Journal of Inequalities and Applications

J Inequal Appl. 2017; 2017(1): 52.
Published online 2017 February 28.
PMCID: PMC5331114

# 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.

Keywords: strong convergence, extragradient-type, multiple-sets split equality problem

## 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

x ∈ CAx ∈ Q
1.1

if such a point exists, where C and Q are nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively, and A:H1 → H2 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 ∈ C and y ∈ Q that satisfy the property

AxBy
1.2

if such points exist, where C and Q are nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively, H3 is also a Hilbert space, A:H1 → H3 and B:H2 → H3 are two bounded and linear operators. When BI, 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

1.3

where rt are positive integers, ${Ci}i=1t∈H1$ and ${Qj}j=1r∈H2$ are nonempty, closed and convex subsets of Hilbert spaces H1 and H2, respectively, H3 is also a Hilbert space, A:H1 → H3, B:H2 → H3 are two bounded and linear operators. Obviously, if BI, the MSSEP is just right MSSFP; if tr = 1, the MSSEP changes into the SEP. Moreover, when BI and tr = 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.

## Preliminaries

Let H be a real Hilbert space whose inner product and norm are denoted by 〈 ⋅ ,  ⋅ 〉 and  ∥  ⋅  ∥ , 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 → H goes by the name of

• (i) nonexpansive if
∥ Tx − Ty ∥  ≤  ∥ x − y ∥ ,  ∀xy ∈ H
• (ii) firmly nonexpansive if
∥ Tx − Ty ∥  ≤ 〈x − yTx − Ty〉,  ∀xy ∈ H
• (iii) contractive on x if there exists 0 < α < 1 such that
∥ Tx − Ty ∥  ≤ α ∥ x − y ∥ ,  ∀xy ∈ H
• (iv) monotone if
Tx − Tyx − y〉 ≥ 0,  ∀xy ∈ H
• (v) β-inverse strongly monotone if there exists β > 0 such that
Tx − Tyx − y〉 ≥ βTxTy2,  ∀xy ∈ 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 PC be a mapping from H onto a closed, convex and nonempty subset C of H if

$PC(x)=argminy∈C∥x−y∥,∀x∈H,$

then PC is called an orthogonal projection from H onto C. Furthermore, for any xy ∈ H and z ∈ C,

• (i) x − PCxz − PCx〉 ≤ 0;
• (ii) PCxPCy2 ≤ 〈PCx − PCyx − y;
• (iii) PCxz2 ≤ ∥xz2 − ∥PCxx2.

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 → H be α-inverse strongly monotone, and let r > 0 be a constant. Then, for any xy ∈ C,

∥(IrT)x−(IrT)y2 ≤ ∥xy2r(r − 2α)∥T(x)−T(y)∥2.

Moreover, when 0 < r < 2α, I − rT is nonexpansive.

### Lemma 2.4

[19]

Let {xk} and {yk} be bounded sequences in a Hilbert space H, and let {βk} be a sequence in [0, 1] which satisfies the condition 0 < lim infk→∞βk ≤ lim supk→∞βk < 1. Suppose that xk+1 = (1 − βk)ykβkxk for all k ≥ 0 and lim supk→∞( ∥ yk+1 − yk ∥ − ∥ xk+1 − xk ∥ ) ≤ 0. Then limk→∞ ∥ yk − xk ∥  = 0.

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

### Lemma 2.5

[20]

Let {ak} be a sequence of nonnegative real numbers satisfying the condition ak+1 ≤ (1 − mk)akmkδk, ∀k ≥ 0,  where {mk}, {δk} are sequences of real numbers such that

• (i) {mk} ∈ [0, 1] and $∑k=0∞mk=∞$ or, equivalently,
$∏k=0∞(1−mk)=limk→∞∏j=0k(1−mj)=0;$
• (ii) lim supk→∞δk ≤ 0 or
• (ii)’ $∑k=0∞δkmk$ is convergent. Then limk→∞ak = 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 Qr+1Qr+2 =  ⋯  = QtH2. Hence, MSSEP (1.3) is equivalent to the following problem:

3.1

Moreover, set SiCi × Qi ∈ HH1 × H2 (i = 1, 2, …, t), $S=⋂i=1tSi$, G = [A, −B]:H → H3, the adjoint operator of G is denoted by G, then the original problem (3.1) reduces to

finding w = (xy) ∈ S such that Gw = 0.
3.2

### Theorem 3.1

Let Ω ≠ ∅ be the solution set of MSSEP (3.2). For an arbitrary initial point w0 ∈ S, the iterative sequence {wn} can be given as follows:

${vn=PS{(1−αn)wn−γnG∗Gwn},wn+1=PS{wn−μnG∗Gvn+λn(vn−wn)},$
3.3

where ${αn}n=0∞$ is a sequence in [0, 1] such that $limn→∞αn=0,and∑n=1∞αn=∞$, and ${γn}n=0∞$, ${λn}n=0∞$, ${μn}n=0∞$ are sequences in H satisfying the following conditions:

${γn∈(0,2ρ(G∗G)),limn→∞(γn+1−γn)=0;λn∈(0,1),limn→∞(λn+1−λn)=0;μn≤2ρ(G∗G)λn,limn→∞(μn+1−μn)=0;∑n=1∞(γnλn)<∞.$
3.4

Then {wn} converges strongly to a solution of MSSEP (3.2).

### Proof

In view of the property of the projection, we infer $wˆ=PS(wˆ−tG∗Gwˆ)$ for any t > 0. Further, from the condition in (3.4), we get that $μn≤2ρ(G∗G)λn$, it follows that $I−μnλnG∗G$ is nonexpansive. Hence,

$∥wn+1−wˆ∥=∥PS{wn−μnG∗Gvn+λn(vn−wn)}−PS{wˆ−tG∗Gwˆ}∥=∥PS{(1−λn)wn+λn(I−μnλnG∗G)vn}−PS{(1−λn)wˆ+λn(I−μnλnG∗G)wˆ}∥≤(1−λn)∥wn−wˆ∥+λn∥(I−μnλnG∗G)vn−(I−μnλnG∗G)wˆ∥≤(1−λn)∥wn−wˆ∥+λn∥vn−wˆ∥.$
3.5

Since αn → 0 as n → ∞ and from the condition in (3.4), $γn∈(0,2ρ(G∗G))$, it follows that $αn≤1−γnρ(G∗G)2$ as n → ∞, that is, $γn1−αn∈(0,2ρ(G∗G))$. We deduce that

$∥vn−wˆ∥=∥PS{(1−αn)wn−γnG∗Gwn}−PS(wˆ−tG∗Gwˆ)∥≤(1−αn)(wn−γn1−αnG∗Gwn)−{αnwˆ+(1−αn)(wˆ−γn1−αnG∗Gwˆ)}≤∥−αnwˆ+(1−αn)[wn−γn1−αnG∗Gwn−wˆ+γn1−αnG∗Gwˆ]∥,$
3.6

which is equivalent to

$∥vn−wˆ∥≤αn∥−wˆ∥+(1−αn)∥wn−wˆ∥.$
3.7

Substituting (3.7) in (3.5), we obtain

$∥wn−wˆ∥≤(1−λn)∥wn−wˆ∥+λn(αn∥−wˆ∥+(1−αn)∥wn−wˆ∥)≤(1−λnαn)∥wn−wˆ∥+λnαn∥−wˆ∥≤max{∥wn−wˆ∥,∥−wˆ∥}.$

By induction,

$∥wn−wˆ∥≤max{∥w0−wˆ∥,∥−wˆ∥}.$

Consequently, {wn} is bounded, and so is {vn}.

Let T = 2PS − I. From Proposition 2.2, one can know that the projection operator PS is monotone and nonexpansive, and 2PS − I is nonexpansive.

Therefore,

$wn+1=I+T2[(1−λn)wn+λn(1−μnλnG∗G)vn]=I−λn2wn+λn2(I−μnλnG∗G)vn+T2[(1−λn)wn+λn(I−μnλnG∗G)vn],$

that is,

$wn+1=1−λn2wn+1+λn2bn,$
3.8

where $bn=λn(I−μnλnG∗G)vn+T[(1−λn)wn+λn(I−μnλnG∗G)vn]1+λn$.

Indeed,

$∥bn+1−bn∥≤λn+11+λn+1∥(I−μn+1λn+1G∗G)vn+1−(I−μnλnG∗G)vn∥+|λn+11+λn+1−λn1+λn|×∥(I−μnλnG∗G)vn∥+T1+λn+1{(1−λn+1)wn+1+λn+1(I−μn+1λn+1G∗G)vn+1−[(1−λn)wn+λn(I−μnλnG∗G)vn]}+|11+λn+1−11+λn|×∥T[(1−λn)wn+λn(I−μnλnG∗G)vn]∥.$
3.9

For convenience, let $cn=(I−μnλnG∗G)vn$. By Lemma 2.5 in Shi et al. [1], it follows that $(I−μnλnG∗G)$ is nonexpansive and averaged. Hence,

$∥bn+1−bn∥≤λn+11+λn+1∥cn+1−cn∥+|λn+11+λn+1−λn1+λn|∥cn∥+T1+λn+1{(1−λn+1)wn+1+λn+1cn+1−[(1−λn)wn+λncn]}+|11+λn+1−11+λn|∥T[(1−λn)wn+λncn]∥≤λn+11+λn+1∥cn+1−cn∥+|λn+11+λn+1−λn1+λn|∥cn∥+1−λn+11+λn+1∥wn+1−wn∥+λn+11+λn+1∥cn+1−cn∥+λn−λn+11+λn+1∥wn∥+λn+1−λn1+λn+1∥cn∥+|11+λn+1−11+λn|∥T[(1−λn)wn+λncn]∥.$
3.10

Moreover,

$∥cn+1−cn∥=∥(I−μn+1λn+1G∗G)vn+1−(I−μnλnG∗G)vn∥≤∥vn+1−vn∥=∥PS[(1−αn+1)wn+1−γnG∗Gwn+1]−PS[(1−αn)wn−γnG∗Gwn]∥≤∥(I−γn+1G∗G)wn+1−(I−γn+1G∗G)wn+(γn−γn+1)G∗Gwn∥+αn+1∥−wn+1∥+αn∥wn∥≤∥wn+1−wn∥+|γn−γn+1|∥G∗Gwn∥+αn+1∥−wn+1∥+αn∥wn∥.$
3.11

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

$∥bn+1−bn∥≤|λn+11+λn+1−λn1+λn|∥cn∥+λn−λn+11+λn+1∥wn∥+λn+1−λn1+λn+1∥cn∥+∥wn+1−wn∥+|11+λn+1−11+λn|∥T[(1−λn)wn+λncn]∥+|γn−γn+1|∥wn∥+αn+1∥−wn+1∥+αn∥wn∥.$
3.12

By virtue of limn→∞(λn+1 − λn) = 0, it follows that $limn→∞|λn+11+λn+1−λn1+λn|=0$. Moreover, {wn} and {vn} are bounded, and so is {cn}. Therefore, (3.12) reduces to

$limsupn→∞(∥bn+1−bn∥−∥wn+1−wn∥)≤0.$
3.13

Applying (3.13) and Lemma 2.4, we get

$limn→∞∥bn−wn∥=0.$
3.14

Combining (3.14) with (3.8), we obtain

$limn→∞∥xn+1−xn∥=0.$

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

$∥wn+1−wˆ∥2≤(1−λn)∥wn−wˆ∥2+λn∥vn−wˆ∥2≤(1−λn)∥wn−wˆ∥2+λn∥−αnwˆ+(1−αn)[wn−γn1−αnG∗Gwn−(wˆ−γn1−αnG∗Gwˆ)]∥2≤(1−λn)∥wn−wˆ∥2+λnαn∥−wˆ∥2+(1−αn)λn[∥wn−wˆ∥2+γn1−αn(γn1−αn−2ρ(G∗G))∥G∗Gwn−G∗Gwˆ∥2]≤∥wn−wˆ∥2+λnαn∥−wˆ∥2+λnγn(γn1−αn−2ρ(G∗G))∥G∗Gwn−G∗Gwˆ∥2,$

which implies that

$λnγn(2ρ(G∗G)−γn1−αn)∥G∗Gwn−G∗Gwˆ∥2≤∥wn−wˆ∥2−∥wn+1−wˆ∥2+λnαn∥−wˆ∥2≤∥wn+1−wn∥(∥wn−wˆ∥+∥wn+1−wˆ∥)+λnαn∥−wˆ∥2.$

Since $liminfn→∞λnγn(2ρ(G∗G)−γn1−αn)>0$, limn→∞αn = 0 and limn→∞ ∥ wn+1 − wn ∥  = 0, we infer that

$limn→∞∥G∗Gwn−G∗Gwˆ∥=0.$
3.15

Applying Proposition 2.2 and the property of the projection PS, one can easily show that

$∥vn−wˆ∥2=∥PS[(1−αn)wn−γnG∗Gwn]−PS[wˆ−γnG∗Gwˆ]∥2≤〈(1−αn)wn−γnG∗Gwn−(wˆ−γnG∗Gwˆ),vn−wˆ〉=12{∥wn−γnG∗Gwn−(wˆ−γnG∗Gwˆ)−αnwn∥2+∥vn−wˆ∥2−∥(1−αn)wn−γnG∗Gwn−(wˆ−γnG∗Gwˆ)−vn+wˆ∥2}≤12{∥wn−wˆ∥2+2αn∥−wn∥∥wn−γnG∗Gwn−(wˆ−γnG∗Gwˆ)−αnwn∥+∥vn−wˆ∥2−∥wn−vn−γnG∗G(wn−wˆ)−αnwn∥2}≤12{∥wn−wˆ∥2+αnM+∥vn−wˆ∥2−∥wn−vn∥2+2γn〈wn−vn,G∗G(wn−wˆ)〉+2αn〈wn,wn−vn〉−∥γnG∗G(wn−wˆ)+αnwn∥2}≤12{∥wn−wˆ∥2+αnM+∥vn−wˆ∥2−∥wn−vn∥2+2γn∥wn−vn∥∥G∗G(wn−wˆ)∥+2αn∥wn∥∥wn−vn∥}≤∥wn−wˆ∥2+αnM−∥wn−vn∥2+4γn∥wn−vn∥∥G∗G(wn−wˆ)∥+4αn∥wn∥∥wn−vn∥,$
3.16

where M > 0 satisfies

$M≥supk{2∥−wn∥∥wn−γnG∗Gwn−(wˆ−γnG∗Gwˆ)−αnwn∥}.$

From (3.5) and (3.16), we get

$∥wn+1−wˆ∥2≤(1−λn)∥wn−wˆ∥2+λn∥vn−wˆ∥2≤∥wn−wˆ∥2−λn∥wn−vn∥2+αnM+4γn∥wn−vn∥∥γnG∗G(wn−wˆ)∥+4αn∥wn∥∥wn−vn∥,$

which means that

$λn∥wn−vn∥2≤∥wn+1−wn∥(∥wn−wˆ∥+∥wn+1−wˆ∥)+αnM+4γn∥wn−vn∥∥γnG∗G(wn−wˆ)∥+4αn∥wn∥∥wn−vn∥.$

Since limn→∞αn = 0, limn→∞ ∥ wn+1 − wn ∥  = 0 and $limn→∞∥G∗Gwn−G∗Gwˆ∥=0$, we infer that

$limn→∞∥wn−vn∥=0.$

Finally, we show that $wn→wˆ$. Using the property of the projection PS, we derive

$∥vn−wˆ∥2=∥PS[(1−αn)(wn−γn1−αnG∗Gwn)]−PS[αnwˆ+(1−αn)(wˆ−γn1−αnG∗Gwˆ)]∥2≤〈(1−αn)(I−γn1−αnG∗G)(wn−wˆ)−αnwˆ,vn−wˆ〉≤(1−αn)∥wn−wˆ∥∥vn−wˆ∥+αn〈wˆ,wˆ−vn〉≤1−αn2(∥wn−wˆ∥2+∥vn−wˆ∥2)+αn〈wˆ,wˆ−vn〉,$

which equals

$∥vn−wˆ∥2≤1−αn1+αn∥wn−wˆ∥2+2αn1−αn〈wˆ,wˆ−vn〉.$
3.17

It follows from (3.5) and (3.17) that

$∥wn+1−wˆ∥2≤(1−λn)∥wn−wˆ∥2+λn∥vn−wˆ∥2≤(1−λn)∥wn−wˆ∥2+λn{1−αn1+αn∥wn−wˆ∥2+2αn1−αn〈wˆ,wˆ−vn〉}≤(1−2αnλn1+αn)∥wn−wˆ∥2+2αnλn1−αn〈wˆ,wˆ−vn〉.$
3.18

Since $γn1−αn∈(0,2ρ(G∗G))$, we observe that $αn∈(0,1−γnρ(G∗G)2)$, then

$2αnλn1−αn∈(0,2λn(2−γnρ(G∗G))γnρ(G∗G)),$

that is to say,

$2αnλn1−αn〈wˆ,wˆ−vn〉≤2λn(2−γnρ(G∗G))γnρ(G∗G)〈wˆ,wˆ−vn〉.$

By virtue of $∑n=1∞(λnγn)<∞$, $γn∈(0,2ρ(G∗G))$ and $〈wˆ,wˆ−vn〉$ is bounded, we obtain $∑n=1∞(2λn(2−γnρ(G∗G))γnρn(G∗G))〈wˆ,wˆ−vn〉<∞$, which implies that

$∑n=1∞2αnλn1−αn〈wˆ,wˆ−vn〉≤∞.$

Moreover,

$∑n=1∞2αnλn1−αn〈wˆ,wˆ−vn〉=∑n=1∞2αnλn1+αn1+αn1−αn〈wˆ,wˆ−vn〉,$
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 $wn→wˆ$. This completes the proof.

## 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 = (aij)P×N, B = (bij)P×M, where aij ∈ [0, 1], bij ∈ [0, 1] are all given randomly, PMN are positive integers. The initial point x0 = (1, 1, …, 1), and y0 = (0, 0, …, 0), αn = 0.1, λn = 0.1, $γn=0.2ρ(G∗G)$, $μn=0.2ρ(G∗G)$ in Theorem 3.1, $ρ1n=ρ2n=0.1$ in Tian’s (3.15)’ and γn = 0.01 in Byrne’s (1.2). The termination condition is  ∥ Ax − By ∥  < ϵ. In Tables Tables11--4,4, the iterative steps and CPU are denoted by n and t, respectively.

ϵ = 10−10P = 3, M = 3, N = 3

ϵ = 10−5P = 10, M = 10, N = 10

ϵ = 10−5P = 3, M = 3, N = 3
ϵ = 10−10P = 10, M = 10, N = 10

## Acknowledgements

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

## Footnotes

Ying Zhao and Luoyi Shi contributed equally to this work.

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.

## Contributor Information

Ying Zhao, moc.qq@655908563.

Luoyi Shi, nc.ude.upjt@iyoulihs.

## References

1. Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms. 1994;8(2-4):221–239. doi: 10.1007/BF02142692.
2. Xu HK. A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 2006;22(6):2021–2034. doi: 10.1088/0266-5611/22/6/007.
3. Lopez G, Martin-Marqnez V, Wang F, Xu HK. Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 2012;28 doi: 10.1088/0266-5611/28/8/085004.
4. Yang Q. The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 2004;20(4):1261–1266. doi: 10.1088/0266-5611/20/4/014.
5. Byrne C. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002;18(2):441–453. doi: 10.1088/0266-5611/18/2/310.
6. Byrne C. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2004;20(1):103–120. doi: 10.1088/0266-5611/20/1/006.
7. Censor Y, Elfving T, Kopf N, Bortfeld T. The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 2005;21(6):2071–2084. doi: 10.1088/0266-5611/21/6/017.
8. Moudafi A. Alternating CQ algorithm for convex feasibility and split fixed point problem. J. Nonlinear Convex Anal. 2013;15(4):809–818.
9. Shi LY, Chen RD, Wu YJ. Strong convergence of iterative algorithms for solving the split equality problems. J. Inequal. Appl. 2014;2014 doi: 10.1186/1029-242X-2014-478.
10. Dong QL, He SN, Zhao J. Solving the split equality problem without prior knowledge of operator norms. Optimization. 2015;64(9):1887–1906. doi: 10.1080/02331934.2014.895897.
11. Korpelevich GM. An extragradient method for finding saddle points and for other problems. Ekonomikai Matematicheskie Metody. 1976;12(4):747–756.
12. Nadezhkina N, Takahashi W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2006;128(1):191–201. doi: 10.1007/s10957-005-7564-z.
13. Ceng LC, Ansari QH, Yao JC. An extragradient method for solving split feasibility and fixed point problems. Comput. Math. Appl. 2012;64(4):633–642. doi: 10.1016/j.camwa.2011.12.074.
14. Yao Y, Postolache M, Liou YC. Variant extragradient-type method for monotone variational inequalities. Fixed Point Theory Appl. 2013;2013 doi: 10.1186/1687-1812-2013-185.
15. Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 2011;148(2):318–335. doi: 10.1007/s10957-010-9757-3. [PubMed]
16. Censor Y, Gibali A, Reich S. Strong convergence of subgradient extragradient method for the variational inequalities in Hilbert space. Optim. Methods Softw. 2011;26(4-5):827–845. doi: 10.1080/10556788.2010.551536.
17. Bauschke HH, Combettes PL. Convex Analysis and Monotone Operator Theory in Hilbert Space. London: Springer; 2011.
18. Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003;118(2):417–428. doi: 10.1023/A:1025407607560.
19. Suzuki T. Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005;2005 doi: 10.1155/FPTA.2005.103.
20. Xu HK. Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002;66(1):240–256. doi: 10.1112/S0024610702003332.
21. Tian D, Shi L, Chen R. Iterative algorithm for solving the multiple-sets split equality problem with split self-adaptive step size in Hilbert spaces. Arch. Inequal. Appl. 2016;2016(1):1–9. doi: 10.1186/s13660-015-0952-5.
22. Byrne, C, Moudafi, A: Extensions of the CQ algorithm for feasibility and split equality problems. hal-00776640-version 1 (2013)

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