# Damped projection method for split common fixed point problems

- Huanhuan Cui
^{1}Email author, - Menglong Su
^{1}and - Fenghui Wang
^{1}

**2013**:123

https://doi.org/10.1186/1029-242X-2013-123

© Cui et al.; licensee Springer 2013

**Received: **20 October 2012

**Accepted: **4 March 2013

**Published: **22 March 2013

## Abstract

The paper deals with the split common fixed-point problem (SCFP) introduced by Censor and Segal. Motivated by Eicke’s damped projection method, we propose a cyclic iterative scheme and prove its strong convergence to a solution of SCFP under some mild assumptions. An application of the proposed method to multiple-set split feasibility problems is also included.

## Keywords

## 1 Introduction

*C*and

*Q*are closed convex subsets in Hilbert spaces ℋ and $\mathcal{K}$, respectively. Moreover, if

*C*and

*Q*are the intersections of finitely many closed convex subsets, then the problem is known as the multiple-set split feasibility problem (MSFP) [2]. Note that SFP and MSFP model image retrieval [3] and intensity-modulated radiation therapy [4], and they have recently been investigated by many researchers (see,

*e.g.*, [5–11]). One method for solving SFP is Byrne’s CQ algorithm [5]: For any initial guess ${x}_{1}\in \mathcal{H}$, define $\{{x}_{n}\}$ recursively by

*C*,

*I*is the identity operator on $\mathcal{K}$ and

*λ*is the step-size satisfying $0<\lambda <\frac{2}{{\parallel A\parallel}^{2}}$. By using Hundal’s counterexample, Xu [12] showed the CQ algorithm does not converge strongly in infinite-dimensional spaces. Motivated by Byrne’s CQ algorithm, Wang and Xu [13] proposed the following iterative method: For any initial guess ${x}_{1}\in \mathcal{H}$, define $\{{x}_{n}\}$ recursively by

where $\{{\alpha}_{n}\}\subset (0,1)$ satisfies (i) ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$; (ii) ${lim}_{n\to \mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|=0$; (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1$. It is clear that such an algorithm is an extension of (3). However, algorithm (4) fails to include the original one (3) because of condition (iii).

*C*and

*Q*in (1) are the intersections of finitely many fixed-point sets of nonlinear operators, problem (1) is called by Censor and Segal [17] the split common fixed-point problem (SCFP). More precisely, SCFP requires to seek an element $\stackrel{\u02c6}{x}\in \mathcal{H}$ satisfying

where $\lambda >0$ is known as the step-size. They proved that if *U* and *T* in (6) are directed operators, then *λ* should be chosen in $(0,\frac{2}{{\parallel A\parallel}^{2}})$. Some further generations of this algorithm were studied by Moudafi [18] for demicontractive operators and by Wang-Xu [19] for finitely many directed operators.

We note that the existing algorithms for SCFP have only weak convergence in the framework of infinite-dimensional spaces (see [18, 19]). However, as pointed by Bauschke and Combettes [20], norm convergence of the algorithm is much more desirable than weak convergence in some applied sciences. It is therefore of interest to seek modifications of these algorithms so that strong convergence is guaranteed. Following the damped projection method, we propose in this paper a new iterative scheme and prove its strong convergence to a solution of SCFP. An application of our method to multiple-set split feasibility problems is also included. This enables us to cover some recent results on split feasibility problems.

## 2 Preliminary and notation

Throughout this paper, *I* denotes the identity operator on ℋ, $Fix(T)$ the set of fixed points of an operator *T*, ‘→’ strong convergence, and ‘⇀’ weak convergence. Given a positive integer *p*, denote by $[n]:=(nmodp)$ the mod function taking values in $\{1,2,\dots ,p\}$.

**Definition 1** An operator $T:\mathcal{H}\to \mathcal{H}$ is called *nonexpansive* if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $, $\mathrm{\forall}x,y\in \mathcal{H}$; *firmly nonexpansive* if ${\parallel Tx-Ty\parallel}^{2}\le {\parallel x-y\parallel}^{2}-{\parallel (I-T)x-(I-T)y\parallel}^{2}$, $\mathrm{\forall}x,y\in \mathcal{H}$.

**Definition 2**Assume that $T:\mathcal{H}\to \mathcal{H}$ is a nonlinear operator. Then $I-T$ is said to be

*demiclosed at zero*, if, for any $\{{x}_{n}\}$ in ℋ, the following implication holds:

Clearly, firm nonexpansiveness implies nonexpansiveness. It is well known that nonexpansive operators are demiclosed at zero (*cf.* [21]).

**Definition 3** Let $T:\mathcal{H}\to \mathcal{H}$ be an operator with $Fix(T)\ne \mathrm{\varnothing}$. Then *T* is called *directed* if $\u3008z-Tx,x-Tx\u3009\le 0$, $\mathrm{\forall}z\in Fix(T)$, $x\in \mathcal{H}$; *ν*-*demicontractive* with $\nu \in (-\mathrm{\infty},1)$ if ${\parallel Tx-z\parallel}^{2}\le {\parallel x-z\parallel}^{2}+\nu {\parallel (I-T)x\parallel}^{2}$, $\mathrm{\forall}z\in Fix(T)$, $x\in \mathcal{H}$.

**Lemma 1** (Bauschke-Combettes [20])

*An operator*$T:\mathcal{H}\to \mathcal{H}$

*is directed if and only if one of following inequalities holds for all*$z\in Fix(T)$

*and*$x\in \mathcal{H}$:

**Lemma 2** (Wang-Xu [19])

*Assume that*$A:\mathcal{H}\to \mathcal{K}$

*is a bounded linear operator and*$T:\mathcal{K}\to \mathcal{K}$

*is a directed operator*.

*Let*${V}_{\lambda}=I-\lambda {A}^{\ast}(I-T)A$

*with*$\lambda >0$.

*Then*

*whenever* ${A}^{-1}(Fix(T)):=\{x\in \mathcal{H}:Ax\in Fix(T)\}$ *is nonempty*.

**Lemma 3**

*Assume that*$A:\mathcal{H}\to \mathcal{K}$

*is a bounded linear operator and*$T:\mathcal{K}\to \mathcal{K}$

*is a directed operator*.

*Let*${V}_{\lambda}=I-\lambda {A}^{\ast}(I-T)A$

*with*$0<\lambda <\frac{2}{{\parallel A\parallel}^{2}}$.

*If*${A}^{-1}(Fix(T))$

*is nonempty*,

*then*

*for all* $z\in {A}^{-1}(Fix(T))$ *and* $x\in \mathcal{H}$.

*Proof*Since $Az\in Fix(T)$, it follows from (8) that

Hence the proof is complete. □

We end this section by a useful lemma.

**Lemma 4** (Xu [22])

*Let*$\{{a}_{n}\}$

*be a nonnegative real sequence satisfying*

*where*$\{{\alpha}_{n}\}\subset (0,1)$

*and*$\{{b}_{n}\}$

*are real sequences*.

*Then*${a}_{n}\to 0$

*provided that*

- (i), ${lim}_{n}{\alpha}_{n}=0$,${\sum}_{n}{\alpha}_{n}=\mathrm{\infty}$
- (ii)${\stackrel{\u203e}{lim}}_{n}{b}_{n}\le 0$
*or*$\sum {\alpha}_{n}|{b}_{n}|<\mathrm{\infty}$.

## 3 Algorithm and its convergence analysis

In this section, we consider the following problem.

**Problem 1**Find an element $\stackrel{\u02c6}{x}\in \mathcal{H}$ satisfying

where *p* is a positive integer and ${({U}_{i})}_{i=1}^{p}$, ${({T}_{i})}_{i=1}^{p}$ are two classes of directed operators such that ${U}_{i}-I$ and ${T}_{i}-I$ are demiclosed at zero for every $i=1,2,\dots ,p$.

where ${U}_{n}:={U}_{[n]}$, ${T}_{n}:={T}_{[n]}$ and $\{{\alpha}_{n}\}\subset (0,1)$, $\{{\beta}_{n}\}\subseteq [0,1]$, $\{{\lambda}_{n}\}\subseteq {\mathbb{R}}^{+}$ are properly chosen real sequences.

**Theorem 1**

*Assume that the following conditions hold*:

- (i),${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}>0$
- (ii), ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$,${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$
- (iii).$0<\underline{\lambda}\le {\lambda}_{n}\le \overline{\lambda}<\frac{2}{{\parallel A\parallel}^{2}}$

*If the solution set of problem* (11) *denoted by* Ω *is nonempty*, *then the sequence* $\{{x}_{n}\}$ *generated by* (12) *converges strongly to* ${P}_{\mathrm{\Omega}}(0)$.

*Proof*We first show the boundedness of $\{{x}_{n}\}$. To see this, let $z={P}_{\mathrm{\Omega}}(0)$ and set ${V}_{n}=I-{\lambda}_{n}{A}^{\ast}(I-{T}_{n})A$, ${y}_{n}=(1-{\alpha}_{n}){V}_{n}{x}_{n}$. Hence

By induction, the sequence $\{{x}_{n}\}$ is bounded, and so is $\{{y}_{n}\}$.

Adding up (14)-(16), we thus get inequality (13).

Finally, we prove ${s}_{n}\to 0$. To see this, let $\{{s}_{{n}_{k}}\}$ be a subsequence such that it includes all elements in $\{{s}_{n}\}$ with the property: each of them is less than or equal to the term after it. Following an idea developed by Maingé [23], we consider two possible cases on such a sequence.

*k*. Since $\parallel (I-{U}_{i}){y}_{{m}_{k}}\parallel =\parallel (I-{U}_{{m}_{k}}){y}_{{m}_{k}}\parallel \to 0$, we thus use the demiclosedness of $I-{U}_{i}$ at zero to conclude that ${y}^{\prime}\in Fix({U}_{i})$. On the other hand, we deduce from (8) that

*A*yields that $A{x}_{{m}_{k}}\rightharpoonup A{y}^{\prime}$, which together with the demiclosedness of $I-{T}_{i}$ at zero enables us to deduce $A{y}^{\prime}\in Fix({T}_{i})$. Since the index

*i*is arbitrary, we therefore conclude ${y}^{\prime}\in \mathrm{\Omega}$. Consequently,

We therefore apply Lemma 4 to conclude ${s}_{n}\to 0$.

which immediately implies ${s}_{{n}_{k}+1}\to 0$. Consequently, ${s}_{n}\to 0$ follows from (17) and the proof is complete. □

where $U:\mathcal{H}\to \mathcal{H}$ and $T:\mathcal{K}\to \mathcal{K}$ are directed operators so that $U-I$ and $T-I$ are demiclosed at zero.

**Corollary 2**

*Suppose that the following conditions hold*:

- (i), ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$,${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$
- (ii).$0<\underline{\lambda}\le {\lambda}_{n}\le \overline{\lambda}<\frac{2}{{\parallel A\parallel}^{2}}$

*Then the sequence*$\{{x}_{n}\}$,

*generated by*

*converges strongly to* ${P}_{\mathrm{\Omega}}(0)$, *whenever such point exists*.

## 4 Some applications

In this section, we extend our result to SCFP for demicontractive operators recently considered by Moudafi [18].

**Problem 2**Find an element $\stackrel{\u02c6}{x}\in \mathcal{H}$ satisfying

where *p* is a positive integer and ${({U}_{i})}_{i=1}^{p}$, ${({T}_{i})}_{i=1}^{p}$ are respectively ${\nu}_{i}$-demicontractive and ${\kappa}_{i}$-demicontractive operator so that ${U}_{i}-I$ and ${T}_{i}-I$ are demiclosed at zero for every $i=1,2,\dots ,p$.

The following lemma states a relation between directed and demicontractive operators.

**Lemma 5** *Let* $\nu \in (-\mathrm{\infty},1)$ *and* $\tau \in (0,\frac{1-\nu}{2}]$. *If* *T* *is* *ν*-*demicontractive*, *then* ${T}_{\tau}:=(1-\tau )I+\tau T$ *is directed*.

*Proof*For $\mathrm{\forall}z\in Fix(T)$, we deduce that

Then the result follows from Lemma 1. □

where $\{{\alpha}_{n}\}\subset (0,1)$, ${U}_{{\tau}_{n}}=(1-{\tau}_{[n]})I+{\tau}_{[n]}{U}_{[n]}$ and ${T}_{{\gamma}_{n}}=(1-{\gamma}_{[n]})I+{\gamma}_{[n]}{T}_{[n]}$. By using the previous lemma, we can easily extend our result to demicontractive operators.

**Theorem 3**

*Let*$0<{\tau}_{i}\le \frac{1-{\nu}_{i}}{2}$

*and*$0<{\gamma}_{i}\le \frac{1-{\kappa}_{i}}{2}$

*for every*$i=1,2,\dots ,p$.

*Assume that the following conditions hold*:

- (i), ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$,${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$
- (ii).$0<\lambda <min\{\frac{1-{\kappa}_{i}}{2{\gamma}_{i}{\parallel A\parallel}^{2}}:1\le i\le p\}$

*If the solution set of problem* (22) *denoted by* Ω *is nonempty*, *then the sequence* $\{{x}_{n}\}$ *generated by* (23) *converges strongly to* ${P}_{\mathrm{\Omega}}(0)$.

**Remark 1** Theorem 3 also holds true if we relax hypothesis (ii) above as $0<\lambda <min\{\frac{1-{\kappa}_{i}}{{\gamma}_{i}{\parallel A\parallel}^{2}}:1\le i\le p\}$.

where ${C}_{n}:={C}_{[n]}$, ${Q}_{n}:={Q}_{[n]}$, and $\{{\alpha}_{n}\}\subset (0,1)$, $\{{\beta}_{n}\}\subseteq [0,1]$, $\{{\lambda}_{n}\}\subseteq {\mathbb{R}}^{+}$ are properly chosen real sequences.

**Theorem 4**

*Assume that the following conditions hold*:

- (i),${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}>0$
- (ii), ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$,${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$
- (iii).$0<\underline{\lambda}\le {\lambda}_{n}\le \overline{\lambda}<\frac{2}{{\parallel A\parallel}^{2}}$

*If the solution set of MSFP denoted by* Ω *is nonempty*, *then the sequence* $\{{x}_{n}\}$ *generated by* (25) *converges strongly to* ${P}_{\mathrm{\Omega}}(0)$.

*Proof* We note that the metric projection ${P}_{C}$ is firmly nonexpansive, which implies ${P}_{C}$ is directed and $I-{P}_{C}$ is demiclosed at zero. Hence, by using Theorem 1, one can immediately get the desired result. □

**Remark 2** Theorem 4 covers [[16], Theorem 3.1], and we relax the condition on $\{{\beta}_{n}\}$ as ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}>0$. Moreover, the choice of variable $\{{\lambda}_{n}\}$ is more flexible than the fixed one. Also, we cover the result of [19] and remove one condition posed on $\{{\alpha}_{n}\}$: either ${\sum}_{n=1}^{\mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|<\mathrm{\infty}$ or ${lim}_{n\to \mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|/{\alpha}_{n}=0$.

## Declarations

### Acknowledgements

We would like to express our sincere thanks to the referees for their valuable suggestions. This work is supported by the National Natural Science Foundation of China, Tianyuan Foundation (11226227), the Basic Science and Technological Frontier Project of Henan (122300410268) and the Foundation of Henan Educational Committee (12A110016).

## Authors’ Affiliations

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