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New algorithms designed for the split common fixed point problem of quasipseudocontractions
Journal of Inequalities and Applications volume 2014, Article number: 304 (2014)
Abstract
In this paper, we study the split common fixed point problem, which is to find a fixed point of a quasipseudocontractive mapping in one space whose image under a linear transformation is a fixed point of anther quasipseudocontractive mapping in the image space. We design and analyze a new iterative algorithm for solving this split common fixed point problem. A weak convergence theorem is given.
MSC:49J53, 49M37, 65K10, 90C25.
1 Background and motivation
Let C and Q be nonempty closed convex subsets of real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively. The split feasibility problem is formulated as finding a point ${x}^{\ast}$ with the property
where $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator. The split feasibility problem 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].
A special case of the split feasibility problem (1.1) is when $Q=\{b\}$ is singleton and then (1.1) is reduced to the convexly constrained linear inverse problem [3]
which has received considerable attention.
The wellknown projected Landweber algorithm [4] is widely used to solve (1.2). This algorithm generates a sequence $\{{x}_{n}\}$ in such a way that we have

initialization: ${x}_{0}$ selected in ${H}_{1}$ arbitrarily, and

iteration:
$${x}_{n+1}={P}_{C}({x}_{n}+\gamma {A}^{T}(bA{x}_{n})),$$(1.3)
where ${P}_{C}$ denotes the nearest point projection from ${H}_{1}$ onto C, $\gamma >0$ is a parameter such that $0<\gamma <2/{\parallel A\parallel}^{2}$, and ${A}^{T}$ is the transpose of A.
When the system (1.2) is reduced to the unconstrained linear system
then the projected Landweber algorithm [4] is turned to the Landweber algorithm:
The simultaneous algebraic reconstruction technique is a typical example of the Landweber algorithm (1.5) when the system (1.4) is finitedimensional.
The first iterative algorithm for solving the split feasibility problem (1.1) in the finitedimensional case is proposed by Censor and Elfving [1] who define a sequence ${x}_{n}$ by the recursion:
where C and Q are closed convex sets of ${\mathbb{R}}^{n}$, and A is an $n\times n$ matrix of full rank. Here $A(C)=\{y\in {\mathbb{R}}^{n}:y=Ax,x\in C\}$ is the image of C under the matrix A.
Because of the presence of the inverse ${A}^{1}$, the algorithm (1.6) has not become popular. A more popular algorithm that solves the split feasibility problem (1.1) is the socalled CQ algorithm introduced by Byrne [2]. This algorithm, which does not involve ${A}^{1}$, generates a sequence $\{{x}_{n}\}$ as follows:
where $0<\gamma <2/{\parallel A\parallel}^{2}$ and ${P}_{Q}$ denotes the nearest point projection from ${H}_{2}$ onto Q. Consequently, Xu [5] extend the above results from the finitedimensional spaces to the infinitedimensional spaces.
In the case where C and Q in (1.1) are the intersections of finitely many fixed point sets of nonlinear operators, problem (1.1) is called by Censor and Segal [6] the split common fixed point problem. More precisely, the split common fixed point problem requires one to seek an element ${x}^{\ast}\in H$ satisfying
where $Fix({S}_{i})$ and $Fix({T}_{j})$ denote the fixed point sets of two classes of nonlinear operators ${S}_{i}:{H}_{1}\to {H}_{1}$ and ${T}_{j}:{H}_{2}\to {H}_{2}$. In this situation, Byrne’s CQ algorithm does not work because the metric projection onto fixed point sets is generally not easy to calculate. To solve the twoset split common fixed point problem, motivated by the algorithms (1.3) and (1.7), Censor and Segal [6] proposed the following iterative method: For any initial guess ${x}_{1}\in {H}_{1}$, define $\{{x}_{n}\}$ recursively by
where U and T are directed operators and $\lambda >0$ is known as the stepsize. They proved that if $\lambda \in (0,\frac{2}{{\parallel A\parallel}^{2}})$, then (1.9) converges to a split common fixed point ${x}^{\ast}\in \mathrm{\Gamma}=\{x\in Fix(U);Ax\in Fix(T)\}$. Consequently, Moudafi [7] extended (1.9) to the following relaxed algorithm:
where U and T are demicontractive operators, $\beta \in (0,1)$, $\gamma \in (0,\frac{1\mu}{\lambda})$ with λ being the spectral radius of the operator ${A}^{\ast}A$ and ${\alpha}_{n}\in (0,1)$ is relaxation parameter. We note that the classes of directed and demicontractive operators are important classes since they include the orthogonal projections and the subgradient projectors. For some other related work, please refer to [8–26] and [27].
In the present paper, our main motivation is to extend the classes of directed and demicontractive operators to the class of quasipseudocontractions because the class of quasipseudocontractions includes the classes of directed and demicontractive operators as special cases. Interest in pseudocontractive mappings stems mainly from their firm connection with the class of monotone operators. We present a unified framework for the study of this problem and this class of operators. We propose an iterative algorithm and study its convergence.
2 Notations and lemmas
Let H be a real Hilbert space with inner product $\u3008\cdot ,\cdot \u3009$ and norm $\parallel \cdot \parallel $, respectively. Let C be a nonempty closed convex subset of H.
Recall that a mapping $T:C\to C$ is called

nonexpansive if $\parallel TxTy\parallel \le \parallel xy\parallel $ for all $x,y\in C$;

quasinonexpansive if $\parallel Tx{x}^{\ast}\parallel \le \parallel x{x}^{\ast}\parallel $ for all $x\in C$ and ${x}^{\ast}\in Fix(T)$;

firmly nonexpansive if ${\parallel TxTy\parallel}^{2}\le {\parallel xy\parallel}^{2}{\parallel (IT)x(IT)y\parallel}^{2}$ for all $x,y\in C$;

firmly quasinonexpansive if ${\parallel Tx{x}^{\ast}\parallel}^{2}\le {\parallel x{x}^{\ast}\parallel}^{2}{\parallel Txx\parallel}^{2}$ for all $x\in C$ and ${x}^{\ast}\in Fix(T)$;

strictly pseudocontractive if ${\parallel TxTy\parallel}^{2}\le {\parallel xy\parallel}^{2}+k{\parallel (IT)x(IT)y\parallel}^{2}$ for all $x,y\in C$, where $k\in [0,1)$;

directed if $\u3008Tx{x}^{\ast},Txx\u3009\le 0$ for all $x\in C$ and ${x}^{\ast}\in Fix(T)$;

demicontractive if ${\parallel Tx{x}^{\ast}\parallel}^{2}\le {\parallel x{x}^{\ast}\parallel}^{2}+k{\parallel Txx\parallel}^{2}$ for all $x\in C$ and ${x}^{\ast}\in Fix(T)$, where $k\in [0,1)$.
The concept of directed operators was introduced by Bauschke and Combettes [28] who proved that $T:C\to C$ is directed if and only if
for all $x\in C$ and ${x}^{\ast}\in Fix(T)$. It can be seen easily that the class of directed operators coincides with that of firmly quasinonexpansive mappings.
From the above definitions, we note that the class of demicontractive operators contains important operators such as the directed operators, the quasinonexpansive operators and the strictly pseudocontractive mappings with fixed points. Such a class of operators is fundamental because they include many types of nonlinear operators arising in applied mathematics and optimization; see for example [29] and references therein.
Recall also that a mapping $T:C\to C$ is called pseudocontractive if
for all $x,y\in C$. It is well known that T is pseudocontractive if and only if
for all $x,y\in C$ and $T:C\to C$ is said to be quasipseudocontractive if
for all $x\in C$ and ${x}^{\ast}\in Fix(T)$.
It is obvious that the class of quasipseudocontractive mappings includes the class of demicontractive mappings.
A mapping $T:C\to C$ is called LLipschitzian if there exists $L>0$ such that
for all $x,y\in C$.
Usually, the convergence of fixed point algorithms requires some additional smoothness properties of the mapping T such as demiclosedness.
Recall that a mapping T is said to be demiclosed if, for any sequence $\{{x}_{n}\}$ which weakly converges to $\tilde{x}$, and if the sequence $\{T({x}_{n})\}$ strongly converges to z, we have $T(\tilde{x})=z$.
Observe also that the nonexpansive operators are both quasinonexpansive and strictly pseudocontractive maps and are well known for being demiclosed. For the pseudocontractions, the following demiclosedness principle is well known.
Lemma 2.1 ([30])
Let H be a real Hilbert space, C a closed convex subset of H. Let $U:C\to C$ be a continuous pseudocontractive mapping. Then

(i)
$Fix(U)$ is a closed convex subset of C,

(ii)
$(IU)$ is demiclosed at zero.
In the next section, we will need to impose the demiclosedness to the quasipseudocontractions.
It is well known that in a real Hilbert space H, the following equality holds:
for all $x,y\in H$ and $t\in [0,1]$.
Lemma 2.2 ([28])
Let H be a Hilbert space and let $\{{u}_{n}\}$ be a sequence in H such that there exists a nonempty set $\mathrm{\Omega}\subset H$ satisfying the following:

(i)
for every $u\in \mathrm{\Omega}$, ${lim}_{n}\parallel {u}_{n}u\parallel $ exists,

(ii)
any weakcluster point of the sequence $\{{u}_{n}\}$ belongs in Ω.
Then there exists ${x}^{\u2020}\in \mathrm{\Omega}$ such that $\{{u}_{n}\}$ weakly converges to ${x}^{\u2020}$.
In the sequel we shall use the following notations:

1.
${\omega}_{w}({u}_{n})=\{x:\mathrm{\exists}{u}_{{n}_{j}}\to x\text{weakly}\}$ denote the weak ωlimit set of $\{{u}_{n}\}$;

2.
${u}_{n}\rightharpoonup x$ stands for the weak convergence of $\{{u}_{n}\}$ to x;

3.
${u}_{n}\to x$ stands for the strong convergence of $\{{u}_{n}\}$ to x.
3 Main results
In this section, we will focus our attention on the following general twooperator split common fixed point problem:
where $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator, $U:{H}_{1}\to {H}_{1}$ is a quasipseudocontractive mapping and $T:{H}_{2}\to {H}_{2}$ is a quasipseudocontractive mapping with nonempty fixed point sets $Fix(U)=C$ and $Fix(T)=Q$, and we denote the solution set of the twooperator split common fixed point problem by
Algorithm 3.1 For ${u}_{0}\in {H}_{1}$, define a sequence $\{{u}_{n}\}$ as follows:
for all $n\in \mathbb{N}$, where γ, ν, η, and β are four constants, $\{{\alpha}_{n}\}$, $\{{\delta}_{n}\}$, and $\{{\xi}_{n}\}$ are three sequences in $[0,1]$.
Now, we demonstrate the convergence analysis of the algorithm (3.1).
Theorem 3.2 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces. Let $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator. Let $U:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ be two LLipschitzian quasipseudocontractions with nonempty $Fix(U)=C$ and $Fix(T)=Q$. Assume $TI$ and $UI$ are demiclosed at 0 and $\mathrm{\Gamma}\ne \mathrm{\varnothing}$. If the parameters γ, ν, η, β, $\{{\alpha}_{n}\}$, $\{{\delta}_{n}\}$, and $\{{\xi}_{n}\}$ satisfy the following control conditions:
(C_{1}): $0<\nu <1$ and $0<\gamma <\frac{1}{\lambda \nu}$, where λ is the spectral radius of the operator ${A}^{\ast}A$;
(C_{2}): $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$;
(C_{3}): $0<1\eta \le \beta <\frac{1}{\sqrt{1+{L}^{2}}+1}$ and $0<a\le 1{\delta}_{n}\le {\xi}_{n}<\frac{1}{\sqrt{1+{L}^{2}}+1}$ for all $n\in \mathbb{N}$.
Then the sequence $\{{u}_{n}\}$ generated by algorithm (3.2) weakly converges to a split common fixed point $\mu \in \mathrm{\Gamma}$.
Remark 3.3 Without loss of generality, we may assume that the Lipschitz constant $L>1$. It is obvious that $\beta <\frac{1}{\sqrt{1+{L}^{2}}+1}<\frac{1}{L}$ for all $n\ge 1$.
Since ${\xi}_{n}<\frac{1}{\sqrt{1+{L}^{2}}+1}$, we have
for all $n\in \mathbb{N}$.
Proposition 3.4 Let the mapping $T:{H}_{2}\to {H}_{2}$ be LLipschitzian with $L>1$. Then
for all $\beta \in (0,\frac{1}{L})$.
Proof As a matter of fact, $Fix(T)\subset Fix(T((1\beta )I+\beta T))$ is obvious.
Next, we show that $Fix(T((1\beta )I+\beta T))\subset Fix(T)$.
Take any ${x}^{\ast}\in Fix(T((1\beta )I+\beta T))$. We have $T((1\beta )I+\beta T){x}^{\ast}={x}^{\ast}$. Set $S=(1\beta )I+\beta T$. We have $TS{x}^{\ast}={x}^{\ast}$. Write $S{x}^{\ast}={y}^{\ast}$. Then $T{y}^{\ast}={x}^{\ast}$. Now we show ${x}^{\ast}={y}^{\ast}$. In fact,
Since $\beta <\frac{1}{L}$, we deduce ${y}^{\ast}={x}^{\ast}\in Fix(S)=Fix(T)$. Thus, ${x}^{\ast}\in Fix(T)$. Hence, $Fix(T((1\beta )I+\beta T))\subset Fix(T)$. Therefore, $Fix(T((1\beta )I+\beta T))=Fix(T)$. □
Proposition 3.5
for all $x\in {H}_{2}$ and all $x\in Fix(T)$.
Proof Since ${x}^{\ast}\in Fix(T)$, we have from (2.1)
and
for all $x\in {H}_{2}$.
By (3.3), (2.2), and (3.4), we obtain
Noting that T is LLipschitzian and $x((1\beta )I+\beta T)x=\beta (xTx)$, we have
Since $\beta <\frac{1}{\sqrt{1+{L}^{2}}+1}$, we have
From (3.5), we can deduce
for all $x\in {H}_{2}$ and ${x}^{\ast}\in Fix(T)$.
Hence,
By (C_{3}) and (3.7), we deduce
□
Proposition 3.6 Let the mapping $T:{H}_{2}\to {H}_{2}$ be LLipschitzian with $L>1$. If $TI$ is demiclosed at 0, then $T((1\beta )I+\beta T)I$ is also demiclosed at 0 when $\beta \in (0,\frac{1}{L})$.
Proof Let the sequence $\{{x}_{n}\}\subset {H}_{2}$ satisfying ${x}_{n}\rightharpoonup \tilde{x}$ and ${x}_{n}T((1\beta )I+\beta T){x}_{n}\to 0$. Next, we will show that $\tilde{x}\in Fix(T((1\beta )I+\beta T))$.
From Proposition 3.4, we only need to prove that $\tilde{x}\in Fix(T)$. As a matter of fact, since T is LLipschitzian, we have
It follows that
Hence,
Applying the demiclosedness of T, we immediately deduce $\tilde{x}\in Fix(T)$. □
Next, we prove Theorem 3.2.
Proof Let ${x}^{\ast}\in \mathrm{\Gamma}$. Then we get ${x}^{\ast}\in Fix(U)$ and $A{x}^{\ast}\in Fix(T)$. From (2.2) and (3.2), we have
Since ${x}^{\ast}\in Fix(U)$, we have from (2.1)
for all $x\in C$.
By a similar argument to that of (3.6), we obtain
Substituting (3.10) to (3.8) and noting that $1{\xi}_{n}\le {\delta}_{n}$, we have
Since λ is the spectral radius of the operator $A{A}^{\ast}$, we deduce
This together with (3.2) implies that
By Proposition 3.5 and noting that $A{x}^{\ast}\in Fix(T)$, we have
At the same time, we have the following equality in Hilbert spaces:
In (3.13), picking up $x=[\eta I+(1\eta )T((1\beta )I+\beta T)I]A{u}_{n}$ and $y=[\eta I+(1\eta )T((1\beta )I+\beta T)]A{u}_{n}A{x}^{\ast}$ we deduce
It follows that
Thus,
From (3.11), (3.12), and (3.14), we get
We deduce immediately that
Hence, ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}{x}^{\ast}\parallel $ exists. This implies that $\{{u}_{n}\}$ is bounded. Consequently, we have
Therefore,
Since $\{{u}_{n}\}$ is bounded, ${\omega}_{w}({u}_{n})\ne \mathrm{\varnothing}$. We can take $\mu \in {\omega}_{w}({u}_{n})$, that is, there exists $\{{u}_{{n}_{j}}\}$ such that $\omega {lim}_{j\to \mathrm{\infty}}{u}_{{n}_{j}}=\mu $. Since $TI$ is demiclosed at 0, by Proposition 3.6, we see that $T((1\beta )I+\beta T)I$ is also demiclosed at 0. Then, from (3.16), we obtain
Thus, $A\mu \in Fix(T((1\beta )I+\beta T))=Fix(T)$.
From (3.15), we deduce
This together with (C_{2}) implies that
Noticing that $1{\delta}_{n}\ge a$, we get immediately
Since U is LLipschitzian, we have
It follows that
Since ${\xi}_{n}<\frac{1}{\sqrt{1+{L}^{2}}+1}<\frac{1}{L}$, we deduce
From (3.2) and (3.16), we have ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{u}_{n}\parallel =0$. Thus, $\omega {lim}_{j\to \mathrm{\infty}}{x}_{{n}_{j}}=\mu $. By the demiclosedness of $UI$ at 0 and (3.17), we get $\mu \in Fix(U)$. Hence, $\mu \in Fix(U)$. Therefore, $\mu \in \mathrm{\Gamma}$.
Note that there is no more than one weakcluster point of $\{{u}_{n}\}$. In fact, if we assume there exists another $\{{u}_{{n}_{k}}\}$ such that $\omega {lim}_{k\to \mathrm{\infty}}{u}_{{n}_{k}}=\tilde{\mu}\ne \mu $, then we can deduce $\tilde{\mu}\in Fix(U)$. Now we show $\tilde{\mu}=\mu $. By the Opial property of Hilbert space, we have
This is a contradiction. Hence, the weak convergence of the whole sequence $\{{u}_{n}\}$ follows by applying Lemma 2.2 with $\mathrm{\Omega}=\mathrm{\Gamma}$. This completes the proof. □
Remark 3.7 Since the class of quasipseudocontractions contains the demicontractive operators, the directed operators, the quasinonexpansive operators and the strictly pseudocontractive mappings with fixed points as special cases, our results present a unified framework for the study of this problem and this class of operators.
Corollary 3.8 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces. Let $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator. Let $U:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ be two LLipschitzian demicontractive mappings with nonempty $Fix(U)=C$ and $Fix(T)=Q$. Assume $TI$ and $UI$ are demiclosed at 0 and $\mathrm{\Gamma}\ne \mathrm{\varnothing}$. If the parameters γ, ν, η, β, $\{{\alpha}_{n}\}$, $\{{\delta}_{n}\}$ and $\{{\xi}_{n}\}$ satisfy the following control conditions:
(C_{1}): $0<\nu <1$ and $0<\gamma <\frac{1}{\lambda \nu}$, where λ is the spectral radius of the operator ${A}^{\ast}A$;
(C_{2}): $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$;
(C_{3}): $0<1\eta \le \beta <\frac{1}{\sqrt{1+{L}^{2}}+1}$ and $0<a\le 1{\delta}_{n}\le {\xi}_{n}<\frac{1}{\sqrt{1+{L}^{2}}+1}$ for all $n\in \mathbb{N}$.
Then the sequence $\{{u}_{n}\}$ generated by algorithm (3.2) weakly converges to a split common fixed point $\mu \in \mathrm{\Gamma}$.
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Acknowledgements
LiJun Zhu was supported in part by NNSF of China (61362033). YeongCheng Liou was supported in part by the Grants NSC 1012628E230001MY3 and NSC 1032923E037001MY3. JenChih Yao was partially supported by the Grant NSC 1032923E037001MY3.
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Zhu, L., Liou, Y., Yao, J. et al. New algorithms designed for the split common fixed point problem of quasipseudocontractions. J Inequal Appl 2014, 304 (2014). https://doi.org/10.1186/1029242X2014304
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Keywords
 split common fixed point
 quasipseudocontractive mappings
 weak convergence