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# Split equality problem and multiple-sets split equality problem for quasi-nonexpansive multi-valued mappings

- Yujing Wu
^{1}, - Rudong Chen
^{2}Email author and - Luo Yi Shi
^{2}

**2014**:428

https://doi.org/10.1186/1029-242X-2014-428

© Wu et al.; licensee Springer. 2014

**Received:**11 September 2014**Accepted:**15 October 2014**Published:**30 October 2014

## Abstract

The multiple-sets split equality problem (MSSEP) requires finding a point $x\in {\bigcap}_{i=1}^{N}{C}_{i}$, $y\in {\bigcap}_{j=1}^{M}{Q}_{j}$, such that $Ax=By$, where *N* and *M* are positive integers, $\{{C}_{1},{C}_{2},\dots ,{C}_{N}\}$ and $\{{Q}_{1},{Q}_{2},\dots ,{Q}_{M}\}$ are closed convex subsets of Hilbert spaces ${H}_{1}$, ${H}_{2}$, respectively, and $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ are two bounded linear operators. When $N=M=1$, the MSSEP is called the split equality problem (SEP). If let $B=I$, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. Recently, some authors proposed many algorithms to solve the SEP and MSSEP. However, to implement these algorithms, one has to find the projection on the closed convex sets, which is not possible except in simple cases. One of the purposes of this paper is to study the SEP and MSSEP for a family of quasi-nonexpansive multi-valued mappings in the framework of infinite-dimensional Hilbert spaces, and propose an algorithm to solve the SEP and MSSEP without the need to compute the projection on the closed convex sets.

## Keywords

- split equality problem
- iterative algorithms
- converge strongly

## 1 Introduction and preliminaries

*H*is a real Hilbert space,

*C*is a subset of

*H*. Denote by $\mathit{CB}(H)$ the collection of all nonempty closed and bounded subsets of

*H*and by $Fix(T)$ the set of the fixed points of a mapping

*T*. The

*Hausdorff*metric $\tilde{H}$ on $\mathit{CB}(H)$ is defined by

where $d(x,K):={inf}_{y\in K}d(x,y)$.

**Definition 1.1**Let $R:H\to \mathit{CB}(H)$ be a multi-valued mapping. An element $p\in H$ is said to be a

*fixed point of*

*R*, if $p\in Rp$. The set of fixed points of

*R*will be denoted by $Fix(R)$.

*R*is said to be

- (i)
nonexpansive, if $\tilde{H}(Rx,Ry)\le \parallel x-y\parallel $, $\mathrm{\forall}x,y\in H$;

- (ii)
quasi-nonexpansive, if $Fix(R)\ne \mathrm{\varnothing}$ and $\tilde{H}(Rx,Ry)\le \parallel x-y\parallel $, $\mathrm{\forall}x\in H$, $y\in Fix(R)$.

*multiple*-

*sets split feasibility problem*(MSSFP) is to find a point

*x*satisfying the property:

if such a point exists. If $N=M=1$, then the MSSFP reduce to the well-known *split feasibility problem* (SFP).

The SFP and MSSFP was first introduced by Censor and Elfving [1], and Censor *et al.* [2], respectively, which attracted many authors’ attention due to the applications in signal processing [1] and intensity-modulated radiation therapy [2]. Various algorithms have been invented to solve it (see [1–8], *etc.*).

*split equality problem*(SEP): Let ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ be real Hilbert spaces, $C\subseteq {H}_{1}$, $Q\subseteq {H}_{2}$ be two nonempty closed convex sets, and let $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators. Find $x\in C$, $y\in Q$ satisfying

When $B=I$, SEP reduces to the well-known SFP.

*multiple*-

*sets split equality problem*(MSSEP) requiring to find a point $x\in {\bigcap}_{i=1}^{N}{C}_{i}$, $y\in {\bigcap}_{j=1}^{M}{Q}_{j}$, such that

where *N* and *M* are positive integers, $\{{C}_{1},{C}_{2},\dots ,{C}_{N}\}$ and $\{{Q}_{1},{Q}_{2},\dots ,{Q}_{M}\}$ are closed convex subsets of Hilbert spaces ${H}_{1}$, ${H}_{2}$, respectively, and $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ are two bounded linear operators.

In the paper [9], Moudafi give the alternating CQ-algorithm and relaxed alternating CQ-algorithm iterative algorithm for solving the split equality problem.

In the paper [10], we use the well-known Tychonov regularization to get some algorithms that converge strongly to the minimum-norm solution of the SEP.

Note that to implement these algorithms, one has to find the projection on the closed convex sets, which is not possible except in simple cases.

*i.e.*, to find $w=(x,y)\in C$ such that

*i.e.*, $\mathrm{\Gamma}=\{(x,y)\in {H}_{1}\times {H}_{2},Ax=By,x\in C,y\in Q\}\ne \mathrm{\varnothing}$. The multiple-sets split equality problem (MSSEP) for a family quasi-nonexpansive multi-valued mappings in infinitely dimensional Hilbert spaces,

*i.e.*, to find $w=(x,y)\in C$ such that

where ${R}_{i}^{j}:{H}_{i}\to \mathit{CB}({H}_{i})$, $i=1,2$, $j=1,2,\dots ,N$ is a family of quasi-nonexpansive multi-valued mappings, $C={\bigcap}_{j=1}^{N}Fix({R}_{1}^{j})$, $Q={\bigcap}_{j=1}^{N}Fix({R}_{2}^{j})$. In the rest of this paper, we use $\overline{\mathrm{\Gamma}}$ to denote the set of solutions of MSSEP (1.4), and assume consistency of MSSEP so that $\overline{\mathrm{\Gamma}}$ is closed, convex, and nonempty, *i.e.*, $\overline{\mathrm{\Gamma}}=\{(x,y)\in {H}_{1}\times {H}_{2},Ax=By,x\in C,y\in Q\}\ne \mathrm{\varnothing}$.

In this paper, we study the SEP and MSSEP for a family of quasi-nonexpansive multi-valued mappings in the framework of infinite-dimensional Hilbert spaces, and propose an algorithm to solve the SEP and MSSEP not requiring to compute the projection on the closed convex sets.

We now collect some definitions and elementary facts which will be used in the proofs of our main results.

**Definition 1.2**Let

*H*be a Banach space.

- (1)
A multi-valued mapping $R:H\to \mathit{CB}(H)$ is said to be

*demi*-*closed at the origin*if, for any sequence $\{{x}_{n}\}\subseteq H$ with ${x}_{n}$ converges weakly to*x*and $d({x}_{n},R{x}_{n})\to 0$, we have $x\in Rx$. - (2)
A multi-valued mapping $R:H\to \mathit{CB}(H)$ is said to be

*semi*-*compact*if, for any bounded sequence $\{{x}_{n}\}\subseteq H$ with $d({x}_{n},R{x}_{n})\to 0$, there exists a subsequence $\{{x}_{{n}_{k}}\}$ such that $\{{x}_{{n}_{k}}\}$ converges strongly to a point $x\in H$.

*Let* *X* *be a Banach space*, *C* *a closed convex subset of* *X*, *and* $T:C\to C$ *a nonexpansive mapping with* $Fix(T)\ne \mathrm{\varnothing}$. *If* $\{{x}_{n}\}$ *is a sequence in* *C* *weakly converging to* *x* *and if* $\{(I-T){x}_{n}\}$ *converges strongly to* *y*, *then* $(I-T)x=y$.

**Lemma 1.4** [13]

*Let*

*H*

*be a Hilbert space and*$\{{w}_{n}\}$

*a sequence in*

*H*

*such that there exists a nonempty set*$S\subseteq H$

*satisfying the following*:

- (i)
*for every*$w\in S$, ${lim}_{n\to \mathrm{\infty}}\parallel {w}_{n}-w\parallel $*exists*; - (ii)
*any weak*-*cluster point of the sequence*$\{{w}_{n}\}$*belongs to**S*.

*Then there exists* $\tilde{w}\in s$ *such that* $\{{w}_{n}\}$ *weakly converges to* $\tilde{w}$.

**Lemma 1.5** [10]

*Let*$T=I-\gamma {G}^{\ast}G$,

*where*$0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$

*with*$\rho ({G}^{\ast}G)$

*being the spectral radius of the self*-

*adjoint operator*${G}^{\ast}G$

*on*

*H*, $S=C\times Q$.

*Then we have the following*:

- (1)
$\parallel T\parallel \le 1$ (

*i*.*e*.,*T**is nonexpansive*)*and averaged*; - (2)
$Fix(T)=\{(x,y)\in H,Ax=By\}$, $Fix({P}_{S}T)=Fix({P}_{S})\cap Fix(T)=\mathrm{\Gamma}$.

## 2 Iterative algorithm for SEP

In this section, we establish an iterative algorithm that converges strongly to a solution of SEP (1.3).

**Algorithm 2.1**For an arbitrary initial point ${w}_{0}=({x}_{0},{y}_{0})$, the sequence $\{{w}_{n}=({x}_{n},{y}_{n})\}$ is generated by the iteration:

*H*, $R:{H}_{1}\times {H}_{2}\to {H}_{1}\times {H}_{2}$ by

and ${R}_{1}$, ${R}_{2}$ are quasi-nonexpansive multi-valued mappings on ${H}_{1}$, ${H}_{2}$, respectively.

To prove its convergence we need the following lemma.

**Lemma 2.2**

*Any sequence*$\{{w}_{n}\}$

*generated by Algorithm*(2.1)

*is Féjer*-

*monotone with respect to*Γ,

*namely for every*$w\in \mathrm{\Gamma}$,

*provided that* ${\alpha}_{n}>0$ *is a sequence in* $(0,1)$ *and* $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$.

*Proof*Let ${u}_{n}=(I-\gamma {G}^{\ast}G){w}_{n}$ and taking $w\in \mathrm{\Gamma}$, by Lemma 1.5, $w\in Fix({P}_{S})\cap Fix(I-\gamma {G}^{\ast}G)$, $Gw=0$ and we have

It follows that $\parallel {w}_{n+1}-w\parallel \le \parallel {w}_{n}-w\parallel $, $\mathrm{\forall}w\in \mathrm{\Gamma}$, $n\ge 1$. □

**Theorem 2.3** *If* $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$ *and* ${R}_{1}$, ${R}_{2}$ *are demi*-*closed at the origin*, *then the sequence* $\{{w}_{n}\}$ *generated by Algorithm* (2.1) *converges weakly to a solution of SEP* (1.3). *In addition*, *if* ${R}_{1}$, ${R}_{2}$ *are semi*-*compact*, *then* $\{{w}_{n}\}$ *converges strongly to a solution of SEP* (1.3).

*Proof*For any solution of SEP

*w*, according to Lemma 2.2, we see that the sequence $\parallel {w}_{n}-w\parallel $ is monotonically decreasing and thus converges to some positive real. Since $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$ and $0<\gamma <\lambda $, by (2.2), we can obtain

Since ${v}_{n}\in R{u}_{n}$, we can get $d({u}_{n},R{u}_{n})\le \parallel {u}_{n}-{v}_{n}\parallel \to 0$.

From the Féjer-monotonicity of $\{{w}_{n}\}$ it follows that the sequence is bounded. Denoting by $\tilde{w}$ a weak-cluster point of $\{{w}_{n}\}$ let $v=0,1,2,\dots $ be the sequence of indices, such that ${w}_{{n}_{v}}$ converges weakly to $\tilde{w}$. Then, by Lemma 1.3, we obtain $G\tilde{w}=0$, and it follows that $\tilde{w}\in Fix(I-\gamma {G}^{\ast}G)$.

Since ${R}_{1}$, ${R}_{2}$ are demi-closed at the origin, it is easy to check that *R* is demi-closed at the origin. Now, by setting ${u}_{n}=(I-\gamma {G}^{\ast}G){w}_{n}$, it follows that ${u}_{{n}_{v}}$ converges weakly to $\tilde{w}$. Since $d({u}_{n},R{u}_{n})\to 0$, and *R* is demi-closed at the origin, we obtain $\tilde{w}\in FixR=C\times Q$, *i.e.*, ${P}_{S}(\tilde{w})=\tilde{w}$. That is to say, $\tilde{w}\in Fix({P}_{S})$.

Hence $\tilde{w}\in Fix({P}_{S})\cap Fix(I-\gamma {G}^{\ast}G)$. By Lemma 1.5, we find that $\tilde{w}$ is a solution of SEP (1.3).

The weak convergence of the whole sequence $\{{w}_{n}\}$ holds true since all conditions of the well-known Opial lemma (Lemma 1.4) are fulfilled with $S=\mathrm{\Gamma}$.

Moreover, if ${R}_{1}$, ${R}_{2}$ are semi-compact, it is easy to prove that *R* is semi-compact, and since $d({u}_{n},R{u}_{n})\to 0$, we get the result that there exists a subsequence of $\{{u}_{{n}_{i}}\}\subseteq \{{u}_{n}\}$ such that ${u}_{{n}_{i}}$ converges strongly to ${w}^{\ast}$. Since ${u}_{{n}_{v}}$ converges weakly to $\tilde{w}$, we have ${w}^{\ast}=\tilde{w}$ and so ${u}_{{n}_{i}}$ converges strongly to $\tilde{w}\in \mathrm{\Gamma}$. From the Féjer-monotonicity of $\{{w}_{n}\}$ and $\parallel {w}_{n+1}-{u}_{n}\parallel =(1-{\alpha}_{n})\parallel {u}_{n}-{v}_{n}\parallel \to 0$, we can find that $\parallel {w}_{n}-\tilde{w}\parallel \to 0$, *i.e.*, $\{{w}_{n}\}$ converges strongly to a solution of the SEP (1.3). □

## 3 Iterative algorithm for MSSEP

In this section, we establish an iterative algorithm that converges strongly to a solution of the following MSSEP (1.4) for a family quasi-nonexpansive multi-valued mappings in infinitely dimensional Hilbert spaces.

Let ${C}_{j}=Fix{R}_{1}^{j}$, ${Q}_{j}=Fix{R}_{2}^{j}$ and ${S}_{j}={C}_{j}\times {Q}_{j}$, $j=1,2,\dots ,N$, $S={\bigcap}_{j=1}^{N}{S}_{j}$. The original problem can now be reformulated as finding $w=(x,y)\in S$ with $Gw=0$, or, more generally, minimizing the function $\parallel Gw\parallel $ over $w\in S$.

**Algorithm 3.1**For an arbitrary initial point ${w}_{0}=({x}_{0},{y}_{0})$, sequence $\{{w}_{n}=({x}_{n},{y}_{n})\}$ is generated by the iteration:

and ${R}_{1}^{i(n)}$, ${R}_{2}^{i(n)}$ are a family of quasi-nonexpansive multi-valued mappings on ${H}_{1}$, ${H}_{2}$, respectively.

The proof of the following lemma is similar to Lemma 1.5, and we omit its proof.

**Lemma 3.2** *Let* $T=I-\gamma {G}^{\ast}G$, *where* $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$. *Then we have* $Fix(T)=\{(x,y)\in H,Ax=By\}$, $Fix({P}_{\bigcap {S}_{j}}T)=Fix({P}_{\bigcap {S}_{j}})\cap Fix(T)=\overline{\mathrm{\Gamma}}$ *and* $\bigcap Fix({P}_{{S}_{j}}T)=\bigcap [Fix({P}_{{S}_{j}})\cap Fix(T)]=\overline{\mathrm{\Gamma}}$.

To prove its convergence we also need the following lemma.

**Lemma 3.3**

*Any sequence*$\{{w}_{n}\}$

*generated by Algorithm*(3.1)

*is the Féjer*-

*monotone with respect to*$\overline{\mathrm{\Gamma}}$,

*namely for every*$w\in \overline{\mathrm{\Gamma}}$,

*provided that* ${\alpha}_{n}>0$ *is a sequence in* $(0,1)$ *and* $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$.

*Proof*Let ${u}_{n}=(I-\gamma {G}^{\ast}G){w}_{n}$ and taking $w\in \overline{\mathrm{\Gamma}}$, by Lemma 3.2, $w\in Fix({P}_{{S}_{j}})\cap Fix(I-\gamma {G}^{\ast}G)$, $\mathrm{\forall}N\ge i\ge 1$, $Gw=0$ and we have

It follows that $\parallel {w}_{n+1}-w\parallel \le \parallel {w}_{n}-w\parallel $, $\mathrm{\forall}w\in \overline{\mathrm{\Gamma}}$, $n\ge 1$. □

**Theorem 3.4** *If* $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$, *then the sequence* $\{{w}_{n}\}$ *generated by Algorithm* (3.1) *converges weakly to a solution of MSSEP* (1.4). *In addition*, *if there exists* $1\le j\le N$ *such that* ${R}_{1}^{j}$, ${R}_{2}^{j}$ *are semi*-*compact*, *then* $\{{w}_{n}\}$ *converges strongly to a solution of MSSEP* (1.4).

*Proof*From (3.2), and the fact that $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$ and $0<\gamma <\lambda =2/\rho ({G}^{\ast}G)$, we obtain

Since ${v}_{n}\in {R}_{i(n)}{u}_{n}$, we get $d({u}_{n},{R}_{i(n)}{u}_{n})\le \parallel {u}_{n}-{v}_{n}\parallel \to 0$.

It follows from the Féjer-monotonicity of $\{{w}_{n}\}$ that the sequence is bounded. Let $\tilde{w}$ be a weak-cluster point of $\{{w}_{n}\}$. Take a subsequence $\{{w}_{{n}_{k}}\}$ of $\{{w}_{n}\}$ such that ${w}_{{n}_{k}}$ converges weakly to $\tilde{w}$. Then, by Lemma 1.3, we obtain $G\tilde{w}=0$, and it follows that $\tilde{w}\in Fix(I-\gamma {G}^{\ast}G)$.

Now, by setting ${u}_{n}=(I-\gamma {G}^{\ast}G){w}_{n}$, it follows that ${u}_{{n}_{k}}$ converges weakly to $\tilde{w}$.

Thus, ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n+1}-{u}_{n}\parallel =0$ and ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n+j}-{u}_{n}\parallel =0$ for all $j=1,2,\dots ,N$.

Hence, ${lim}_{n\to \mathrm{\infty}}d({u}_{n},{R}_{j}{u}_{n})=0$ for all $j=1,2,\dots ,N$. Since ${R}_{1}^{j}$, ${R}_{2}^{j}$ are demi-closed at the origin, it is easy to check that ${R}_{j}$ is demi-closed at the origin, and it follows that $\tilde{w}\in {\bigcap}_{j=1}^{N}Fix{R}_{j}=C\times Q$, *i.e.*, ${P}_{S}(\tilde{w})=\tilde{w}$. That is to say $\tilde{w}\in Fix({P}_{S})$. Hence $\tilde{w}\in Fix({P}_{S})\cap Fix(I-\gamma {G}^{\ast}G)$. By Lemma 3.2, we get that $\tilde{w}$ is a solution of the MSSEP (1.4).

The weak convergence of the whole sequence $\{{w}_{n}\}$ holds true since all conditions of the well-known Opial lemma (Lemma 1.4) are fulfilled with $S=\overline{\mathrm{\Gamma}}$.

Moreover, if ${R}_{1}^{j}$, ${R}_{2}^{j}$ are semi-compact, it is easy to prove that ${R}_{j}$ is semi-compact, since $d({u}_{n},{R}_{j}{u}_{n})\to 0$, we find that there exists a subsequence of $\{{u}_{{n}_{i}}\}\subseteq \{{u}_{n}\}$ such that ${u}_{{n}_{i}}$ converges strongly to ${w}^{\ast}$. Since ${u}_{{n}_{v}}$ converges weakly to $\tilde{w}$, we have ${w}^{\ast}=\tilde{w}$ and so ${u}_{{n}_{i}}$ converges strongly to $\tilde{w}\in \mathrm{\Gamma}$. From the Féjer-monotonicity of $\{{w}_{n}\}$ and $\parallel {w}_{n+1}-{u}_{n}\parallel =(1-{\alpha}_{n})\parallel {u}_{n}-{v}_{n}\parallel \to 0$, we can see that $\parallel {w}_{n}-\tilde{w}\parallel \to 0$, *i.e.*, $\{{w}_{n}\}$ converges strongly to a solution of the MSSEP (1.4). □

## Declarations

### Acknowledgements

This research was supported by NSFC Grants No. 11071279; No. 11226125; No. 11301379.

## Authors’ Affiliations

## References

- Censor Y, Elfving T:
**A multiprojection algorithm using Bregman projections in a product space.***Numer. Algorithms*1994,**8:**221–239. 10.1007/BF02142692MathSciNetView ArticleMATHGoogle Scholar - Censor Y, Elfving T, Kopf N, Bortfeld T:
**The multiple-sets split feasibility problem and its applications for inverse problems.***Inverse Probl.*2005,**21:**2071–2084. 10.1088/0266-5611/21/6/017MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
**Iterative oblique projection onto convex sets and the split feasibility problem.***Inverse Probl.*2002,**18**(2):441–453. 10.1088/0266-5611/18/2/310MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
**A unified treatment of some iterative algorithms in signal processing and image reconstruction.***Inverse Probl.*2004,**20**(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleMATHGoogle Scholar - Qu B, Xiu N:
**A note on the CQ algorithm for the split feasibility problem.***Inverse Probl.*2005,**21**(5):1655–1665. 10.1088/0266-5611/21/5/009MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem.***Inverse Probl.*2006,**22**(6):2021–2034. 10.1088/0266-5611/22/6/007View ArticleMathSciNetMATHGoogle Scholar - Yang Q:
**The relaxed CQ algorithm solving the split feasibility problem.***Inverse Probl.*2004,**20**(4):1261–1266. 10.1088/0266-5611/20/4/014View ArticleMathSciNetMATHGoogle Scholar - Yang Q, Zhao J:
**Generalized KM theorems and their applications.***Inverse Probl.*2006,**22**(3):833–844. 10.1088/0266-5611/22/3/006View ArticleMathSciNetMATHGoogle Scholar - Moudafi A:
**A relaxed alternating CQ-algorithms for convex feasibility problems.***Nonlinear Anal., Theory Methods Appl.*2013,**79:**117–121.MathSciNetView ArticleMATHGoogle Scholar - Shi, LY, Chen, RD, Wu, YJ: Strong convergence of iterative algorithms for the split equality problem. J. Inequal. Appl. (submitted)Google Scholar
- Geobel K, Kirk WA
**Cambridge Studies in Advanced Mathematics 28.**In*Topics in Metric Fixed Point Theory*. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar - Geobel K, Reich S:
*Uniform Convexity, Nonexpansive Mappings, and Hyperbolic Geometry*. Dekker, New York; 1984.Google Scholar - Schöpfer F, Schuster T, Louis AK:
**An iterative regularization method for the solution of the split feasibility problem in Banach spaces.***Inverse Probl.*2008.,**24:**Article ID 055008Google Scholar

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