Iterative methods for the split common fixed point problem in Hilbert spaces
 Huanhuan Cui^{1} and
 Fenghui Wang^{1}Email author
https://doi.org/10.1186/16871812201478
© Cui and Wang; licensee Springer. 2014
Received: 30 September 2013
Accepted: 11 March 2014
Published: 25 March 2014
Abstract
The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed point set. Recently Censor and Segal proposed an efficient algorithm for solving such a problem. However, to employ their algorithm, one needs to know prior information on the norm of the bounded linear operator. In this paper we propose a new algorithm that does not need any prior information of the operator norm, and we establish the weak convergence of the proposed algorithm under some mild assumptions.
MSC:47J25, 47J20, 49N45, 65J15.
Keywords
1 Introduction
where $Fix(U)$ and $Fix(T)$ stand for, respectively, the fixed point sets of $U:H\to H$ and $T:K\to K$.
solving problem (1.2) for directed operators. Subsequently, this algorithm was extended to the case of quasinonexpansive [6] operators, demicontractive operators [7], and finitely many directed operators [8].
algorithm (1.4) converges to a solution to problem (1.2) whenever such a solution exists. However, in order to implement this algorithm, one has first to compute (or, at least, estimate) the norm of A, which is in general not an easy work in practice. A natural question thus arises: Does there exist a way to select the step ${\rho}_{n}$ in algorithm (1.4) that does not depend on the operator norm $\parallel A\parallel $?
It is the purpose of this paper to answer the above question affirmatively. By introducing a new way of selections of the step, we obtain a method in a way that the implementation of algorithm (1.4) does not need any prior information of the operator norm. By using the Fejér monotonicity, we state the weak convergence of the new algorithm for demicontractive operators. Particular cases such as quasinonexpansive and directed operators are also considered.
2 Preliminary and notation
where $A:H\to K$ is a linear bounded operator.
It is well known that nonexpansive operators are demiclosed at zero (cf. [9]). Recall that an operator $T:H\to H$ is called nonexpansive if $\parallel TxTy\parallel \le \parallel xy\parallel $, $\mathrm{\forall}x,y\in H$.
 (i)$T:H\to H$ is called directed if$\u3008zTx,xTx\u3009\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}z\in Fix(T),x\in H;$
 (ii)$T:H\to H$ is called quasinonexpansive if$\parallel Txz\parallel \le \parallel xz\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}z\in Fix(T),x\in H;$
 (iii)$T:H\to H$ is called τdemicontractive with $\tau <1$, if${\parallel Txz\parallel}^{2}\le {\parallel xz\parallel}^{2}+\tau {\parallel (IT)x\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}z\in Fix(T),x\in H,$
The sequence with Fejér monotonicity has the following property.
Lemma 2.3 [10]
where $x\in H$, $z\in Fix(U)$, and ${U}_{\lambda}:=(1\lambda )I+\lambda U$ ($0<\lambda <1\kappa $).
which is the inequality as desired. □
where the inequality follows from (2.1), so that $Ax=T(Ax)$. Hence the proof is complete. □
where the inequality follows from (2.1). □
3 A new iterative algorithm
Let us first consider the SCFP (1.2) for demicontractive operators. More specifically, we make use of the following assumptions:

$U:H\to H$ is κdemicontractive with $\kappa <1$;

$T:K\to K$ is τdemicontractive with $\tau <1$;

both $IU$ and $IT$ are demiclosed at zero;

it is consistent, i.e., its solution set, denoted by S, is nonempty.
Under these conditions, we propose the following algorithm.
Remark 3.2 By Lemma 2.5, we see that $A{x}_{n}=T(A{x}_{n})$ if and only if ${A}^{\ast}(IT)A{x}_{n}=0$. So Algorithm 3.1 is well defined.
Theorem 3.3 Let $({x}_{n})$ be the sequence generated by Algorithm 3.1. Then $({x}_{n})$ converges weakly to a solution ${x}^{\ast}\in S$.
Hence we have shown in both cases that $\parallel {x}_{n+1}z\parallel \le \parallel {x}_{n}z\parallel $. Consequently $({x}_{n})$ is Féjermonotone w.r.t. S and $({\parallel {x}_{n}z\parallel}^{2})$ is therefore a convergent sequence.
Having in mind that $\parallel (IU){y}_{n}\parallel \to 0$, we conclude that $\parallel (IU){x}_{n}\parallel \to 0$. Consequently (3.3) holds in both cases.
Finally, we show that $({x}_{n})$ converges weakly to ${x}^{\ast}\in S$. By Lemma 2.3, it remains to show that ${\omega}_{w}({x}_{n})\subseteq S$. To see this let $\stackrel{\u02c6}{x}\in {\omega}_{w}({x}_{n})$ and let $\{{x}_{{n}_{k}}\}$ be a subsequence of $({x}_{n})$ converging weakly to $\stackrel{\u02c6}{x}$. By noting that $\parallel (IU){x}_{{n}_{k}}\parallel \to 0$, we then make use of the demiclosedness of $IU$ at zero to deduce that $\stackrel{\u02c6}{x}\in Fix(U)$; on the other hand, since, by weak continuity of A, $A{x}_{{n}_{k}}$ converges weakly to $A\stackrel{\u02c6}{x}$ and $\parallel (IT)A{x}_{{n}_{k}}\parallel \to 0$, this, together with the demiclosedness of $IT$ at zero, yields $A\stackrel{\u02c6}{x}\in Fix(T)$. Altogether $\stackrel{\u02c6}{x}\in S$, and therefore the proof is complete. □
Hence, ${x}^{\ast}=\stackrel{\u02c6}{x}$ and therefore ${x}^{\ast}={lim}_{n\to \mathrm{\infty}}{P}_{S}{x}_{n}$.
4 Some special cases
4.1 The case for quasinonexpansive operators
Consider now the SCFP (1.2) under the following assumptions:

$U:H\to H$ and $T:K\to K$ are both quasinonexpansive;

both $IU$ and $IT$ are demiclosed at zero;

it is consistent, i.e., its solution set, denoted by S, is nonempty.
Since every quasinonexpansive operator is clearly 0demicontractive, we can state the following result by using Algorithm 3.1.
Corollary 4.2 Let $({x}_{n})$ be the sequence generated by Algorithm 4.1. Then $({x}_{n})$ converges weakly to a solution ${x}^{\ast}\in S$.
4.2 The case for directed operators
Let us consider the SCFP (1.2) under the following assumptions:

$U:H\to H$ and $T:K\to K$ are both directed;

$IU$ and $IT$ are both demiclosed at zero;

it is consistent, i.e., its solution set, denoted by S, is nonempty.
A simple calculation shows that every directed operator is −1demicontractive. Thus we can state the following result by using Algorithm 3.1.
Corollary 4.4 Let $({x}_{n})$ be the sequence generated by Algorithm 4.3. Then $({x}_{n})$ converges weakly to a solution ${x}^{\ast}\in S$.
Remark 4.5 Algorithm 4.3 covers the algorithm studied in [11] for solving the SFP. One can further apply the above result to the split variational inequality problem [12, 13] and the split common null point problem [14].
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11301253, 11271112).
Authors’ Affiliations
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