fixed point of G. Conversely, if $u^*\in H_1 \times H_1$ is a fixed point of G with $u^*= (v^*, x^*)$, then $x^*$ is a solution of the SFP (1.1).

Remark 2.1

For any two positive numbers $\lambda$ and $\gamma$, G is defined by (2.7). Then

$$\begin{aligned} F(G)=\{(0,x^*):x^*\in \Gamma \}=\{(0,x^*):x^*\in C,\ \ Ax^*\in Q\}. \end{aligned}$$

So we can denote $\Omega :=F(G)$ for any two positive numbers $\lambda$ and $\gamma$.

Indeed, for any two positive numbers $\lambda$ and $\gamma$, let $u^*=(v^*, x^*)\in H_1\times H_1$ be a fixed point of G defined in (2.7). We have $v^*=T(v^*,x^*)$ and $x^*=S(v^*, x^*)$. Using Lemma 2.4, we have $x^*$ is a solution of (1.1) and hence $x^*\in C$, $Ax^*\in Q$. It follows that $g(x^*)=x^*$ and $x^*=S(v^*, x^*)=x^*-\lambda v^*$. It follows that $v^*=0$, which implies that $F(G)\subseteq \{(0,x^*):x^*\in \Gamma \}$. Conversely, we easily obtain $\{(0,x^*):x^*\in \Gamma \}\subseteq F(G)$. So, $F(G)=\{(0,x^*):x^*\in \Gamma \}$ for any two positive numbers $\lambda$ and $\gamma$. $\square$

Lemma 2.5

[25] Assume $\{s_k\}$ is a sequence of nonnegative real numbers such that

$$\begin{aligned} {\left\{ \begin{array}{ll} s_{k+1}\le (1-\lambda _k)s_k+\lambda _k\delta _k,\ \ k\ge 0,\\ s_{k+1}\le s_k-\eta _k +\mu _k,\ \ k\ge 0, \end{array}\right. } \end{aligned}$$

where $\{\lambda _k\}$ is a sequence in (0, 1), $\{\eta _k\}$ is a sequence of nonnegative real numbers and $\{\delta _k\}$ and $\{\mu _k\}$ are two sequences in $\mathbb {R}$ such that

• (i) $\Sigma _{k=1}^\infty \lambda _k=\infty$,

• (ii) $\lim _{k\rightarrow \infty }\mu _k = 0$,

• (iii) $\lim _{l\rightarrow \infty }\eta _{k_l}= 0$ implies $\limsup _{l\rightarrow \infty }\delta _{k_l}\le 0$ for any subsequence $\{k_l\}\subset \{k\}$.

Then $\lim _{k\rightarrow \infty }s_k= 0$.

3 Weak convergence for SFP (1.1)

Now we can use fixed point equations to introduce a new iterative algorithm for the SFP (1.1) with variable step-sizes in infinite-dimensional Hilbert space.

Algorithm 3.1

Initialization: Let $x_0,v_0\in H_1$ be arbitrary.

Iterative step: For $k\in N$, let

$$\begin{aligned} {\left\{ \begin{array}{ll} x_{k+\frac{1}{2}}=x_k-\gamma _k A^*(I-P_Q)Ax_k,\\ v_{k+1}=(I-P_C)(x_{k+\frac{1}{2}}+ (1-\lambda )v_k),\\ x_{k+1}=x_{k+\frac{1}{2}}-\lambda v_{k+1}, \end{array}\right. } \end{aligned}$$

(3.1)

where $\{\gamma _k\}\subset (0,\frac{2}{\Vert A\Vert ^2})$, $\lambda \in (0,1]$.

For the convergence analysis of Algorithm 3.1, we will first prove a key inequality for general cases (cf Theorem 3.1). Denote

$$\begin{aligned} {\left\{ \begin{array}{ll} g_k(x)=x-\gamma _k A^*(I-P_Q)Ax,\\ T_k(v, x) = (I-P_C)(g_k(x)+ (1-\lambda )v), \\ S_k(v, x) = g_k(x)-\lambda T_k(v,x),\\ G_k(v, x) = (T_k(v, x), S_k(v, x)), \end{array}\right. } \end{aligned}$$

(3.2)

where v, $x\in H_1$. For an element $u = (v, x)\in H_1\times H_1$, let

$$\begin{aligned} \Vert u\Vert _\lambda =(\Vert x\Vert ^2+\lambda \Vert v\Vert ^2)^{\frac{1}{2}}. \end{aligned}$$

(3.3)

We can easily see that $\Vert \cdot \Vert _\lambda$ is a norm over the produce space $H_1\times H_1$ whenever $\lambda > 0$.

Theorem 3.1

For any two elements $u_1=(v_1,x_1)$, $u_2=(v_2,x_2)$ in $H_1\times H_1$, there holds

$$\begin{aligned}&\Vert G_k(u_1)-G_k(u_2)\Vert _\lambda ^2\le \Vert u_1-u_2\Vert _\lambda ^2\nonumber \\&\quad -\gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)Ax_1-(I-P_Q)Ax_2\Vert ^2\nonumber \\&\quad -\Vert \lambda (v_1-v_2)\Vert ^2-\lambda (1-\lambda )\Vert (T_ku_1-T_ku_2)-(v_1-v_2)\Vert ^2. \end{aligned}$$

(3.4)

Proof

By Lemma 2.1, $I-P_C$ is a firmly nonexpansive operator. This together with (3.2) yields

$$\begin{aligned} \Vert T_k(u_1)-T_k(u_2)\Vert ^2\le & {} \langle T_k(u_1)-T_k(u_2), g_k(x_1)-g_k(x_2)\nonumber \\&\quad +\, (1-\lambda )(v_1-v_2)\rangle . \end{aligned}$$

(3.5)

It follows from (3.2) that

$$\begin{aligned}&\Vert S_k(u_1)-S_k(u_2)\Vert ^2\nonumber \\&\quad =\Vert g_k(x_1)-g_k(x_2)-\lambda (T_k(u_1)-T_k(u_2))\Vert ^2\nonumber \\&\quad =\Vert g_k(x_1)-g_k(x_2)\Vert ^2-2\lambda \langle g_k(x_1)-g_k(x_2),T_k(u_1)\nonumber \\&\qquad -T_k(u_2)\rangle +\lambda ^2\Vert T_k(u_1)-T_k(u_2)\Vert ^2. \end{aligned}$$

(3.6)

Observing the definitions in (3.2) and (3.3), by (3.5) and (3.6), we have

$$\begin{aligned}&\Vert G_k(u_1)-G_k(u_2)\Vert _\lambda ^2\nonumber \\&\quad =\Vert S_k(u_1)-S_k(u_2)\Vert ^2+\lambda \Vert T_k(u_1)-T_k(u_2)\Vert ^2\nonumber \\&\quad =\Vert g_k(x_1)-g_k(x_2)\Vert ^2-2\lambda \langle g_k(x_1)-g_k(x_2),T_k(u_1)-T_k(u_2)\rangle \nonumber \\&\qquad +(\lambda +\lambda ^2)\Vert T_k(u_1)-T_k(u_2)\Vert ^2\nonumber \\&\quad =\Vert g_k(x_1)-g_k(x_2)\Vert ^2-2\lambda \langle g_k(x_1)-g_k(x_2),T_k(u_1)-T_k(u_2)\rangle \nonumber \\&\qquad +2\lambda \Vert T_k(u_1)-T_k(u_2)\Vert ^2 -\lambda (1-\lambda )\Vert T_k(u_1)-T_k(u_2)\Vert ^2\nonumber \\&\quad \le \Vert g_k(x_1)-g_k(x_2)\Vert ^2+2\lambda (1-\lambda )\langle T_k(u_1)-T_k(u_2),v_1-v_2\rangle \nonumber \\&\qquad -\lambda (1-\lambda )\Vert T_k(u_1)-T_k(u_2)\Vert ^2. \end{aligned}$$

(3.7)

Since

$$\begin{aligned}&\lambda (1-\lambda )\Vert T_k(u_1)-T_k(u_2)-(v_1-v_2)\Vert ^2\nonumber \\&\quad =\lambda (1-\lambda )(\Vert T_k(u_1)-T_k(u_2)\Vert ^2-2\langle T_k(u_1)\nonumber \\&\qquad -T_k(u_2),v_1-v_2\rangle +\Vert v_1-v_2\Vert ^2), \end{aligned}$$

(3.8)

we have

$$\begin{aligned}&\Vert G_k(u_1)-G_k(u_2)\Vert _\lambda ^2\nonumber \\&\quad \le \Vert g_k(x_1)-g_k(x_2)\Vert ^2+\lambda (1-\lambda )\Vert v_1-v_2\Vert ^2\nonumber \\&\qquad -\lambda (1-\lambda )\Vert T_k(u_1)-T_k(u_2)-(v_1-v_2)\Vert ^2. \end{aligned}$$

(3.9)

It follows from Lemma 2.1 and the definition of firmly nonexpansive operators that

$$\begin{aligned}&\langle x_1-x_2, A^*(I-P_Q)Ax_1-A^*(I-P_Q)Ax_2\rangle \nonumber \\&\quad =\langle Ax_1-Ax_2, (I-P_Q)Ax_1-(I-P_Q)Ax_2\rangle \nonumber \\&\quad \ge \Vert (I-P_Q)Ax_1-(I-P_Q)Ax_2\Vert ^2, \end{aligned}$$

(3.10)

which implies that

$$\begin{aligned}&\Vert g_k(x_1)-g_k(x_2)\Vert ^2\nonumber \\&\quad =\Vert x_1-x_2-\gamma _k(A^*(I-P_Q)Ax_1-A^*(I-P_Q)Ax_2)\Vert ^2\nonumber \\&\quad =\Vert x_1-x_2\Vert ^2-2\gamma _k\langle x_1-x_2, A^*(I-P_Q)Ax_1-A^*(I-P_Q)Ax_2\rangle \nonumber \\&\qquad +\gamma _k^2\Vert A^*(I-P_Q)Ax_1-A^*(I-P_Q)Ax_2\Vert ^2\nonumber \\&\quad \le \Vert x_1-x_2\Vert ^2-2\gamma _k\Vert (I-P_Q)Ax_1-(I-P_Q)Ax_2\Vert ^2\nonumber \\&\qquad +\gamma _k^2\Vert A^*\Vert ^2\Vert (I-P_Q)Ax_1-(I-P_Q)Ax_2\Vert ^2\nonumber \\&\quad =\Vert x_1-x_2\Vert ^2-\gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)Ax_1-(I-P_Q)Ax_2\Vert ^2. \end{aligned}$$

(3.11)

Recalling the definition in (3.3), we easily see that (3.4) is a direct consequence of (3.9) and (3.11). $\square$

Remark 3.1

If $0<\gamma <\frac{2}{\Vert A\Vert ^2}$, $0 <\lambda \le 1$, then G is nonexpansive defined in (2.7) under the norm $\Vert \cdot \Vert _\lambda$.

Theorem 3.2

Suppose $0<\liminf _{k\rightarrow \infty }\gamma _k \le \limsup _{k\rightarrow \infty }\gamma _k<\frac{2}{\Vert A\Vert ^2}$, $0 <\lambda \le 1$. Let $u_k = (v_k, x_k )$ be the sequence generated by Algorithm 3.1. Then the sequence $\{u_k\}$ weakly converges to a point in $\Omega$ (i.e. F(G)) under the norm $\Vert \cdot \Vert _\lambda$ and the sequence $\{x_k\}$ weakly converges to a solution of the SFP (1.1).

Proof

From Algorithm 3.1, we have $u_{k+1}=G_k(u_k)$. Let $u^*\in \Omega$, we have $u^*=(v^*, x^*)\in H_1\times H_1$ be a fixed point of G defined in (2.7) for any two positive numbers $\lambda$ and $\gamma$. So, $x^*\in \Gamma$, $v^*=0$ and $u^*=(v^*, x^*)$ is a fixed point of $G_k$ defined in (3.2) for any $k\ge 1$. Using Theorem 3.1, $x^*\in C$, $Ax^*\in Q$ and $v^*=0$, we have

$$\begin{aligned}&\Vert u_{k+1}-u^*\Vert _\lambda ^2\nonumber \\&\quad =\Vert G_k(u_k)-G_k(u^*)\Vert _\lambda ^2\nonumber \\&\quad \le \Vert u_k-u^*\Vert _\lambda ^2 -\gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)Ax_k-(I-P_Q)Ax^*\Vert ^2\nonumber \\&\qquad -\Vert \lambda (v_k-v^*)\Vert ^2-\lambda (1-\lambda )\Vert (T_ku_k-T_ku^*)-(v_k-v^*)\Vert ^2\nonumber \\&\quad =\Vert u_k-u^*\Vert _\lambda ^2-\gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)Ax_k\Vert ^2-\lambda ^2\Vert v_k\Vert ^2\nonumber \\&\qquad -\lambda (1-\lambda )\Vert v_{k+1}-v_k\Vert ^2. \end{aligned}$$

(3.12)

By $0<\liminf _{k\rightarrow \infty }\gamma _k \le \limsup _{k\rightarrow \infty }\gamma _k<\frac{2}{\Vert A\Vert ^2}$ and $0 <\lambda \le 1$, we get the sequence $\{\Vert u_{k}-u^*\Vert _\lambda \}$ is non-increasing and lower bounded by 0. Consequently, the sequence $\{\Vert u_{k}-u^*\Vert _\lambda \}$ converges to some finite limit and the sequence $\{\Vert u_k\Vert _\lambda \}$ is bounded. So the sequence $\{x_k\}$ is bounded and

$$\begin{aligned}&\lim _{k\rightarrow \infty }\left\{ \gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)Ax_k\Vert ^2+\lambda ^2\Vert v_k\Vert ^2\right. \nonumber \\&\quad \left. +\lambda (1-\lambda )\Vert v_{k+1}-v_k\Vert ^2\right\} =0, \end{aligned}$$

(3.13)

which together with $0<\liminf _{k\rightarrow \infty }\gamma _k \le \limsup _{k\rightarrow \infty }\gamma _k<\frac{2}{\Vert A\Vert ^2}$ and $0 <\lambda \le 1$ implies

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert (I-P_Q)Ax_k\Vert =0 \end{aligned}$$

(3.14)

and

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert v_k\Vert =0. \end{aligned}$$

(3.15)

So, we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert v_{k+1}-v_k\Vert =0. \end{aligned}$$

(3.16)

From (3.1), we have

$$\begin{aligned} x_{k+1}-x_k=-\gamma _kA^*(I-P_Q)Ax_k-\lambda v_{k+1}. \end{aligned}$$

Now, using (3.14) and (3.15) we immediately obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert x_{k+1}-x_k\Vert =0. \end{aligned}$$

(3.17)

By the definition in (3.3) and (3.16)–(3.17), we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert G_ku_k-u_k\Vert _\lambda ^2=\lim _{k\rightarrow \infty }\Vert u_{k+1}-u_k\Vert _\lambda ^2=0. \end{aligned}$$

(3.18)

Since $\{\Vert u_k\Vert _\lambda \}$ is bounded, we see that the set of weak limit points of $\{u_k\}$ under the norm $\Vert \cdot \Vert _\lambda$, $\omega _w(u_k)$, is nonempty. Next, we prove that $\omega _w(u_k)\subseteq \Omega$. Assume that $u^\circ \in \omega _w(u_k)$ and $\{u_{k_j}\}$ is a subsequence of $\{u_{k}\}$ which converges weakly to $u^\circ$ under the norm $\Vert \cdot \Vert _\lambda$. It follows from the boundedness of real sequence $\{\gamma _{k}\}$ that $\{\gamma _{k_j}\}$ is a bounded subsequence, which has convergent subsequence. Without loss of generality, we assume that $\gamma _{k_j}\rightarrow \gamma ^\circ$ as $j\rightarrow \infty$. Let g, T, S and G be defined in (2.4)–(2.7) for $\gamma =\gamma ^\circ$, we have

$$\begin{aligned}&\Vert G_{k_j}u_{k_j}-Gu_{k_j}\Vert _\lambda ^2\nonumber \\&\quad =\Vert g_{k_j}(x_{k_j})-\lambda T_{k_j}(u_{k_j})-g(x_{k_j})-\lambda T(u_{k_j})\Vert ^2+\lambda \Vert T_{k_j}(u_{k_j})-T(u_{k_j})\Vert ^2\nonumber \\&\quad \le 2\Vert g_{k_j}(x_{k_j})-g(x_{k_j})\Vert ^2+2\lambda ^2\Vert T_{k_j}(u_{k_j})-T(u_{k_j})\Vert ^2+\lambda \Vert T_{k_j}(u_{k_j})-T(u_{k_j})\Vert ^2\nonumber \\&\quad =2\Vert g_{k_j}(x_{k_j})-g(x_{k_j})\Vert ^2\nonumber \\&\qquad +(2\lambda ^2+\lambda )\Vert (I-P_C)(g_{k_j}(x_{k_j})+(1-\lambda )v_{k_j})\nonumber \\&\qquad -(I-P_C)(g(x_{k_j})+(1-\lambda )v_{k_j})\Vert ^2\nonumber \\&\quad \le 2\Vert g_{k_j}(x_{k_j})-g(x_{k_j})\Vert ^2+(2\lambda ^2+\lambda )\Vert g_{k_j}(x_{k_j})-g(x_{k_j})\Vert ^2\nonumber \\&\quad =(2+2\lambda ^2+\lambda )\Vert x_{k_j}-\gamma _{k_j} A^*(I-P_Q)Ax_{k_j}-x_{k_j}+\gamma ^\circ A^*(I-P_Q)Ax_{k_j}\Vert ^2\nonumber \\&\quad \le (2+2\lambda ^2+\lambda )|\gamma _{k_j}-\gamma ^\circ |\cdot \Vert A^*(I-P_Q)Ax_{k_j}\Vert ^2. \end{aligned}$$

(3.19)

By the boundedness of $\{x_k\}$ and $\gamma _{k_j}\rightarrow \gamma ^\circ$ as $j\rightarrow \infty$, we get

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert G_{k_j}u_{k_j}-Gu_{k_j}\Vert _\lambda =0. \end{aligned}$$

(3.20)

Since

$$\begin{aligned} \Vert u_{k_j}-Gu_{k_j}\Vert _\lambda \le \Vert u_{k_j}-G_{k_j}u_{k_j}\Vert _\lambda +\Vert G_{k_j}u_{k_j}-Gu_{k_j}\Vert _\lambda , \end{aligned}$$

combining with (3.18) and (3.20), we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert u_{k_j}-Gu_{k_j}\Vert _\lambda =0. \end{aligned}$$

(3.21)

By Lemma 2.2, (3.21) implies that $u^\circ \in F(G)$ with $\gamma =\gamma ^\circ$. So, $u^\circ \in \Omega$ and $\omega _w(u_k)\subseteq \Omega$. Lemma 2.3 ensures that $\{u_k\}$ weakly converges to a point in $\Omega$ under the norm $\Vert \cdot \Vert _\lambda$. So, the sequence $\{x_k\}$ weakly converges to a solution of problem (1.1). $\square$

Remark 3.2

Algorithm 3.1 generalizes CQ algorithm (1.3) for solving the SFP (1.1). Indeed, if we take $\lambda =1$ in Algorithm 3.1, then we can obtain (1.3) with $\gamma _k\equiv \gamma$ under the same conditions.

4 Strong convergence for SFP (1.1)

As we see from the above section, the sequence generated by Algorithm 3.1 has only weak convergence. So we aim to modify the proposed algorithm so that it has strongly convergence. It is known that the viscosity approximation method [26] is often used to approximate a fixed point of a nonexpansive mapping U in a Hilbert space with strong convergence and it is defined by

$$\begin{aligned} x_{k+1}=\alpha _k f(x_k)+(1-\alpha _k)U(x_k),\ \ k\ge 1, \end{aligned}$$

(4.1)

where $\{\alpha _n\}\subseteq [0,1]$ and f is a contractive operator.

In [27], Yang and He proposed a general alternative regularization method for finding fixed point of firmly nonexpansive operator U

$$\begin{aligned} x_{k+1}=U(\alpha _k f(x_k)+(1-\alpha _k)x_k),\ \ k\ge 1, \end{aligned}$$

(4.2)

where $\{\alpha _k\}\subseteq [0,1]$ and f is a Lipschitzian strong pseudo-contraction operator. They obtained the strong convergence of algorithm (4.2) and gave a simple example to show that if f in algorithm (4.1) is replaced with a Lipschitzian and strongly pseudo-contractive mapping, then strong convergence (even boundedness) of the iteration sequence $\{x_k\}$ may not be guaranteed, in general. We now adapt (4.2) to modify Algorithm 3.1.

Algorithm 4.1

Let $h:H_1\rightarrow H_1$ be an L-Lipschitzian and $\theta$-strongly pseudo-contractive mapping. Initialization: Let $x_0,v_0\in H_1$ be arbitrary.

Iterative step: For $k\in N$, let

$$\begin{aligned} {\left\{ \begin{array}{ll} \overline{v}_{k+1}=\alpha _kh(v_k)+(1-\alpha _k)v_k, \\ \overline{x}_{k+1}=\alpha _kh(x_k)+(1-\alpha _k)x_k,\\ x_{k+\frac{1}{2}}=g_k(\overline{x}_{k+1})=\overline{x}_{k+1}-\gamma _kA^*(I-P_Q)A\overline{x}_{k+1},\\ v_{k+1}=T_k(\overline{v}_{k+1},\overline{x}_{k+1})=(I-P_C)(x_{k+\frac{1}{2}}+(1-\lambda )\overline{v}_{k+1}),\\ x_{k+1}=S_k(\overline{v}_{k+1},\overline{x}_{k+1})=x_{k+\frac{1}{2}}-\lambda v_{k+1}, \end{array}\right. } \end{aligned}$$

(4.3)

where $\{\alpha _k\}\subset (0,1),$ $\{\gamma _k\}\subset (0,\frac{2}{\Vert A\Vert ^2})$, $\lambda \in (0,1]$.

Theorem 4.1

If the sequences $\{\alpha _k\}$, $\{\gamma _{k}\}$ and the constant $\lambda$ satisfy the following conditions:

$$\begin{aligned}&\mathrm{(i)}\quad \alpha _k\rightarrow 0 (k\rightarrow \infty );\\&\mathrm{(ii)}\quad \sum _{k=0}^\infty \alpha _k=\infty ;\\&\mathrm{(iii)}\quad 0<\liminf _{k\rightarrow \infty } \gamma _k\le \limsup _{k\rightarrow \infty }\gamma _k<\frac{2}{\Vert A\Vert ^2},\quad 0<\lambda <1, \end{aligned}$$

then the sequence $u_{k+1}=(v_{k+1},x_{k+1})$ generated by Algorithm 4.1 strongly converges to a point in $\Omega$ under the norm $\Vert \cdot \Vert _\lambda$ and the sequence $\{x_k\}$ strongly converges to a solution $x^*$ of the SFP (1.1). Moreover, $x^*$ also solves the following variational inequality problem: finding a point $\hat{x}\in \Gamma$ such that

$$\begin{aligned} \langle \hat{x}-h(\hat{x}),x-\hat{x} \rangle \ge 0,\quad \forall x \in \Gamma . \end{aligned}$$

(4.4)

Proof

Let $\overline{u}_{k+1}=(\overline{v}_{k+1},\overline{x}_{k+1})$. From Algorithm 4.1, we have $u_{k+1}=G_k(\overline{u}_{k+1})$. Since h is an L-Lipschitzian and $\theta$-strongly pseudo-contractive mapping, so the variational inequality (4.4) has a unique solution $x^*$. Letting $u^*=(0,x^*)$, we have $u^*\in \Omega$. Similar to the proof of Theorem 3.2, we obtain that

$$\begin{aligned}&\Vert u_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad =\Vert G_k(\overline{u}_{k+1})-G_k(u^*)\Vert ^2_\lambda \nonumber \\&\quad \le \Vert \overline{u}_{k+1}-u^*\Vert ^2_\lambda -\gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)A\overline{x}_{k+1}\Vert ^2 \nonumber \\&\qquad -\lambda ^2\Vert \overline{v}_{k+1}\Vert ^2-\lambda (1-\lambda )\Vert v_{k+1}-\overline{v}_{k+1}\Vert ^2. \end{aligned}$$

(4.5)

From (4.3), we have

$$\begin{aligned}&\Vert \overline{u}_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad =\Vert (\overline{v}_{k+1},\overline{x}_{k+1})-(0,x^*)\Vert ^2_\lambda \nonumber \\&\quad =\Vert \alpha _kh(x_k)+(1-\alpha _k)x_k-x^*\Vert ^2+\lambda \Vert \alpha _kh(v_k)+(1-\alpha _k)v_k\Vert ^2\nonumber \\&\quad ={\alpha _k}^2\Vert h(x_k)-x^*\Vert ^2+2\alpha _k(1-\alpha _k)<h(x_k)-x^*,x_k-x^*>+(1-\alpha _k)^2\Vert x_k-x^*\Vert ^2\nonumber \\&\qquad +\lambda [{\alpha _k}^2\Vert h(v_k)\Vert ^2+2\alpha _k(1-\alpha _k)<h(v_k),v_k>+(1-\alpha _k)^2\Vert v_k\Vert ^2]\nonumber \\&\quad \le 2{\alpha _k}^2[\Vert h(x_k)-h(x^*)\Vert ^2+\Vert h(x^*)-x^*\Vert ^2]+(1-\alpha _k)^2\Vert x_k-x^*\Vert ^2\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)[<h(x_k)-h(x^*),x_k-x^*>+<h(x^*)-x^*,x_k-x^*>]\nonumber \\&\qquad +\lambda [2{\alpha _k}^2(\Vert h(v_k)-h(0)\Vert ^2+\Vert h(0)\Vert ^2)+(1-\alpha _k)^2\Vert v_k\Vert ^2\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)(<h(v_k)-h(0),v_k>+<h(0),v_k>)]. \end{aligned}$$

(4.6)

Obviously, we have

$$\begin{aligned}&<h(x^*)-x^*,x_k-x^*> \le \Vert h(x^*)-x^*\Vert \cdot \Vert x_k-x^*\Vert \nonumber \\&\quad \le \beta \Vert x_k-x^*\Vert ^2+\frac{1}{4\beta }\Vert h(x^*)-x^*\Vert ^2 \end{aligned}$$

(4.7)

and

$$\begin{aligned}&<h(0),v_k>\le \Vert h(0)\Vert \cdot \Vert v_k\Vert \le \beta \Vert v_k\Vert ^2+\frac{1}{4\beta }\Vert h(0)\Vert ^2, \end{aligned}$$

(4.8)

where $\beta$ is a positive constant such that $\theta +\beta <1.$ Thus, we get

$$\begin{aligned}&\Vert \overline{u}_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad \le 2{\alpha _k}^2[L^2\Vert x_k-x^*\Vert ^2+\Vert h(x^*)-x^*\Vert ^2]+(1-\alpha _k)^2\Vert x_k-x^*\Vert ^2\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)[\theta \Vert x_k-x^*\Vert ^2+\beta \Vert x_k-x^*\Vert ^2+\frac{1}{4\beta }\Vert h(x^*)-x^*\Vert ^2]\nonumber \\&\qquad +\lambda [2{\alpha _k}^2(L^2\Vert v_k\Vert ^2+\Vert h(0)\Vert ^2)+2\alpha _k(1-\alpha _k)(\theta \Vert v_k\Vert ^2+\beta \Vert v_k\Vert ^2\nonumber \\&\qquad +\frac{1}{4\beta }\Vert h(0)\Vert ^2)+(1-\alpha _k)^2\Vert v_k\Vert ^2]\nonumber \\&\quad =[(1-\alpha _k)^2+2{\alpha _k}^2L^2+2\alpha _k(1-\alpha _k)(\theta +\beta )](\Vert x_k-x^*\Vert ^2+\lambda \Vert v_k\Vert ^2)\nonumber \\&\qquad +2{\alpha _k}^2\Vert h(x^*)-x^*\Vert ^2\nonumber \\&\qquad +\frac{2\alpha _k(1-\alpha _k)}{4\beta }\Vert h(x^*)-x^*\Vert ^2+\lambda [2{\alpha _k}^2\Vert h(0)\Vert ^2+\frac{2\alpha _k(1-\alpha _k)}{4\beta }\Vert h(0)\Vert ^2]\nonumber \\&\quad =[1-2\alpha _k\big (1-\alpha _k(\frac{1}{2}+L^2)-(1-\alpha _k)(\theta +\beta )\big )]\Vert u_k-u^*\Vert _\lambda ^2\nonumber \\&\qquad +2\alpha _k(\alpha _k+\frac{1-\alpha _k}{4\beta })(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)\nonumber \\&\quad \le [1-2\alpha _k\big (1-\alpha _k(\frac{1}{2}+L^2)-(1-\alpha _k)(\theta +\beta )\big )]\Vert u_k-u^*\Vert _\lambda ^2\nonumber \\&\qquad +2\alpha _k(1+\frac{1}{4\beta })(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2). \end{aligned}$$

(4.9)

Noting the fact that $\alpha _k\rightarrow 0$ and $\lim _{k\rightarrow \infty }[1-\alpha _k(\frac{1}{2}+L^2)-(1-\alpha _k)(\theta +\beta )]=1-(\theta +\beta )>0$, we assert that there exists some integer $k_0$ such that $2\alpha _k<1$ and

$$\begin{aligned} 1-\alpha _k\left( \frac{1}{2}+L^2\right) -(1-\alpha _k)(\theta +\beta )>\frac{1}{2}(1-(\theta +\beta )) \end{aligned}$$

(4.10)

hold for all $k\ge k_0.$ So, from (4.5), (4.9) and (4.10), we see that

$$\begin{aligned}&\Vert u_{k+1}-u^*\Vert _\lambda ^2\nonumber \\&\quad \le \Vert \overline{u}_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad \le [1-\alpha _k(1-(\theta +\beta ))]\Vert u_k-u^*\Vert _\lambda ^2\nonumber \\&\qquad +2\alpha _k(1+\frac{1}{4\beta })(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)\nonumber \\&\quad =[1-\alpha _k(1-(\theta +\beta ))]\Vert u_k-u^*\Vert ^2_\lambda \nonumber \\&\qquad +\alpha _k(1-(\theta +\beta ))[\frac{2(1+\frac{1}{4\beta })}{1-(\theta +\beta )}(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)], \end{aligned}$$

(4.11)

for all $k\ge k_0$. Consequently,

$$\begin{aligned} \Vert u_{k+1}-u^*\Vert ^2_\lambda \le \max \{\Vert u_k-u^*\Vert ^2_\lambda ,\frac{2\left( 1+\frac{1}{4\beta }\right) }{1-(\theta +\beta )}(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)\} \end{aligned}$$

and inductively

$$\begin{aligned}&\Vert u_{k+1}-u^*\Vert ^2_\lambda \le \max \{\Vert u_{k_0}-u^*\Vert _\lambda ^2,\\&\quad \times \,\frac{2\left( 1+\frac{1}{4\beta }\right) }{1-(\theta +\beta )}(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)\}, \ \ k\ge k_0. \end{aligned}$$

This means $\{\Vert u_k\Vert _\lambda \}$ is bounded. So $\{x_k\},$ $\{v_k\},$ $\{h(x_k)\}$ and $\{h(v_k)\}$ are bounded, too. Moreover, from (4.6), we have

$$\begin{aligned}&\Vert u_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad \le \Vert \overline{u}_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad \le 2{\alpha _k}^2[L^2\Vert x_k-x^*\Vert ^2+\Vert h(x^*)-x^*\Vert ^2]+(1-\alpha _k)^2\Vert x_k-x^*\Vert ^2\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)[\theta \Vert x_k-x^*\Vert ^2+<h(x^*)-x^*,x_k-x^*>]\nonumber \\&\qquad +\lambda [2{\alpha _k}^2(L^2\Vert v_k\Vert ^2+\Vert h(0)\Vert ^2)+2\alpha _k(1-\alpha _k)(\theta \Vert v_k\Vert ^2\nonumber \\&\qquad \,\,+<h(0),v_k>)+(1-\alpha _k)^2\Vert v_k\Vert ^2]\nonumber \\&\quad =[1-2\alpha _k+{\alpha _k}^2+2{\alpha _k}^2L^2+2\alpha _k(1-\alpha _k)\theta ]\Vert u_k-u^*\Vert _\lambda ^2+2{\alpha _k}^2\Vert h(x^*)-x^*\Vert ^2\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)<h(x^*)-x^*,x_k-x^*>\nonumber \\&\qquad +2\lambda {\alpha _k}^2\Vert h(0)\Vert ^2+2\lambda \alpha _k(1-\alpha _k)<h(0),v_k>\nonumber \\&\quad =[1-\alpha _k\big (2-\alpha _k(1+2L^2)-2(1-\alpha _k)\theta \big )]\Vert u_k-u^*\Vert ^2_\lambda \nonumber \\&\qquad +2{\alpha _k}^2[\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2]\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)[<h(x^*)-x^*,x_k-x^*>+\lambda <h(0),v_k>]. \end{aligned}$$

(4.12)

Set

$$\begin{aligned} s_k=\Vert u_k-u^*\Vert ^2_\lambda ,\ \ \sigma _k=2-\alpha _k(1+2L^2)-2(1-\alpha _k)\theta ,\ \ \lambda _k=\alpha _k\sigma _k \end{aligned}$$

and

$$\begin{aligned} \delta _k= & {} \frac{2}{\sigma _k}[\alpha _k(\Vert h(x^*)-x^*\Vert ^2 \lambda \Vert h(0)\Vert ^2)\\&\quad +(1-\alpha _k)(\langle h(x^*)-x^*,x_k-x^* \rangle \\&\quad + \lambda \langle h(0),v_k \rangle )]. \end{aligned}$$

Then (4.12) can be rewritten as the form

$$\begin{aligned} s_{k+1}\le (1-\lambda _k)s_k+\lambda _k\delta _k. \end{aligned}$$

(4.13)

On the other hand, there exists some integer $k_1$ such that $2\alpha _k<1$ and

$$\begin{aligned} 1-\alpha _k\left( \frac{1}{2}+L^2\right) -(1-\alpha _k)\theta >\frac{1}{2}(1-\theta ) \end{aligned}$$

hold for all $k\ge k_1$. From (4.12), we have

$$\begin{aligned}&\Vert \overline{u}_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad \le [1-\alpha _k(1-\theta )]\Vert u_k-u^*\Vert ^2_\lambda +2{\alpha _k}^2[\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2]\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)[\langle h(x^*)-x^*,x_k-x^* \rangle +\lambda \langle h(0),v_k \rangle ]\nonumber \\&\quad \le \Vert u_k-u^*\Vert ^2_\lambda +2{\alpha _k}^2(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)[\Vert h(x^*)-x^*\Vert \cdot \Vert x_k-x^*\Vert +\lambda \Vert h(0)\Vert \cdot \Vert v_k\Vert ], \end{aligned}$$

(4.14)

for all $k\ge k_1$. So, it follows from (4.5) that

$$\begin{aligned}&\Vert u_{k+1}-u^*\Vert ^2_\lambda \nonumber \\&\quad \le \Vert u_k-u^*\Vert ^2_\lambda +2{\alpha _k}^2(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)\nonumber \\&\qquad +2\alpha _k(1-\alpha _k)(\Vert h(x^*)-x^*\Vert \cdot \Vert x_k-x^*\Vert +\lambda \Vert h(0)\Vert \cdot \Vert v_k\Vert )\nonumber \\&\qquad -\gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)A\overline{x}_{k+1}\Vert ^2-\lambda ^2\Vert \overline{v}_{k+1}\Vert ^2-\lambda (1-\lambda )\Vert v_{k+1}-\overline{v}_{k+1}\Vert ^2. \end{aligned}$$

(4.15)

Set

$$\begin{aligned} \mu _k&=2{\alpha _k}^2(\Vert h(x^*)-x^*\Vert ^2+\lambda \Vert h(0)\Vert ^2)\\ {}&\qquad +2\alpha _k(1-\alpha _k)(\Vert h(x^*)-x^*\Vert \cdot \Vert x_k-x^*\Vert \\ {}&\qquad +\lambda \Vert h(0)\Vert \cdot \Vert v_k\Vert ), \end{aligned}$$

and

$$\begin{aligned} \eta _k&=\gamma _k(2-\gamma _k\Vert A\Vert ^2)\Vert (I-P_Q)A\overline{x}_{k+1}\Vert ^2\\&\quad +\lambda ^2\Vert \overline{v}_{k+1}\Vert ^2+\lambda (1-\lambda )\Vert v_{k+1}-\overline{v}_{k+1}\Vert ^2. \end{aligned}$$

We can rewritten (4.15) as the form

$$\begin{aligned} s_{k+1}\le s_k+\mu _k-\eta _k. \end{aligned}$$

(4.16)

Since $\lim _{k\rightarrow \infty }(2-\alpha _k(1+2L^2)-2(1-\alpha _k)\theta )=2(1-\theta )>0$, from conditions (i) and (ii), we get $\lambda _k\rightarrow 0,$ $\sum _{k=0}^\infty \lambda _k=\infty$ and $\mu _k\rightarrow 0$. So to complete the proof using Lemma 2.5, it suffices to verify $\eta _{k_l}\rightarrow 0(l\rightarrow \infty )$ implies that $\limsup _{l\rightarrow \infty }\delta _{k_l}\le 0$ for any subsequence $\{k_l\}\subset \{k\}.$ Indeed, due to $\eta _{k_l}\rightarrow 0(l\rightarrow \infty )$ and the condition (iii), we can obtain

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert (I-P_Q)A\overline{x}_{{k_l}+1}\Vert =\lim _{l\rightarrow \infty }\Vert \overline{v}_{{k_l}+1}\Vert =\lim _{l\rightarrow \infty }\Vert v_{{k_l}+1}-\overline{v}_{{k_l}+1}\Vert =0.\nonumber \\ \end{aligned}$$

(4.17)

So, we have

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert v_{{k_l}+1}\Vert =0 \end{aligned}$$

(4.18)

and

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert {x_{{k_l}+\frac{1}{2}}}-\overline{x}_{{k_l}+1}\Vert =\lim _{l\rightarrow \infty }\gamma _{k_l}\Vert A^*(I-P_Q)A\overline{x}_{{k_l}+1}\Vert =0. \end{aligned}$$

(4.19)

Furthermore, due to condition (i), we have

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert \overline{v}_{{k_l}+1}-v_{k_l}\Vert =\lim _{l\rightarrow \infty }\Vert \overline{x}_{{k_l}+1}-x_{k_l}\Vert =0. \end{aligned}$$

(4.20)

From (4.3) and (4.18), we have

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert x_{{k_l}+1}-{x_{{k_l}+\frac{1}{2}}}\Vert =\lim _{l\rightarrow \infty }\lambda \Vert v_{{k_l}+1}\Vert =0. \end{aligned}$$

(4.21)

Combining with (4.19) and (4.20), we can obtain

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert x_{{k_l}+1}-x_{k_l}\Vert =0. \end{aligned}$$

(4.22)

It follows from (4.17) and (4.20) that

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert v_{{k_l}+1}-v_{k_l}\Vert =0. \end{aligned}$$

(4.23)

By (4.22), (4.23) and the definition in (3.3), we have

$$\begin{aligned}&\lim _{l\rightarrow \infty }\Vert G_{k_l}(\overline{u}_{{k_l}+1})-u_{k_l}\Vert ^2_\lambda \nonumber \\&\quad = \lim _{l\rightarrow \infty }\Vert {u}_{{k_l}+1}-u_{k_l}\Vert ^2_\lambda \nonumber \\&\quad = \lim _{l\rightarrow \infty }\Vert (v_{{k_l}+1},x_{{k_l}+1})-(v_{k_l},x_{k_l})\Vert ^2_\lambda =0. \end{aligned}$$

(4.24)

Since $G_{k}$ is nonexpansive, from (4.20), we have

$$\begin{aligned} \Vert G_{k_l}(u_{k_l})-G_{k_l}(\overline{u}_{{k_l}+1})\Vert ^2_\lambda \le \Vert (v_{k_l},x_{k_l})-(\overline{v}_{{k_l}+1},\overline{x}_{{k_l}+1})\Vert ^2_\lambda \rightarrow 0 \end{aligned}$$

(4.25)

as $l\rightarrow \infty$. From (4.24) and (4.25), we get

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert G_{k_l}(u_{k_l})-u_{k_l}\Vert _\lambda =0. \end{aligned}$$

(4.26)

Since $\{x_{k_{l}}\}$ is bounded, we see that the set $\omega _w(x_{k_{l}})$ of weak limit points of $\{x_{k_{l}}\}$ is nonempty. Next, we prove that $\omega _w(x_{k_{l}})\subseteq \Gamma$. Assume that $\hat{x}\in \omega _w(x_{k_{l}})$ and $\{x_{k_{l_{j}}}\}$ is a subsequence of $\{x_{k_{l}}\}$ which converges weakly to $\hat{x}$. Since $\{v_{k_{l_j}}\}$ is also bounded, we know that $\{v_{k_{l_j}}\}$ has a weak convergent subsequence. Without loss of generality, we can assume that $\{v_{k_{l_{j}}}\}$ converges weakly to $\hat{v}$ as $j\rightarrow \infty$. Then $\{u_{k_{l_j}}\}=\{(v_{k_{l_{j}}},x_{k_{l_{j}}})\}$ converges weakly to $\hat{u}=(\hat{v},\hat{x})$ under the norm $\Vert \cdot \Vert _\lambda$. We prove that $\hat{u}\in \Omega$. It follows from the boundedness of real sequence $\{\gamma _{k_{l}}\}$, without loss of generality, we can assume that $\gamma _{k_{l_{j}}}\rightarrow \hat{\gamma }$ as $j\rightarrow \infty$. Let g, T, S and G be defined in (2.4)–(2.7) for $\gamma =\hat{\gamma }$, similar to (3.19), we have

$$\begin{aligned} \Vert G_{k_{l_{j}}}(u_{k_{l_{j}}})-G(u_{k_{l_{j}}})\Vert _\lambda ^2&\le (2+2\lambda ^2+\lambda )|\gamma _{k_{l_{j}}}\nonumber \\&\quad -\hat{\gamma }|\cdot \Vert A^*(I-P_Q)A\overline{x}_{k_{l_j}}\Vert ^2. \end{aligned}$$

(4.27)

By the boundedness of $\{\overline{x}_{k_{l}}\}$ and $\gamma _{k_{l_{j}}}\rightarrow \hat{\gamma }$ as $j\rightarrow \infty$, we get

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert G_{k_{l_{j}}}(u_{k_{l_{j}}})-G(u_{k_{l_{j}}})\Vert _\lambda =0. \end{aligned}$$

(4.28)

From (4.26), (4.28) and

$$\begin{aligned} \Vert u_{k_{l_{j}}}-G(u_{k_{l_{j}}})\Vert _\lambda \le \Vert u_{k_{l_{j}}}-G_{k_{l_{j}}}(u_{k_{l_{j}})\Vert _\lambda +\Vert G_{k_{l_{j}}}(u_{k_{l_{j}}})--G(u_{k_{l_{j}}}})\Vert _\lambda , \end{aligned}$$

we obtain

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert u_{k_{l_{j}}}-G(u_{k_{l_{j}}})\Vert _\lambda =0. \end{aligned}$$

(4.29)

By Lemma 2.2 and Remark 3.1, (4.29) implies that $\hat{u}=(\hat{v},\hat{x})\in F(G)$ with $\gamma =\hat{\gamma }.$ So $(\hat{v},\hat{x})\in \Omega$ and $\hat{x}\in \Gamma$. Hence, we have $\omega _w(x_{k_{l}})\subseteq \Gamma$. To get $\limsup _{l\rightarrow \infty }\delta _{k_l}\le 0$, from $\alpha _k\rightarrow 0$, (4.18) and (4.23) we only need to verify

$$\begin{aligned} \limsup _{l\rightarrow \infty } \langle h(x^*)-x^*,x_{k_l}-x^* \rangle \le 0. \end{aligned}$$

We can take a subsequence $\{x_{k_{l_j}}\}$ of $\{x_{k_l}\}$ such that $\{x_{k_{l_j}}\}$ converges weakly to $\hat{x}$ as $j\rightarrow \infty$ and

$$\begin{aligned}&\limsup _{l\rightarrow \infty } \langle h(x^*)-x^*,x_{k_l}-x^* \rangle \nonumber \\&\quad =\lim _{j\rightarrow \infty } \langle h(x^*)-x^*,x_{k_{l_{j}}}-x^* \rangle \nonumber \\&\quad =\langle h(x^*)-x^*,\widehat{x}-x^* \rangle . \end{aligned}$$

(4.30)

Since $\omega _w(x_{k_l})\subset \Gamma$ and $x^*$ is the solution of the variational inequality problem (4.4), from (4.30), we obtain

$$\begin{aligned} \limsup _{l\rightarrow \infty }\langle h(x^*)-x^*,x_{k_l}-x^* \rangle \le 0. \end{aligned}$$

From Lemma 2.5, it follows that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert u_k-u^*\Vert _\lambda ^2=0, \end{aligned}$$

which implies that $x_k\rightarrow x^*$ as $k\rightarrow \infty$. $\square$

5 Split equality problem

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $C\subset H_1$, $Q\subset H_2$ be two nonempty closed convex sets, let $A:\ H_1\rightarrow H_3$, $B:\ H_2\rightarrow H_3$ be two bounded linear operators. The split equality problem introduced by Moudafi in [18] is to find

$$\begin{aligned} x \in C, y \in Q\ \ \text {such that}\ \ Ax = By. \end{aligned}$$

(5.1)

In [28], Moudafi and Al-Shemas proposed the following simultaneous iterative algorithm for the split equality problem (5.1):

$$\begin{aligned} {\left\{ \begin{array}{ll} x_{k+1}=P_C(x_k-\gamma _kA^*(Ax_k-By_k)),\\ y_{k+1}=P_Q(y_k+\gamma _kB^*(Ax_{k}-By_k)), \end{array}\right. } \end{aligned}$$

(5.2)

where $\varepsilon \le \gamma _k\le \frac{2}{\lambda _A+\lambda _B}-\varepsilon$, $\lambda _A$ and $\lambda _B$ stand for the spectral radius of $A^*A$ and $B^*B$, respectively.

We will transform the split equality problem (5.1) into the original SFP. Let us denote

$$\begin{aligned}&\mathbf{H_1}:= H_1\times H_2,\\&\mathbf{H_2}:= H_3\times H_3,\\&\mathbf{{C}}:=C\times Q\subset \mathbf {H_1},\\&\mathbf{{Q}}:=\{(z,w)\in \mathbf{{H_2}}:z=w\}.\\ \end{aligned}$$

Denote a linear operator $\mathbf {A}:\mathbf {H_1}\rightarrow \mathbf {H_2}$ by

$$\begin{aligned} {\mathbf {A}}(x,y)=(Ax,By) \end{aligned}$$

for all $(x,y)\in \mathbf {H_1}$. If the set ${\mathbf {\Gamma }}:=\{(x,y)\in {\mathbf {C}}:{\mathbf {A}}(x,y)\in \mathbf {Q}\}$ is nonempty, then $(x,y)\in \mathbf {H_1}$ solves (5.1) if and only if

$$\begin{aligned} (x,y)\in {\mathbf {C}}\ \ \text {such that}\ \ {\mathbf {A}}(x,y)\in \mathbf {Q}. \end{aligned}$$

Note that:

$$\begin{aligned} P_{\mathbf {C}}(x,y)&=(P_Cx,P_Qy)\ \ \ \ \ \ \ \text {for all}\ \ (x,y)\in \mathbf {H_1},\\ P_{\mathbf {Q}}(z,w)&=(\frac{z+w}{2},\frac{z+w}{2})\ \ \text {for all}\ \ (z,w)\in \mathbf {H_2},\\ {\mathbf {A}}^*(z,w)&=(A^*z,B^*w)\ \ \ \ \ \ \ \text {for all}\ \ (z,w)\in \mathbf {H_2}.\\ \end{aligned}$$

As a consequence of our Theorem 3.2, the following iterative sequence$\{x_k,y_k\}$ defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} v_{k+1}=(I-P_C)(x_{k}-\frac{\gamma _k}{2}A^*(Ax_k-By_k)+(1-\lambda )v_k),\\ w_{k+1}=(I-P_Q)(y_{k}+\frac{\gamma _k}{2}B^*(Ax_k-By_k)+(1-\lambda )w_k),\\ x_{k+1}=x_{k}-\frac{\gamma _k}{2}A^*(Ax_k-By_k)-\lambda v_{k+1},\\ y_{k+1}=y_{k}+\frac{\gamma _k}{2}B^*(Ax_k-By_k)-\lambda w_{k+1}, \end{array}\right. } \end{aligned}$$

(5.3)

converges weakly to $(x^*,y^*)$ which solves split equality problem (5.1), where $0 <\lambda \le 1$ and

$$\begin{aligned} 0<\liminf _{k\rightarrow \infty }\gamma _k \le \limsup _{k\rightarrow \infty }\gamma _k<\frac{2}{\Vert \mathbf{A}\Vert ^2}=\frac{2}{\Vert A\Vert ^2+\Vert B\Vert ^2}. \end{aligned}$$

We remark that (5.3) generalizes (5.2), that is, if we take $\lambda =1$ in (5.3), then we can obtain (5.2).

Similarly, as a consequence of our Theorem 4.1, the following iterative sequence$\{x_k,y_k\}$ defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} \overline{v}_{k+1}=\alpha _kh_1(v_k)+(1-\alpha _k)v_k, \\ \overline{w}_{k+1}=\alpha _kh_2(w_k)+(1-\alpha _k)w_k, \\ \overline{x}_{k+1}=\alpha _kh_1(x_k)+(1-\alpha _k)x_k,\\ \overline{y}_{k+1}=\alpha _kh_2(y_k)+(1-\alpha _k)y_k,\\ v_{k+1}=(I-P_C)(\overline{x}_{k}-\frac{\gamma _k}{2}A^*(A\overline{x}_{k+1}-B\overline{y}_{k+1})+(1-\lambda )\overline{v}_{k+1}),\\ w_{k+1}=(I-P_Q)(\overline{y}_{k+1}+\frac{\gamma _k}{2}B^*(A\overline{x}_{k+1}-B\overline{y}_{k+1})+(1-\lambda )\overline{w}_{k+1}),\\ x_{k+1}=\overline{x}_{k+1}-\frac{\gamma _k}{2}A^*(A\overline{x}_{k+1}-B\overline{y}_{k+1})-\lambda v_{k+1},\\ y_{k+1}=\overline{y}_{k+1}+\frac{\gamma _k}{2}B^*(A\overline{x}_{k+1}-B\overline{y}_{k+1})-\lambda w_{k+1}, \end{array}\right. } \end{aligned}$$

(5.4)

converges strongly to $(x^*,y^*)$ which solves split equality problem (5.1), where $h_1:H_1\rightarrow H_1$ and $h_2:H_2\rightarrow H_2$ are L-Lipschitzian and $\theta$-strongly pseudo-contractive mappings.

6 Numerical results

To verify the theoretical assertions, we present some numerical results in this section for the SFP (1.1). To give some insight into the behavior of Algorithm 3.1 presented in this paper, we implemented them in MATLAB to solve an example. In the implementation, we take $p(x)<\varepsilon =10^{-6}$ as the stopping criterion, where

$$\begin{aligned} p(x)=\Vert x-P_Cx\Vert +\Vert Ax-P_QAx\Vert . \end{aligned}$$

Fig. 1

Comparison of the number of iterations of Algorithm 3.1 with different $\lambda$ and initial values

Example

[29]. The SFP (1.1) is to find $x\in C=\{x\in R^5|\ \Vert x\Vert \le 0.25\}$ such that $Ax\in Q=\{y=(y_1,y_2,y_3,y_4)^T\in R^4|\ 0.6\le y_j\le 1,j=1,2,3,4\}$, where the matrix

$$\begin{aligned} A= \left( \begin{array}{ccccc} 2&{}\quad -1&{}\quad 3&{}\quad 2&{}\quad 3\\ 1&{}\quad 2&{}\quad 5&{}\quad 2&{}\quad 1\\ 2&{}\quad 0&{}\quad 2&{}\quad 1&{}\quad -2\\ 2&{}\quad -1&{}\quad 0&{}\quad -3&{}\quad 5 \end{array} \right) . \end{aligned}$$

We take parameters $\gamma _k\equiv \gamma \in (0, \frac{2}{\Vert A\Vert ^2})$ and $0<\lambda \le 1$ in Algorithm 3.1. Since a larger step size is more efficient than a smaller one (as illustrated in [29]), we take $\gamma =\frac{1.95}{\Vert A\Vert ^2}$ in the experiment. The parameter $\lambda \in (0,1]$ is important in Algorithm 3.1, we try different values of $\lambda$ for solving this example. When the parameter $\lambda =1$, Algorithm 3.1 becomes the exactly CQ algorithm (1.3).

We take different $x_0$, $v_0$ as initial points. In case 1, we take $x_0=(1,1,1,1,1)$ and $v_0=(2,-2,2,-2,2)$. In case 2, we take $x_0=(1,1,1,1,1)$ and $v_0=(0,0,0,0,0)$. In case 3, we take $x_0=v_0(1,5,1,5,1)$. In case 4, we take $x_0=(1,-5,1,-5,1)$ and $v_0=(1,-2,1,-2,1)$. We report the numerical results in Fig. 1.

Elementary experiments showed that for some examples Algorithm 3.1 approaches to an approximate solution faster than the algorithm (1.3) ($\lambda =1$), which indicates that a proper choice of the parameter $\lambda \in (0,1)$ may accelerate the convergence. But for different initial points, we failed to obtained the stability of $\lambda$ for Algorithm 3.1. This still needs further study.