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Alternating mann iterative algorithms for the split common fixedpoint problem of quasinonexpansive mappings
 Jing Zhao^{1}Email author and
 Songnian He^{1}
https://doi.org/10.1186/168718122013288
© Zhao and He; licensee Springer. 2013
 Received: 29 May 2013
 Accepted: 25 September 2013
 Published: 8 November 2013
Abstract
Very recently, Moudafi (Alternating CQalgorithms for convex feasibility and split fixedpoint problems, J. Nonlinear Convex Anal. ) introduced an alternating CQalgorithm with weak convergence for the following split common fixedpoint problem. Let ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ be real Hilbert spaces, let $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators.
where $U:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ are two firmly quasinonexpansive operators with nonempty fixedpoint sets $F(U)=\{x\in {H}_{1}:Ux=x\}$ and $F(T)=\{x\in {H}_{2}:Tx=x\}$. Note that by taking ${H}_{2}={H}_{3}$ and $B=I$, we recover the split common fixedpoint problem originally introduced in Censor and Segal (J. Convex Anal. 16:587600, 2009) and used to model many significant realworld inverse problems in sensor networks and radiation therapy treatment planning. In this paper, we will continue to consider the split common fixedpoint problem (1) governed by the general class of quasinonexpansive operators. We introduce two alternating Mann iterative algorithms and prove the weak convergence of algorithms. At last, we provide some applications. Our results improve and extend the corresponding results announced by many others.
MSC:47H09, 47H10, 47J05, 54H25.
Keywords
 split common fixedpoint problem
 quasinonexpansive mapping
 weak convergence
 Mann iterative algorithm
 Hilbert space
1 Introduction
where $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator. The SFP in finitedimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. The SFP attracts many authors’ attention due to its application in signal processing. Various algorithms have been invented to solve it (see [3–12] and references therein).
That is, ${x}^{\ast}$ solves the SFP (1.1) if and only if ${x}^{\ast}$ solves the fixed point equation (1.2) (see [13] for the details). This implies that we can use fixed point algorithms (see [6, 13–15]) to solve SFP. A popular algorithm that solves the SFP (1.1) is due to Byrne’s CQ algorithm [2], which is found to be a gradientprojection method (GPM) in convex minimization. Subsequently, Byrne [3] applied KM iteration to the CQ algorithm and Zhao [16] applied KM iteration to the perturbed CQ algorithm to solve the SFP.
which allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, for instance, in decomposition methods for PDEs, applications in game theory and in intensitymodulated radiation therapy (IMRT). In decision sciences, this allows to consider agents who interplay only via some components of their decision variables (see [18]). In (IMRT), this amounts to envisaging a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity (see [19]). If ${H}_{2}={H}_{3}$ and $B=I$, then the convex feasibility problem (1.3) reduces to the split feasibility problem (1.1).
where ${\gamma}_{k}\in (\epsilon ,min(\frac{1}{{\lambda}_{A}},\frac{1}{{\lambda}_{B}})\epsilon )$, ${\lambda}_{A}$ and ${\lambda}_{B}$ are the spectral radiuses of ${A}^{\ast}A$ and ${B}^{\ast}B$, respectively.
where ${u}_{k}={x}_{k}+\gamma \beta {A}^{\ast}(TI)A{x}_{k}$, $\beta \in (0,1)$, ${\alpha}_{k}\in (0,1)$ and $\gamma \in (0,\frac{1}{\lambda \beta})$, with λ being the spectral radius of the operator ${A}^{\ast}A$. Moudafi proved weak convergence result of the algorithm in Hilbert spaces.
for firmly quasinonexpansive operators U and T. Moudafi [17] obtained the following result.
Theorem 1.1 Let ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ be real Hilbert spaces, let $U:{H}_{1}\to {H}_{1}$, $T:{H}_{2}\to {H}_{2}$ be two firmly quasinonexpansive operators such that $IU$, $IT$ are demiclosed at 0. Let $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators. Assume that the solution set Γ is nonempty, $({\gamma}_{k})$ is a positive nondecreasing sequence such that ${\gamma}_{k}\in (\epsilon ,min(\frac{1}{{\lambda}_{A}},\frac{1}{{\lambda}_{B}})\epsilon )$, where ${\lambda}_{A}$, ${\lambda}_{B}$ stand for the spectral radiuses of ${A}^{\ast}A$ and ${B}^{\ast}B$, respectively. Then the sequence $({x}_{k},{y}_{k})$ generated by (1.9) weakly converges to a solution $(\overline{x},\overline{y})$ of (1.8). Moreover, $\parallel A{x}_{k}B{y}_{k}\parallel \to 0$, $\parallel {x}_{k}{x}_{k+1}\parallel \to 0$, and $\parallel {y}_{k}{y}_{k+1}\parallel \to 0$ as $k\to \mathrm{\infty}$.
In this paper, inspired and motivated by the works mentioned above, firstly, we introduce the following alternating Mann iterative algorithm for solving the SCFP (1.8) for the general class of quasinonexpansive operators.
By taking $B=I$, we recover (1.8) clearly the classical SCFP (1.6). In addition, if ${\gamma}_{k}=1$ and ${\beta}_{k}=\beta \in (0,1)$ in Algorithm 1.1, we have ${v}_{k+1}=A{x}_{k+1}$ and ${y}_{k+1}={\beta}_{k}A{x}_{k+1}+(1{\beta}_{k})T(A{x}_{k+1})$. Thus, Algorithm 1.1 reduces to ${u}_{k}={x}_{k}+(1\beta ){A}^{\ast}(TI)A{x}_{k}$ and ${x}_{k+1}={\alpha}_{k}{u}_{k}+(1{\alpha}_{k})U({u}_{k})$, which is algorithm (1.7) proposed by Moudafi [22].
The CQ algorithm is a special case of the KM algorithm. Due to the fixed point formulation (1.4) of the CFP (1.3), we can apply the KM algorithm to obtain the following other alterative Mann iterative sequence for solving the SCFP (1.8) for quasinonexpansive operators.
The organization of this paper is as follows. Some useful definitions and results are listed for the convergence analysis of the iterative algorithm in Section 2. In Section 3, we prove the weak convergence of the alternating Mann iterative Algorithms 1.1 and 1.2. At last, we provide some applications of Algorithms 1.1 and 1.2.
2 Preliminaries

A mapping $T:H\to H$ belongs to the general class ${\mathrm{\Phi}}_{Q}$ of (possibly discontinuous) quasinonexpansive mappings if$\parallel Txq\parallel \le \parallel xq\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,q)\in H\times F(T).$

A mapping $T:H\to H$ belongs to the set ${\mathrm{\Phi}}_{N}$ of nonexpansive mappings if$\parallel TxTy\parallel \le \parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,y)\in H\times H.$

A mapping $T:H\to H$ belongs to the set ${\mathrm{\Phi}}_{FN}$ of firmly nonexpansive mappings if${\parallel TxTy\parallel}^{2}\le {\parallel xy\parallel}^{2}{\parallel (xy)(TxTy)\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,y)\in H\times H.$

A mapping $T:H\to H$ belongs to the set ${\mathrm{\Phi}}_{FQ}$ of firmly quasinonexpansive mappings if${\parallel Txq\parallel}^{2}\le {\parallel xq\parallel}^{2}{\parallel xTx\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,q)\in H\times F(T).$
It is easily observed that ${\mathrm{\Phi}}_{FN}\subset {\mathrm{\Phi}}_{N}\subset {\mathrm{\Phi}}_{Q}$ and that ${\mathrm{\Phi}}_{FN}\subset {\mathrm{\Phi}}_{FQ}\subset {\mathrm{\Phi}}_{Q}$. Furthermore, ${\mathrm{\Phi}}_{FN}$ is well known to include resolvents and projection operators, while ${\mathrm{\Phi}}_{FQ}$ contains subgradient projection operators (see, for instance, [23] and the reference therein).
A mapping $T:H\to H$ is called demiclosed at the origin if, for any sequence $\{{x}_{n}\}$ which weakly converges to x, and if the sequence $\{T{x}_{n}\}$ strongly converges to 0, then $Tx=0$.
In what follows, we give some key properties of the relaxed operator ${T}_{\alpha}=\alpha I+(1\alpha )T$ which will be needed in the convergence analysis of our algorithms.
Lemma 2.1 ([22])
 (i)
$\u3008xTx,xq\u3009\ge \frac{1}{2}{\parallel xTx\parallel}^{2}$ and $\u3008xTx,qTx\u3009\le \frac{1}{2}{\parallel xTx\parallel}^{2}$;
 (ii)
${\parallel {T}_{\alpha}xq\parallel}^{2}\le {\parallel xq\parallel}^{2}\alpha (1\alpha ){\parallel Txx\parallel}^{2}$;
 (iii)
$\u3008x{T}_{\alpha}x,xq\u3009\ge \frac{1\alpha}{2}{\parallel xTx\parallel}^{2}$.
Remark 2.2 Let ${T}_{\alpha}=\alpha I+(1\alpha )T$, where $T:H\to H$ is a quasinonexpansive mapping and $\alpha \in [0,1)$. We have $F({T}_{\alpha})=F(T)$ and ${\parallel {T}_{\alpha}xx\parallel}^{2}={(1\alpha )}^{2}{\parallel Txx\parallel}^{2}$. It follows from (ii) of Lemma 2.1 that ${\parallel {T}_{\alpha}xq\parallel}^{2}\le {\parallel xq\parallel}^{2}\frac{\alpha}{1\alpha}{\parallel {T}_{\alpha}xx\parallel}^{2}$, which implies that ${T}_{\alpha}$ is firmly quasinonexpansive when $\alpha =\frac{1}{2}$. On the other hand, if $\stackrel{\u02c6}{T}$ is a firmly quasinonexpansive mapping, we can obtain $\stackrel{\u02c6}{T}=\frac{1}{2}I+\frac{1}{2}T$, where T is quasinonexpansive. This is proved by the following inequalities.
where $\stackrel{\u02c6}{T}$ is firmly quasinonexpansive.
Lemma 2.3 ([24])
3 Convergence of the alternating Mann iterative Algorithms 1.1 and 1.2
Theorem 3.1 Let ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ be real Hilbert spaces. Given two bounded linear operators $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$, let $U:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ be quasinonexpansive mappings with nonempty fixed point set $F(U)$ and $F(T)$. Assume that $UI$, $TI$ are demiclosed at origin, and the solution set Γ of (1.8) is nonempty. Let $\{{\gamma}_{k}\}$ be a positive nondecreasing sequence such that ${\gamma}_{k}\in (\epsilon ,min(\frac{1}{{\lambda}_{A}},\frac{1}{{\lambda}_{B}})\epsilon )$, where ${\lambda}_{A}$, ${\lambda}_{B}$ stand for the spectral radiuses of ${A}^{\ast}A$ and ${B}^{\ast}B$, respectively, and ε is small enough. Then, the sequence $\{({x}_{k},{y}_{k})\}$ generated by Algorithm 1.1 weakly converges to a solution $({x}^{\ast},{y}^{\ast})$ of (1.8), provided that $\{{\alpha}_{k}\}\subset (\delta ,1\delta )$ and $\{{\beta}_{k}\}\subset (\sigma ,1\sigma )$ for small enough $\delta ,\sigma >0$. Moreover, $\parallel A{x}_{k}B{y}_{k}\parallel \to 0$, $\parallel {x}_{k}{x}_{k+1}\parallel \to 0$ and $\parallel {y}_{k}{y}_{k+1}\parallel \to 0$ as $k\to \mathrm{\infty}$.
which ensures that both sequences $\{{x}_{k}\}$ and $\{{y}_{k}\}$ are bounded thanks to the fact that $\{{\rho}_{k}(x,y)\}$ converges to a finite limit.
hence $({x}^{\ast},{y}^{\ast})\in \mathrm{\Gamma}$.
Hence ${x}^{\ast}=\overline{x}$ and ${y}^{\ast}=\overline{y}$, this implies that the whole sequence $\{({x}_{k},{y}_{k})\}$ weakly converges to a solution of problem (1.8), which completes the proof. □
Remark 3.2 Taking ${\alpha}_{n}={\beta}_{n}=\frac{1}{2}$ in Algorithm 1.1, it follows from Remark 2.2 that Theorem 3.1 becomes Theorem 1.1, which is proved by Moudafi [17].
Theorem 3.3 Let ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ be real Hilbert spaces. Given two bounded linear operators $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$, let $U:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ be quasinonexpansive mappings with nonempty fixed point set $F(U)$ and $F(T)$. Assume that $UI$, $TI$ are demiclosed at origin, and the solution set Γ of (1.8) is nonempty. Let $\{{\gamma}_{k}\}$ be a positive nondecreasing sequence such that ${\gamma}_{k}\in (\epsilon ,min(\frac{1}{{\lambda}_{A}},\frac{1}{{\lambda}_{B}})\epsilon )$, where ${\lambda}_{A}$, ${\lambda}_{B}$ stand for the spectral radiuses of ${A}^{\ast}A$ and ${B}^{\ast}B$, respectively, and ε is small enough. Then the sequence $\{({x}_{k},{y}_{k})\}$ generated by Algorithm 1.2 weakly converges to a solution $({x}^{\ast},{y}^{\ast})$ of (1.8), provided that $\{{\alpha}_{k}\}$ is an nonincreasing sequence such that $\{{\alpha}_{k}\}\subset (\delta ,1\delta )$ for small enough $\delta >0$. Moreover, $\parallel A{x}_{k}B{y}_{k}\parallel \to 0$, $\parallel {x}_{k}{x}_{k+1}\parallel \to 0$ and $\parallel {y}_{k}{y}_{k+1}\parallel \to 0$ as $k\to \mathrm{\infty}$.
Proof Taking $(x,y)\in \mathrm{\Gamma}$, i.e., $x\in F(U)$; $y\in F(T)$ and $Ax=By$. By repeating the proof of Theorem 3.1, we have that (3.6) is true.
that $\{{x}_{k}\}$ is asymptotically regular. Similarly, ${lim}_{k\to \mathrm{\infty}}\parallel {v}_{k+1}{y}_{k}\parallel =0$ and $\{{y}_{k}\}$ is asymptotically regular, too.
The rest of the proof is analogous to that of Theorem 3.1. □
4 Applications
We now turn our attention to providing some applications relying on some convex and nonlinear analysis notions, see, for example, [25].
4.1 Convex feasibility problem (1.3)
Taking $U={P}_{C}$ and $T={P}_{Q}$, we have the following alterative Mann iterative algorithms for CFP (1.3).
4.2 Variational problems via resolvent mappings
and the algorithms take the following equivalent form.
Declarations
Acknowledgements
The research was supported by the Fundamental Research Funds for the Central Universities (Program No. 3122013k004), it was also supported by the science research foundation program in the Civil Aviation University of China (2012KYM04). The authors would also like to thank the referees for careful reading of the manuscript.
Authors’ Affiliations
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