Open Access

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

Contributed equally
Journal of Inequalities and Applications20172017:52

https://doi.org/10.1186/s13660-017-1326-y

Received: 23 December 2016

Accepted: 20 February 2017

Published: 28 February 2017

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.

Keywords

strong convergenceextragradient-typemultiple-sets split equality problem

1 Introduction

The split feasibility problem (SFP) was first presented by Censor et al. [1]; 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 [24] 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. [7] 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 [8]. 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 [11]. 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. [12]. Subsequently, Ceng et al. in [13] presented an extragradient method, and Yao et al. in [14] 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.

2 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 [17], and they will be powerful tools in our analysis.

Proposition 2.2

[17]

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

[18]

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

[19]

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

[20]

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 \(\lim_{k\rightarrow\infty}a_{k}=0\).

     

3 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. [1], 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. □

4 Numerical experiments

In this section, we provide several numerical results and compare them with Tian’s [21] algorithm (3.15)’ and Byrne’s [22] 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.
Table 1

\(\pmb{\epsilon=10^{-5}, P=3, M=3, N=3}\)

 

n

t

Sequence (3.3)

60

0.078

Tian’s (3.15)’

117

0.093

Byrne’s (1.2)

1,845

1.125

Table 2

\(\pmb{\epsilon=10^{-10}, P=3, M=3, N=3}\)

 

n

t

Sequence (3.3)

120

0.156

Tian’s (3.15)

294

0.29

Byrne’s (1.2)

8,533

2.734

Table 3

\(\pmb{\epsilon=10^{-5}, P=10, M=10, N=10}\)

 

n

t

Sequence (3.3)

63

0.093

Tian’s (3.15)

426

0.469

Byrne’s (1.2)

2,287

1.313

Table 4

\(\pmb{\epsilon=10^{-10}, P=10, M=10, N=10}\)

 

n

t

Sequence (3.3)

123

0.25

Tian’s (3.15)

948

0.906

Byrne’s (1.2)

13,496

2.437

Notes

Declarations

Acknowledgements

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University, Tianjin, P.R. China

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