New algorithms designed for the split common fixed point problem of quasipseudocontractions
 LiJun Zhu^{1},
 YeongCheng Liou^{2, 3}Email author,
 JenChih Yao^{3, 4} and
 Yonghong Yao^{5}
https://doi.org/10.1186/1029242X2014304
© Zhu et al.; licensee Springer. 2014
Received: 22 April 2014
Accepted: 15 July 2014
Published: 19 August 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.
Keywords
1 Background and motivation
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].
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.
The simultaneous algebraic reconstruction technique is a typical example of the Landweber algorithm (1.5) when the system (1.4) is finitedimensional.
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.
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.
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 $k\in [0,1)$.
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.
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.
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])
 (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.
for all $x,y\in H$ and $t\in [0,1]$.
Lemma 2.2 ([28])
 (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 ${x}^{\u2020}$.
 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
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 $n\ge 1$.
for all $n\in \mathbb{N}$.
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 $Fix(T((1\beta )I+\beta T))\subset Fix(T)$.
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, $Fix(T((1\beta )I+\beta T))=Fix(T)$. □
for all $x\in {H}_{2}$ and all $x\in Fix(T)$.
for all $x\in {H}_{2}$.
for all $x\in {H}_{2}$ and ${x}^{\ast}\in Fix(T)$.
□
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 $\tilde{x}\in Fix(T((1\beta )I+\beta T))$.
Applying the demiclosedness of T, we immediately deduce $\tilde{x}\in Fix(T)$. □
Next, we prove Theorem 3.2.
for all $x\in C$.
Thus, $A\mu \in Fix(T((1\beta )I+\beta T))=Fix(T)$.
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, $\mu \in \mathrm{\Gamma}$.
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 $\mu \in \mathrm{\Gamma}$.
Declarations
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.
Authors’ Affiliations
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