# Strong convergence of an extragradient-type algorithm for the multiple-sets split equality problem

## Abstract

This paper introduces a new extragradient-type method to solve the multiple-sets split equality problem (MSSEP). Under some suitable conditions, the strong convergence of an algorithm can be verified in the infinite-dimensional Hilbert spaces. Moreover, several numerical results are given to show the effectiveness of our algorithm.

## Introduction

The split feasibility problem (SFP) was first presented by Censor et al. ; it is an inverse problem that arises in medical image reconstruction, phase retrieval, radiation therapy treatment, signal processing etc. The SFP can be mathematically characterized by finding a point x that satisfies the property

\begin{aligned} x\in{C},\quad Ax\in{Q}, \end{aligned}
(1.1)

if such a point exists, where C and Q are nonempty closed convex subsets of Hilbert spaces $$H_{1}$$ and $$H_{2}$$, respectively, and $$A:H_{1}\rightarrow{H_{2}}$$ is a bounded and linear operator.

There are various algorithms proposed to solve the SFP, see  and the references therein. In particular, Byrne [5, 6] introduced the CQ-algorithm motivated by the idea of an iterative scheme of fixed point theory. Moreover, Censor et al.  introduced an extension upon the form of SFP in 2005 with an intersection of a family of closed and convex sets instead of the convex set C, which is the original of the multiple-sets split feasibility problem (MSSFP).

Subsequently, an important extension, which goes by the name of split equality problem (SEP), was made by Moudafi . It can be mathematically characterized by finding points $$x\in{C}$$ and $$y\in{Q}$$ that satisfy the property

\begin{aligned} Ax=By, \end{aligned}
(1.2)

if such points exist, where C and Q are nonempty closed convex subsets of Hilbert spaces $$H_{1}$$ and $$H_{2}$$, respectively, $$H_{3}$$ is also a Hilbert space, $$A:H_{1}\rightarrow{H_{3}}$$ and $$B:H_{2}\rightarrow{H_{3}}$$ are two bounded and linear operators. When $$B=I$$, the SEP reduces to SFP. For more information about the methods for solving SEP, see [9, 10].

This paper considers the multiple-sets split equality problem (MSSEP) which generalizes the MSSFP and SEP and can be mathematically characterized by finding points x and y that satisfy the property

\begin{aligned} x\in{\bigcap_{i=1}^{t}}C_{i} \quad\mbox{and}\quad y\in{\bigcap_{j=1}^{r}}Q_{j} \quad\mbox{such that } Ax=By, \end{aligned}
(1.3)

where $$r, t$$ are positive integers, $$\{C_{i}\}_{i=1}^{t}\in{H_{1}}$$ and $$\{ Q_{j}\}_{j=1} ^{r}\in{H_{2}}$$ are nonempty, closed and convex subsets of Hilbert spaces $$H_{1}$$ and $$H_{2}$$, respectively, $$H_{3}$$ is also a Hilbert space, $$A:H_{1}\rightarrow{H_{3}}$$, $$B:H_{2}\rightarrow{H_{3}}$$ are two bounded and linear operators. Obviously, if $$B=I$$, the MSSEP is just right MSSFP; if $$t=r=1$$, the MSSEP changes into the SEP. Moreover, when $$B=I$$ and $$t=r=1$$, the MSSEP reduces to the SFP.

One of the most important methods for computing the solution of a variational inequality and showing the quick convergence is an extragradient algorithm, which was first introduced by Korpelevich . Moreover, this method was applied for finding a common element of the set of solutions for a variational inequality and the set of fixed points of a nonexpansive mapping, see Nadezhkina et al. . Subsequently, Ceng et al. in  presented an extragradient method, and Yao et al. in  proposed a subgradient extragradient method to solve the SFP. However, all these methods to solve the problem have only weak convergence in a Hilbert space. On the other hand, a variant extragradient-type method and a subgradient extragradient method introduced by Censor et al. [15, 16] possess strong convergence for solving the variational inequality.

Motivated and inspired by the above works, we introduce an extragradient-type method to solve the MSSEP in this paper. Under some suitable conditions, the strong convergence of an algorithm can be verified in the infinite-dimensional Hilbert spaces. Finally, several numerical results are given to show the feasibility of our algorithm.

## Preliminaries

Let H be a real Hilbert space whose inner product and norm are denoted by $$\langle\cdot,\cdot\rangle$$ and $$\Vert \cdot \Vert$$, respectively. Let I denote the identity operator on H.

Next, we recall several definitions and basic results that will be available later.

### Definition 2.1

A mapping $$T:H\rightarrow{H}$$ goes by the name of

1. (i)

nonexpansive if

$$\Vert Tx-Ty\Vert \leq \Vert x-y\Vert , \quad \forall x,y\in{H};$$
2. (ii)

firmly nonexpansive if

$$\Vert Tx-Ty\Vert \leq\langle{x-y,Tx-Ty}\rangle,\quad \forall x,y\in{H};$$
3. (iii)

contractive on x if there exists $$0<\alpha<1$$ such that

$$\Vert Tx-Ty\Vert \leq\alpha \Vert x-y\Vert ,\quad \forall x,y\in{H};$$
4. (iv)

monotone if

$$\langle{Tx-Ty,x-y}\rangle\geq0,\quad \forall x,y\in{H};$$
5. (v)

β-inverse strongly monotone if there exists $$\beta>0$$ such that

$$\langle{Tx-Ty,x-y}\rangle\geq\beta \Vert Tx-Ty\Vert ^{2},\quad \forall x,y\in{H}.$$

The following properties of an orthogonal projection operator were introduced by Bauschke et al. in , and they will be powerful tools in our analysis.

### Proposition 2.2



Let $$P_{C}$$ be a mapping from H onto a closed, convex and nonempty subset C of H if

$$P_{C}(x)=\arg\min_{y\in{C}} \Vert x-y\Vert ,\quad \forall x\in{H},$$

then $$P_{C}$$ is called an orthogonal projection from H onto C. Furthermore, for any $$x,y\in{H}$$ and $$z\in{C}$$,

1. (i)

$$\langle{x-P_{C}x,z-P_{C}x}\rangle\leq0$$;

2. (ii)

$$\Vert P_{C}x-P_{C}y\Vert ^{2}\leq\langle{P_{C}x-P_{C}y,x-y}\rangle$$;

3. (iii)

$$\Vert P_{C}x-z\Vert ^{2}\leq \Vert x-z\Vert ^{2}-\Vert P_{C}x-x\Vert ^{2}$$.

The following lemmas provide the main mathematical results in the sequel.

### Lemma 2.3



Let C be a nonempty closed convex subset of a real Hilbert space H, let $$T:C\rightarrow{H}$$ be α-inverse strongly monotone, and let $$r>0$$ be a constant. Then, for any $$x,y\in{C}$$,

$$\bigl\Vert (I-rT)x-(I-rT)y\bigr\Vert ^{2}\leq \Vert x-y\Vert ^{2}+r(r-2\alpha)\bigl\Vert T(x)-T(y)\bigr\Vert ^{2}.$$

Moreover, when $$0< r<2\alpha$$, $$I-rT$$ is nonexpansive.

### Lemma 2.4



Let $$\{x^{k}\}$$ and $$\{y^{k}\}$$ be bounded sequences in a Hilbert space H, and let $$\{\beta_{k}\}$$ be a sequence in $$[0,1]$$ which satisfies the condition $$0<\lim\inf_{k\rightarrow\infty}\beta_{k}\leq{\limsup_{k\rightarrow\infty}\beta_{k}}<1$$. Suppose that $$x^{k+1}=(1-\beta_{k})y^{k}+\beta_{k}x^{k}$$ for all $$k\geq0$$ and $$\lim\sup_{k\rightarrow\infty}(\Vert y^{k+1}-y^{k}\Vert -\Vert x^{k+1}-x^{k}\Vert )\leq0$$. Then $$\lim_{k\rightarrow\infty} \Vert y^{k}-x^{k}\Vert =0$$.

The lemma below will be a powerful tool in our analysis.

### Lemma 2.5



Let $$\{a_{k}\}$$ be a sequence of nonnegative real numbers satisfying the condition $$a_{k+1}\leq(1-m_{k})a_{k}+m_{k}\delta_{k}, \forall{k\geq0,}$$ where $$\{m_{k}\}$$, $$\{\delta_{k}\}$$ are sequences of real numbers such that

1. (i)

$$\{m_{k}\}\in[0,1]$$ and $$\sum_{k=0}^{\infty}{m_{k}}=\infty$$ or, equivalently,

$$\prod_{k=0}^{\infty}(1-m_{k})=\lim _{k\rightarrow\infty}\prod_{j=0}^{k}(1-m_{j})=0;$$
2. (ii)

$$\lim\sup_{k\rightarrow\infty}\delta_{k}\leq0$$ or

3. (ii)’

$$\sum_{k=0}^{\infty}\delta_{k}m_{k}$$ is convergent. Then $$\lim_{k\rightarrow\infty}a_{k}=0$$.

## Main results

In this section, we propose a formal statement of our present algorithm. Review the multiple-sets split equality problem (MSSEP), without loss of generality, suppose $$t>r$$ in (1.3) and define $$Q_{r+1}=Q_{r+2}=\cdots =Q_{t}=H_{2}$$. Hence, MSSEP (1.3) is equivalent to the following problem:

\begin{aligned} \mbox{find}\quad x\in{\bigcap_{i=1}^{t}}C_{i}\quad \mbox{and}\quad y\in{\bigcap_{j=1}^{t}}Q_{j} \quad\mbox{such that } Ax=By. \end{aligned}
(3.1)

Moreover, set $$S_{i}=C_{i}\times{Q_{i}}\in{H}=H_{1}\times{H_{2}}\ (i=1,2,\ldots ,t)$$, $$S={\bigcap_{i=1}^{t}}S_{i}$$, $$G=[A,-B]:H\rightarrow{H_{3}}$$, the adjoint operator of G is denoted by $$G^{*}$$, then the original problem (3.1) reduces to

\begin{aligned} \mbox{finding}\quad w=(x,y)\in{S}\quad \mbox{such that } Gw=0. \end{aligned}
(3.2)

### Theorem 3.1

Let $$\Omega\neq\emptyset$$ be the solution set of MSSEP (3.2). For an arbitrary initial point $$w_{0}\in{S}$$, the iterative sequence $$\{w_{n}\}$$ can be given as follows:

\begin{aligned} \textstyle\begin{cases} v_{n}=P_{S}\{(1-\alpha_{n})w_{n}-\gamma_{n}G^{*}Gw_{n}\},\\ w_{n+1}=P_{S}\{w_{n}-\mu_{n}G^{*}Gv_{n}+\lambda_{n}(v_{n}-w_{n})\}, \end{cases}\displaystyle \end{aligned}
(3.3)

where $$\{\alpha_{n}\}_{n=0}^{\infty}$$ is a sequence in $$[0,1]$$ such that $$\lim_{n\rightarrow\infty}\alpha_{n}=0, and \sum_{n=1}^{\infty}\alpha _{n}=\infty$$, and $$\{\gamma_{n}\}_{n=0}^{\infty}$$, $$\{\lambda_{n}\}_{n=0}^{\infty}$$, $$\{\mu_{n}\}_{n=0}^{\infty}$$ are sequences in H satisfying the following conditions:

\begin{aligned} \textstyle\begin{cases} {\gamma_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}, \qquad\lim_{n\rightarrow\infty }(\gamma_{n+1}-\gamma_{n})=0;\\ \lambda_{n}\in{(0,1)},\qquad \lim_{n\rightarrow\infty}(\lambda_{n+1}-\lambda _{n})=0;\\ {\mu_{n}}\leq\frac{2}{\rho(G^{*}G)}\lambda_{n}, \qquad\lim_{n\rightarrow\infty}(\mu _{n+1}-\mu_{n})=0;\\ \sum_{n=1}^{\infty}(\frac{\gamma_{n}}{\lambda_{n}})< \infty. \end{cases}\displaystyle \end{aligned}
(3.4)

Then $$\{w_{n}\}$$ converges strongly to a solution of MSSEP (3.2).

### Proof

In view of the property of the projection, we infer $$\hat{w}=P_{S}(\hat {w}-tG^{*}G\hat{w})$$ for any $$t>0$$. Further, from the condition in (3.4), we get that $${\mu_{n}}\leq\frac{2}{\rho(G^{*}G)}\lambda_{n}$$, it follows that $$I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G$$ is nonexpansive. Hence,

\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert \\ &\quad=\bigl\Vert P_{S}\bigl\{ w_{n}-\mu_{n}G^{*}Gv_{n}+ \lambda_{n}(v_{n}-w_{n})\bigr\} -P_{S} \bigl\{ \hat {w}-tG^{*}G\hat{w}\bigr\} \bigr\Vert \\ &\quad=\biggl\Vert P_{S}\biggl\{ (1-\lambda_{n})w_{n}+ \lambda_{n}\biggl(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G\biggr)v_{n}\biggr\} -P_{S}\biggl\{ (1-\lambda_{n})\hat{w} +\lambda_{n}\biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)\hat{w}\biggr\} \biggr\Vert \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert + \lambda_{n}\biggl\Vert \biggl(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G \biggr)v_{n}-\biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)\hat{w}\biggr\Vert \\ &\quad \leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert + \lambda_{n}\Vert v_{n}-\hat{w}\Vert . \end{aligned}
(3.5)

Since $$\alpha_{n}\rightarrow{0}$$ as $$n\rightarrow\infty$$ and from the condition in (3.4), $${\gamma_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}$$, it follows that $$\alpha_{n}\leq{1-\frac{\gamma_{n}\rho(G^{*}G)}{2}}$$ as $$n\rightarrow\infty$$, that is, $$\frac{\gamma_{n}}{1-\alpha_{n}}\in{(0,\frac {2}{\rho{(G^{*}G)}})}$$. We deduce that

\begin{aligned} & \Vert v_{n}-\hat{w}\Vert \\ &\quad=\bigl\Vert P_{S}\bigl\{ (1-\alpha_{n})w_{n}- \gamma_{n}G^{*}Gw_{n}\bigr\} -P_{S}\bigl(\hat{w}-tG^{*}G \hat {w}\bigr)\bigr\Vert \\ &\quad\leq(1-\alpha_{n}) \biggl(w_{n}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}Gw_{n} \biggr)-\biggl\{ \alpha _{n}\hat{w}+(1-\alpha_{n}) \biggl( \hat{w}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr)\biggr\} \\ &\quad\leq\biggl\Vert -\alpha_{n}\hat{w}+(1-\alpha_{n}) \biggl[w_{n}-\frac{\gamma_{n}}{1-\alpha _{n}}G^{*}Gw_{n}-\hat{w}+ \frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr]\biggr\Vert , \end{aligned}
(3.6)

which is equivalent to

\begin{aligned} \Vert v_{n}-\hat{w}\Vert \leq\alpha_{n}\Vert {-\hat{w}} \Vert +(1-\alpha_{n})\Vert w_{n}-\hat{w}\Vert . \end{aligned}
(3.7)

Substituting (3.7) in (3.5), we obtain

\begin{aligned} \Vert w_{n}-\hat{w}\Vert &\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert + \lambda_{n}\bigl(\alpha_{n}\Vert {-\hat{w}}\Vert +(1- \alpha_{n})\Vert w_{n}-\hat{w}\Vert \bigr) \\ &\leq(1-\lambda_{n}\alpha_{n})\Vert w_{n}- \hat{w}\Vert +\lambda_{n}\alpha_{n}\Vert {-\hat{w}}\Vert \\ &\leq\max\bigl\{ \Vert w_{n}-\hat{w}\Vert ,\Vert {-\hat{w}}\Vert \bigr\} . \end{aligned}

By induction,

$$\Vert w_{n}-\hat{w}\Vert \leq{\max\bigl\{ \Vert w_{0}- \hat{w}\Vert ,\Vert {-\hat{w}}\Vert \bigr\} }.$$

Consequently, $$\{w_{n}\}$$ is bounded, and so is $$\{v_{n}\}$$.

Let $$T=2P_{S}-I$$. From Proposition 2.2, one can know that the projection operator $$P_{S}$$ is monotone and nonexpansive, and $$2P_{S}-I$$ is nonexpansive.

Therefore,

\begin{aligned} w_{n+1} =&\frac{I+T}{2}\biggl[(1-\lambda_{n})w_{n}+ \lambda_{n}\biggl(1-\frac{\mu _{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr] \\ =&\frac{I-\lambda_{n}}{2}w_{n}+\frac{\lambda_{n}}{2}\biggl(I- \frac{\mu_{n}}{\lambda _{n}}G^{*}G\biggr)v_{n}+\frac{T}{2}\biggl[(1- \lambda_{n})w_{n}+\lambda_{n}\biggl(I- \frac{\mu _{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr], \end{aligned}

that is,

\begin{aligned} w_{n+1}=\frac{1-\lambda_{n}}{2}w_{n}+\frac{1+\lambda_{n}}{2}b_{n}, \end{aligned}
(3.8)

where $$b_{n}=\frac{\lambda_{n}(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G)v_{n}+T[(1-\lambda_{n})w_{n}+\lambda_{n}(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G)v_{n}]}{1+\lambda_{n}}$$.

Indeed,

\begin{aligned} & \Vert b_{n+1}-b_{n}\Vert \\ &\quad\leq \frac{\lambda_{n+1}}{1+\lambda_{n+1}}\biggl\Vert \biggl(I-\frac{\mu_{n+1}}{\lambda _{n+1}}G^{*}G \biggr)v_{n+1}-\biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr\Vert +\biggl\vert \frac{\lambda _{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda_{n}}\biggr\vert \\ &\qquad{}\times \biggl\Vert \biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr\Vert +\frac{T}{1+\lambda _{n+1}}\biggl\{ (1-\lambda_{n+1})w_{n+1}+ \lambda_{n+1}\biggl(I-\frac{\mu _{n+1}}{\lambda_{n+1}}G^{*}G\biggr)v_{n+1} \\ & \qquad{}-\biggl[(1-\lambda_{n})w_{n}+\lambda_{n} \biggl(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr]\biggr\} +\biggl\vert \frac{1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \\ & \qquad{}\times \biggl\Vert T\biggl[(1-\lambda_{n})w_{n}+ \lambda_{n}\biggl(I-\frac{\mu_{n}}{\lambda _{n}}G^{*}G\biggr)v_{n}\biggr] \biggr\Vert . \end{aligned}
(3.9)

For convenience, let $$c_{n}=(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G)v_{n}$$. By Lemma 2.5 in Shi et al. , it follows that $$(I-\frac{\mu_{n}}{\lambda_{n}}G^{*}G)$$ is nonexpansive and averaged. Hence,

\begin{aligned} & \Vert b_{n+1}-b_{n}\Vert \\ &\quad\leq\frac{\lambda_{n+1}}{1+\lambda_{n+1}}\Vert c_{n+1}-c_{n}\Vert +\biggl\vert \frac {\lambda_{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda_{n}}\biggr\vert \Vert c_{n} \Vert \\ & \qquad{}+\frac{T}{1+\lambda_{n+1}}\bigl\{ (1-\lambda_{n+1})w_{n+1}+\lambda _{n+1}c_{n+1}-\bigl[(1-\lambda_{n})w_{n}+ \lambda_{n}c_{n}\bigr]\bigr\} \\ & \qquad{}+\biggl\vert \frac{1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \bigl\Vert T\bigl[(1-\lambda _{n})w_{n}+\lambda_{n}c_{n} \bigr]\bigr\Vert \\ &\quad\leq\frac{\lambda_{n+1}}{1+\lambda_{n+1}}\Vert c_{n+1}-c_{n}\Vert +\biggl\vert \frac {\lambda_{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda_{n}}\biggr\vert \Vert c_{n} \Vert \\ &\qquad{}+\frac{1-\lambda_{n+1}}{1+\lambda_{n+1}}\Vert w_{n+1}-w_{n}\Vert + \frac{\lambda _{n+1}}{1+\lambda_{n+1}}\Vert c_{n+1}-c_{n}\Vert + \frac{\lambda_{n}-\lambda _{n+1}}{1+\lambda_{n+1}}\Vert w_{n}\Vert \\ & \qquad{}+\frac{\lambda_{n+1}-\lambda_{n}}{1+\lambda_{n+1}}\Vert c_{n}\Vert +\biggl\vert \frac {1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \bigl\Vert T\bigl[(1- \lambda_{n})w_{n}+\lambda _{n}c_{n}\bigr] \bigr\Vert . \end{aligned}
(3.10)

Moreover,

\begin{aligned} & \Vert c_{n+1}-c_{n}\Vert \\ &\quad=\biggl\Vert \biggl(I-\frac{\mu_{n+1}}{\lambda_{n+1}}G^{*}G\biggr)v_{n+1}- \biggl(I-\frac{\mu _{n}}{\lambda_{n}}G^{*}G\biggr)v_{n}\biggr\Vert \\ &\quad\leq \Vert v_{n+1}-v_{n}\Vert \\ &\quad=\bigl\Vert P_{S}\bigl[(1-\alpha_{n+1})w_{n+1}- \gamma_{n}G^{*}Gw_{n+1}\bigr]-P_{S}\bigl[(1-\alpha _{n})w_{n}-\gamma_{n}G^{*}Gw_{n}\bigr]\bigr\Vert \\ &\quad\leq\bigl\Vert \bigl(I-\gamma_{n+1}G^{*}G\bigr)w_{n+1}- \bigl(I-\gamma_{n+1}G^{*}G\bigr)w_{n}+(\gamma _{n}- \gamma_{n+1})G^{*}Gw_{n}\bigr\Vert \\ & \qquad{}+\alpha_{n+1}\Vert {-w_{n+1}}\Vert +\alpha_{n} \Vert w_{n}\Vert \\ &\quad\leq \Vert w_{n+1}-w_{n}\Vert +\vert \gamma_{n}-\gamma_{n+1}\vert \bigl\Vert G^{*}Gw_{n} \bigr\Vert +\alpha_{n+1}\Vert { -w_{n+1}}\Vert + \alpha_{n}\Vert w_{n}\Vert . \end{aligned}
(3.11)

Substituting (3.11) in (3.10), we infer that

\begin{aligned} & \Vert b_{n+1}-b_{n}\Vert \\ &\quad\leq\biggl\vert \frac{\lambda_{n+1}}{1+\lambda_{n+1}}-\frac{\lambda_{n}}{1+\lambda _{n}}\biggr\vert \Vert c_{n}\Vert +\frac{\lambda_{n}-\lambda_{n+1}}{1+\lambda_{n+1}}\Vert w_{n}\Vert + \frac {\lambda_{n+1}-\lambda_{n}}{1+\lambda_{n+1}}\Vert c_{n}\Vert \\ & \qquad{}+\Vert w_{n+1}-w_{n}\Vert +\biggl\vert \frac{1}{1+\lambda_{n+1}}-\frac{1}{1+\lambda_{n}}\biggr\vert \bigl\Vert T\bigl[(1- \lambda_{n})w_{n}+\lambda_{n}c_{n}\bigr] \bigr\Vert \\ & \qquad{}+\vert \gamma_{n}-\gamma_{n+1}\vert \Vert w_{n}\Vert +\alpha_{n+1}\Vert {-w_{n+1}}\Vert + \alpha_{n}\Vert w_{n}\Vert . \end{aligned}
(3.12)

By virtue of $$\lim_{n\rightarrow\infty}(\lambda_{n+1}-\lambda_{n})=0$$, it follows that $$\lim_{n\rightarrow\infty} \vert \frac{\lambda_{n+1}}{1+\lambda_{n+1}}-\frac {\lambda_{n}}{1+\lambda_{n}}\vert =0$$. Moreover, $$\{w_{n}\}$$ and $$\{v_{n}\}$$ are bounded, and so is $$\{c_{n}\}$$. Therefore, (3.12) reduces to

\begin{aligned} \lim\sup_{n\rightarrow\infty}\bigl(\Vert b_{n+1}-b_{n} \Vert -\Vert w_{n+1}-w_{n}\Vert \bigr)\leq {0}. \end{aligned}
(3.13)

Applying (3.13) and Lemma 2.4, we get

\begin{aligned} \lim_{n\rightarrow\infty} \Vert b_{n}-w_{n}\Vert =0. \end{aligned}
(3.14)

Combining (3.14) with (3.8), we obtain

$$\lim_{n\rightarrow\infty} \Vert x_{n+1}-x_{n}\Vert =0.$$

Using the convexity of the norm and (3.5), we deduce that

\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\biggl\Vert -\alpha_{n}\hat {w}\\ &\qquad{}+(1-\alpha_{n})\biggl[w_{n}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}Gw_{n}- \biggl(\hat{w}-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr)\biggr]\biggr\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\alpha_{n}\Vert {-\hat{w}}\Vert ^{2}\\ &\qquad{}+(1-\alpha_{n})\lambda_{n}\biggl[\Vert w_{n}-\hat{w}\Vert ^{2} +\frac{\gamma_{n}}{1-\alpha_{n}}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}-\frac {2}{\rho(G^{*}G)}\biggr)\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat{w}\bigr\Vert ^{2}\biggr] \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}+ \lambda_{n}\alpha_{n}\Vert {-\hat{w}}\Vert ^{2}+ \lambda_{n}\gamma _{n}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}- \frac{2}{\rho(G^{*}G)}\biggr)\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat {w}\bigr\Vert ^{2}, \end{aligned}

which implies that

\begin{aligned} & \lambda_{n}\gamma_{n}\biggl(\frac{2}{\rho(G^{*}G)}- \frac{\gamma_{n}}{1-\alpha_{n}}\biggr)\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat{w}\bigr\Vert ^{2} \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}-\Vert w_{n+1}-\hat{w}\Vert ^{2}+\lambda_{n} \alpha_{n}\Vert {-\hat {w}}\Vert ^{2} \\ &\quad \leq \Vert w_{n+1}-w_{n}\Vert \bigl(\Vert w_{n}-\hat{w}\Vert +\Vert w_{n+1}-\hat{w}\Vert \bigr)+ \lambda _{n}\alpha_{n}\Vert {-\hat{w}}\Vert ^{2}. \end{aligned}

Since $$\lim\inf_{n\rightarrow\infty}\lambda_{n}\gamma_{n}(\frac{2}{\rho (G^{*}G)}-\frac{\gamma_{n}}{1-\alpha_{n}})>0$$, $$\lim_{n\rightarrow\infty }\alpha_{n}=0$$ and $$\lim_{n\rightarrow\infty} \Vert w_{n+1}-w_{n}\Vert =0$$, we infer that

\begin{aligned} \lim_{n\rightarrow\infty}\bigl\Vert G^{*}Gw_{n}-G^{*}G\hat{w}\bigr\Vert =0. \end{aligned}
(3.15)

Applying Proposition 2.2 and the property of the projection $$P_{S}$$, one can easily show that

\begin{aligned} & \Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad=\bigl\Vert P_{S}\bigl[(1-\alpha_{n})w_{n}- \gamma_{n}G^{*}Gw_{n}\bigr]-P_{S}\bigl[\hat{w}-\gamma _{n}G^{*}G\hat{w}\bigr]\bigr\Vert ^{2} \\ &\quad\leq\bigl\langle {(1-\alpha_{n})w_{n}-\gamma_{n}G^{*}Gw_{n}- \bigl(\hat{w}-\gamma_{n}G^{*}G\hat {w}\bigr),v_{n}-\hat{w}}\bigr\rangle \\ &\quad=\frac{1}{2}\bigl\{ \bigl\Vert w_{n}-\gamma_{n}G^{*}Gw_{n}- \bigl(\hat{w}-\gamma_{n}G^{*}G\hat {w}\bigr)-\alpha_{n}w_{n} \bigr\Vert ^{2}+\Vert v_{n}-\hat{w}\Vert ^{2} \\ & \qquad{}-\bigl\Vert (1-\alpha_{n})w_{n}-\gamma_{n}G^{*}Gw_{n}- \bigl(\hat{w}-\gamma_{n}G^{*}G\hat {w}\bigr)-v_{n}+\hat{w}\bigr\Vert ^{2}\bigr\} \\ &\quad\leq\frac{1}{2}\bigl\{ \Vert w_{n}-\hat{w}\Vert ^{2}+2\alpha_{n}\Vert {-w_{n}}\Vert \bigl\Vert w_{n}-\gamma _{n}G^{*}Gw_{n}-\bigl(\hat{w}- \gamma_{n}G^{*}G\hat{w}\bigr)-\alpha_{n}w_{n}\bigr\Vert \\ & \qquad{}+\Vert v_{n}-\hat{w}\Vert ^{2}-\bigl\Vert w_{n}-v_{n}-\gamma_{n}G^{*}G(w_{n}- \hat{w})-\alpha_{n}w_{n}\bigr\Vert ^{2}\bigr\} \\ &\quad \leq\frac{1}{2}\bigl\{ \Vert w_{n}-\hat{w}\Vert ^{2}+\alpha_{n}M+\Vert v_{n}-\hat{w}\Vert ^{2}-\Vert w_{n}-v_{n}\Vert ^{2} \\ &\qquad{}+2 \gamma_{n}\bigl\langle {w_{n}-v_{n},G^{*}G(w_{n}- \hat{w})}\bigr\rangle \\ & \qquad{}+2\alpha_{n}\langle{w_{n},w_{n}-v_{n}} \rangle-\bigl\Vert \gamma_{n}G^{*}G(w_{n}-\hat {w})+ \alpha_{n}w_{n}\bigr\Vert ^{2}\bigr\} \\ &\quad\leq\frac{1}{2}\bigl\{ \Vert w_{n}-\hat{w}\Vert ^{2}+\alpha_{n}M+\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\qquad{}-\Vert w_{n}-v_{n}\Vert ^{2}+2 \gamma_{n}\Vert w_{n}-v_{n}\Vert \bigl\Vert G^{*}G(w_{n}-\hat{w})\bigr\Vert \\ & \qquad{}+2\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert \bigr\} \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}+ \alpha_{n}M-\Vert w_{n}-v_{n}\Vert ^{2}+4\gamma_{n}\Vert w_{n}-v_{n} \Vert \bigl\Vert G^{*}G(w_{n}-\hat{w})\bigr\Vert \\ & \qquad{}+4\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert , \end{aligned}
(3.16)

where $$M>0$$ satisfies

$$M\geq{\sup_{k}\bigl\{ 2\Vert {-w_{n}}\Vert \bigl\Vert w_{n}-\gamma_{n}G^{*}Gw_{n}-\bigl(\hat{w}- \gamma_{n}G^{*}G\hat {w}\bigr)-\alpha_{n}w_{n}\bigr\Vert \bigr\} }.$$

From (3.5) and (3.16), we get

\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad\leq \Vert w_{n}-\hat{w}\Vert ^{2}- \lambda_{n}\Vert w_{n}-v_{n}\Vert ^{2}+\alpha_{n}M+4\gamma_{n}\Vert w_{n}-v_{n}\Vert \bigl\Vert \gamma_{n}G^{*}G(w_{n}- \hat{w})\bigr\Vert \\ & \qquad{}+4\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert , \end{aligned}

which means that

\begin{aligned} \lambda_{n}\Vert w_{n}-v_{n}\Vert ^{2}\leq{}&\Vert w_{n+1}-w_{n}\Vert \bigl(\Vert w_{n}-\hat{w}\Vert +\Vert w_{n+1}-\hat{w}\Vert \bigr)+ \alpha_{n}M \\ & {}+4\gamma_{n}\Vert w_{n}-v_{n}\Vert \bigl\Vert \gamma_{n}G^{*}G(w_{n}-\hat{w})\bigr\Vert \\ & {}+4\alpha_{n}\Vert w_{n}\Vert \Vert w_{n}-v_{n}\Vert . \end{aligned}

Since $$\lim_{n\rightarrow\infty}\alpha_{n}=0$$, $$\lim_{n\rightarrow\infty }\Vert w_{n+1}-w_{n}\Vert =0$$ and $$\lim_{n\rightarrow\infty} \Vert G^{*}Gw_{n}-G^{*}G\hat {w}\Vert =0$$, we infer that

$$\lim_{n\rightarrow\infty} \Vert w_{n}-v_{n}\Vert =0.$$

Finally, we show that $$w_{n}\rightarrow{\hat{w}}$$. Using the property of the projection $$P_{S}$$, we derive

\begin{aligned} & \Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad=\biggl\Vert P_{S}\biggl[(1-\alpha_{n}) \biggl(w_{n}-\frac{\gamma_{n}}{1-\alpha _{n}}G^{*}Gw_{n}\biggr) \biggr]\\ &\qquad{}-P_{S}\biggl[\alpha_{n}\hat{w}+(1- \alpha_{n}) \biggl(\hat{w}-\frac{\gamma _{n}}{1-\alpha_{n}}G^{*}G\hat{w}\biggr)\biggr] \biggr\Vert ^{2} \\ &\quad\leq\biggl\langle (1-\alpha_{n}) \biggl(I-\frac{\gamma_{n}}{1-\alpha_{n}}G^{*}G \biggr) (w_{n}-\hat {w})-\alpha_{n}\hat{w},v_{n}- \hat{w}\biggr\rangle \\ &\quad\leq(1-\alpha_{n})\Vert w_{n}-\hat{w}\Vert \Vert v_{n}-\hat{w}\Vert +\alpha_{n}\langle\hat {w}, \hat{w}-v_{n}\rangle \\ &\quad\leq\frac{1-\alpha_{n}}{2}\bigl(\Vert w_{n}-\hat{w}\Vert ^{2}+\Vert v_{n}-\hat{w}\Vert ^{2}\bigr)+\alpha _{n}\langle\hat{w},\hat{w}-v_{n}\rangle, \end{aligned}

which equals

\begin{aligned} \Vert v_{n}-\hat{w}\Vert ^{2}\leq\frac{1-\alpha_{n}}{1+\alpha_{n}} \Vert w_{n}-\hat{w}\Vert ^{2}+\frac{2\alpha_{n}}{1-\alpha_{n}}\langle \hat{w},\hat{w}-v_{n}\rangle. \end{aligned}
(3.17)

It follows from (3.5) and (3.17) that

\begin{aligned} & \Vert w_{n+1}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\quad\leq(1-\lambda_{n})\Vert w_{n}-\hat{w}\Vert ^{2}+\lambda_{n}\biggl\{ \frac{1-\alpha _{n}}{1+\alpha_{n}}\Vert w_{n}-\hat{w}\Vert ^{2}+\frac{2\alpha_{n}}{1-\alpha_{n}}\langle\hat {w}, \hat{w}-v_{n}\rangle\biggr\} \\ &\quad\leq\biggl(1-\frac{2\alpha_{n}\lambda_{n}}{1+\alpha_{n}}\biggr)\Vert w_{n}-\hat{w}\Vert ^{2}+\frac {2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle\hat{w},\hat{w}-v_{n}\rangle. \end{aligned}
(3.18)

Since $$\frac{\gamma_{n}}{1-\alpha_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}$$, we observe that $$\alpha_{n}\in{(0,1-\frac{\gamma_{n}\rho(G^{*}G)}{2})}$$, then

$$\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\in{\biggl(0,\frac{2\lambda_{n}(2-\gamma _{n}\rho(G^{*}G))}{\gamma_{n}\rho(G^{*}G)}\biggr)},$$

that is to say,

$$\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle\hat{w},\hat{w}-v_{n}\rangle \leq \frac{2\lambda_{n}(2-\gamma_{n}\rho(G^{*}G))}{\gamma_{n}\rho(G^{*}G)}\langle \hat{w},\hat{w}-v_{n}\rangle.$$

By virtue of $$\sum_{n=1}^{\infty}(\frac{\lambda_{n}}{\gamma_{n}})<\infty$$, $${\gamma_{n}}\in{(0,\frac{2}{\rho(G^{*}G)})}$$ and $$\langle\hat{w},\hat {w}-v_{n}\rangle$$ is bounded, we obtain $$\sum_{n=1}^{\infty}(\frac{2\lambda _{n}(2-\gamma_{n}\rho(G^{*}G))}{\gamma_{n}\rho_{n}(G^{*}G)})\langle\hat{w},\hat {w}-v_{n}\rangle<\infty$$, which implies that

$$\sum_{n=1}^{\infty}\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle \hat {w},\hat{w}-v_{n}\rangle\leq\infty.$$

Moreover,

\begin{aligned} \sum_{n=1}^{\infty}\frac{2\alpha_{n}\lambda_{n}}{1-\alpha_{n}}\langle \hat {w},\hat{w}-v_{n}\rangle=\sum_{n=1}^{\infty}\frac{2\alpha_{n}\lambda _{n}}{1+\alpha_{n}}\frac{1+\alpha_{n}}{1-\alpha_{n}}\langle\hat{w},\hat {w}-v_{n} \rangle, \end{aligned}
(3.19)

it follows that all the conditions of Lemma 2.5 are satisfied. Combining (3.18), (3.19) and Lemma 2.5, we can show that $$w_{n}\rightarrow \hat{w}$$. This completes the proof. □

## Numerical experiments

In this section, we provide several numerical results and compare them with Tian’s  algorithm (3.15)’ and Byrne’s  algorithm (1.2) to show the effectiveness of our proposed algorithm. Moreover, the sequence given by our algorithm in this paper has strong convergence for the multiple-sets split equality problem. The whole program was written in Wolfram Mathematica (version 9.0). All the numerical results were carried out on a personal Lenovo computer with Intel(R)Pentium(R) N3540 CPU 2.16 GHz and RAM 4.00 GB.

In the numerical results, $$A=(a_{ij})_{P\times{N}}$$, $$B=(b_{ij})_{P\times{M}}$$, where $$a_{ij}\in[0,1]$$, $$b_{ij}\in[0,1]$$ are all given randomly, $$P, M, N$$ are positive integers. The initial point $$x_{0}=(1,1,\ldots,1)$$, and $$y_{0}=(0,0,\ldots,0)$$, $$\alpha_{n}=0.1$$, $$\lambda _{n}=0.1$$, $$\gamma_{n}=\frac{0.2}{\rho{(G^{*}G)}}$$, $$\mu_{n}=\frac{0.2}{\rho {(G^{*}G)}}$$ in Theorem 3.1, $$\rho_{1}^{n}=\rho_{2}^{n}=0.1$$ in Tian’s (3.15)’ and $$\gamma_{n}=0.01$$ in Byrne’s (1.2). The termination condition is $$\Vert Ax-By\Vert <\epsilon$$. In Tables 1-4, the iterative steps and CPU are denoted by n and t, respectively.

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## Acknowledgements

This research was supported by NSFC Grants No: 11226125; No: 11301379; No: 11671167.

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Correspondence to Luoyi Shi.

### Competing interests

The authors declare that there are no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Ying Zhao and Luoyi Shi contributed equally to this work.

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