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- Open Access
Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces
- Shih-sen Chang^{1},
- Jong Kyu Kim^{2}Email author,
- Yeol Je Cho^{3} and
- Jae Yull Sim^{4}
https://doi.org/10.1186/1687-1812-2014-11
© Chang et al.; licensee Springer. 2014
- Received: 1 October 2013
- Accepted: 17 December 2013
- Published: 9 January 2014
Abstract
The purpose of this article is to study the weak- and strong-convergence theorems of solutions to split a feasibility problem for a family of nonspreading-type mapping in Hilbert spaces. The main result presented in this paper improves and extends some recent results of Censor et al., Byrne, Yang, Moudafi, Xu, Censor and Segal, Masad and Reich, and others. As an application, we solve the hierarchical variational inequality problem by using the main theorem.
MSC:47J05, 47H09, 49J25.
Keywords
- split feasibility problem
- convex feasibility problem
- k-strictly pseudo-nonspreading mapping
- demicloseness
- Opial’s condition
1 Introduction
Throughout this paper, we assume that H is a real Hilbert space, D is a nonempty and closed convex subset of H. In the sequel, we denote by ‘${x}_{n}\to x$’ and ‘${x}_{n}\rightharpoonup x$’ the strong and weak convergence of $\{{x}_{n}\}$, respectively. Denote by the set of all positive integers and by $F(T)$ the set of fixed points of a mapping $T:D\to D$.
- (1)$T:D\to D$ is said to be nonexpansive if$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D.$
- (2)T is said to be quasi-nonexpansive if $F(T)$ is nonempty and$\parallel Tx-p\parallel \le \parallel x-p\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T).$(1.1)
- (3)T is said to be nonspreading if$2{\parallel Tx-Ty\parallel}^{2}\le {\parallel Tx-y\parallel}^{2}+{\parallel Ty-x\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D.$(1.2)
- (4)T is said to be k-strictly pseudo-nonspreading [1], if there exists a constant $k\in [0,1)$ such that${\parallel Tx-Ty\parallel}^{2}\le {\parallel x-y\parallel}^{2}+k{\parallel x-Tx-(y-Ty)\parallel}^{2}+2\u3008x-Tx,y-Ty\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D.$(1.4)
- (1)
if T is nonspreading and $F(T)\ne \mathrm{\varnothing}$, then T is quasi-nonexpansive;
- (2)
if T is nonspreading, then it is k-strictly pseudo-nonspreading with $k=0$. But the converse is not true from the following example. Thus, we know that the class of k-strictly pseudo-nonspreading mappings is more general than the class of nonspreading mappings.
Example 1.3 [1]
Then T is a k-strictly pseudo-nonspreading mapping, but it is not nonspreading.
In 2010, Kurokawa and Takahashi [2] obtained a weak mean ergodic theorem of Baillon’s type [3] for nonspreading mappings in Hilbert spaces. They further proved a strong-convergence theorem somewhat related to Halpern’s type [4] for this class of mappings using the idea of mean convergence in Hilbert spaces.
In 2011, Osilike and Isiogugu [1] first introduced the concept of k-strictly pseudo-nonspreading mappings and proved a weak mean convergence theorem of Baillon’s type similar to the ones obtained in [2]. Furthermore, using the idea of mean convergence, a strong-convergence theorem similar to the one obtained in [2] is proved which extends and improves the main theorems of [2] and an affirmative answer given to an open problem posed by Kurokawa and Takahashi [2] for the case where the mapping T is averaged.
On the other hand, the split feasibility problem (SFP) in finitely dimensional spaces was first introduced by Censor and Elfving [5] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [6]. Recently, it has been found that the (SFP) can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [7–9].
The split feasibility problem in an infinitely dimensional Hilbert space can be found in [6, 8, 10–12].
where ${S}_{i,\beta}:=\beta I+(1-\beta ){S}_{i}$, $\beta \in (0,1)$ is a constant.
2 Preliminaries
For this purpose, we first recall some definitions, notations and conclusions which will be needed in proving our main results.
- (1)
$I-T$ is said to be demiclosed at 0, if, for any sequence $\{{x}_{n}\}\subset H$ with ${x}_{n}\rightharpoonup {x}^{\ast}$, $\parallel (I-T){x}_{n}\parallel \to 0$, then ${x}^{\ast}=T{x}^{\ast}$.
- (2)
T is said to be semicompact, if, for any bounded sequence $\{{x}_{n}\}\subset E$, ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$, then there exists a subsequence $\{{x}_{{n}_{i}}\}\subset \{{x}_{n}\}$ such that $\{{x}_{{n}_{i}}\}$ converges strongly to some point ${x}^{\ast}\in E$.
Lemma 2.2 [1]
- (1)
If $F(T)\ne \mathrm{\varnothing}$, then $F(T)$ is closed and convex;
- (2)
$I-T$ is demiclosed at zero.
- (1)For all $x,y\in H$ and for all $t\in [0,1]$,${\parallel tx+(1-t)y\parallel}^{2}=t{\parallel x\parallel}^{2}+(1-t){\parallel y\parallel}^{2}-t(1-t){\parallel x-y\parallel}^{2}.$(2.1)
- (2)For all $x,y\in H$,${\parallel x+y\parallel}^{2}\le {\parallel x\parallel}^{2}+2\u3008y,x+y\u3009.$
Lemma 2.4 [13]
Lemma 2.5 [10]
If ${\sum}_{i=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty}$ and ${\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}$, then the limit ${lim}_{n\to \mathrm{\infty}}{a}_{n}$ exists.
- (1)
$F(T)=F({T}_{\beta})$;
- (2)
$I-{T}_{\beta}$ is demiclosed at zero;
- (3)
${\parallel {T}_{\beta}x-{T}_{\beta}y\parallel}^{2}\le {\parallel x-y\parallel}^{2}+\frac{2}{1-\beta}\u3008x-{T}_{\beta}x,y-{T}_{\beta}y\u3009$;
- (4)
${T}_{\beta}$ is a quasi-nonexpansive mapping.
Proof Since $(I-{T}_{\beta})=(1-\beta )(I-T)$, the conclusions (1), (2) are obvious.
This completes the proof of Lemma 2.6. □
Lemma 2.7 [14]
- (1)
for every $w\in W$, ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}-w\parallel $ exists;
- (2)
each weak-cluster point of the sequence $\{{w}_{n}\}$ is in W.
Then there exists ${w}^{\ast}\in W$ such that $\{{u}_{n}\}$ weakly converges to ${w}^{\ast}$.
3 Weak- and strong-convergence theorems
- (1)
${H}_{1}$ and ${H}_{2}$ are two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator and ${A}^{\ast}:{H}_{2}\to {H}_{1}$ is the adjoint of A;
- (2)
${\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}}:{H}_{1}\to {H}_{1}$ is an infinite family of ${k}_{i}$-strictly pseudo-nonspreading mappings with $k:={sup}_{i\ge 1}{k}_{i}\in (0,1)$;
- (3)
${\{{T}_{i}\}}_{i=1}^{N}:{H}_{2}\to {H}_{2}$ is a finite family of ${\rho}_{i}$-strictly pseudo-nonspreading mappings with $\rho =max\{{\rho}_{i}:i=1,2,\dots ,N\}\in (0,1)$;
- (4)
$C:={\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i})\ne \mathrm{\varnothing}$ and $Q:={\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing}$.
Now we are in a position to give the following main theorem.
- (a)
${\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{i,n}=1$, for each $n\ge 1$;
- (b)
for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty}}{\alpha}_{0,n}{\alpha}_{i,n}>0$;
- (c)
$\gamma \in (0,\frac{1-\rho}{{\parallel A\parallel}^{2}})$.
- (I)
both $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge weakly to some point ${x}^{\ast}\in \mathrm{\Gamma}$;
- (II)
in addition, if there exists some positive integer m such that ${S}_{m}$ is semicompact, then both $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge strongly to ${x}^{\ast}\in \mathrm{\Gamma}$.
Proof First we prove the conclusion (I).
Hence $\{{S}_{i,\beta}{y}_{n}\}$ is also bounded.
Hence conclusion (3.10) is proved.
Step 4. Now we show that every weak-cluster point ${x}^{\ast}$ of the sequence $\{{x}_{n}\}$ is in Γ.
These show that ${x}^{\ast}\in \mathrm{\Gamma}$.
Step 5. Summing up the above arguments, we have proved that: (i) for each $p\in \mathrm{\Gamma}$, the limits ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-p\parallel $ and ${lim}_{n\to \mathrm{\infty}}\parallel {y}_{n}-p\parallel $ exist (see equation (3.2)); (ii) every weak-cluster point ${x}^{\ast}$ of the sequence $\{{x}_{n}\}$ (or $\{{y}_{n}\}$) is in Γ. Taking $W=\mathrm{\Gamma}$ and $\{{u}_{n}\}=\{{x}_{n}\}$ (or $\{{y}_{n}\}$) in Lemma 2.7, therefore all conditions in Lemma 2.7 are satisfied. By using Lemma 2.7, ${x}_{n}\rightharpoonup {x}^{\ast}$, ${y}_{n}\rightharpoonup {x}^{\ast}$ and ${x}^{\ast}\in \mathrm{\Gamma}$. This completes the proof of the conclusion (I).
Next we prove the conclusion (II).
i.e., $\{{y}_{n}\}$ and $\{{x}_{n}\}$ both converge strongly to the point ${x}^{\ast}\in \mathrm{\Gamma}$. This completes the proof of Theorem 3.1. □
- (a)
for the mappings, we extend the mappings from nonexpansive mappings, or demi-contractive mappings, to the more general family of k-strictly pseudo-nonspreading mappings;
- (b)
for the algorithms, we propose some new hybrid iterative algorithms which are different from the ones given in [5–7, 9, 17, 18, 21, 22]. Under suitable conditions, some weak- and strong-convergence results for the algorithms are proved.
If we put $\gamma =0$ in Theorem 3.1, we immediately get the following.
- (a)
${\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{i,n}=1$, for each $n\ge 1$;
- (b)for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty}}{\alpha}_{0,n}{\alpha}_{i,n}>0$. Let$\mathcal{F}:=\bigcap _{i=1}^{\mathrm{\infty}}F({S}_{i})\ne \mathrm{\varnothing}.$
- (I)
the sequence $\{{x}_{n}\}$ converges weakly to some point ${x}^{\ast}\in \mathcal{F}$;
- (II)
in addition, if there exists some positive integer m such that ${S}_{m}$ is semicompact, then the sequence $\{{x}_{n}\}$ converges strongly to ${x}^{\ast}\in \mathcal{F}$.
4 Applications
In this section we utilize the results presented in Section 3 to study the hierarchical variational inequality problem.
Hence from Theorem 3.1 we have the following theorem.
- (a)
${\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{i,n}=1$, for each $n\ge 1$;
- (b)
for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty}}{\alpha}_{0,n}{\alpha}_{i,n}>0$;
- (c)
$\gamma \in (0,1)$.
If $C\cap Q\ne \mathrm{\varnothing}$, then $\{{x}_{n}\}$ converges weakly to a solution of the hierarchical variational inequality problem (4.1). In addition, if one of the mappings ${S}_{i}$ is semicompact, then both $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge strongly to a solution of the hierarchical variational inequality problem (4.1).
Proof In fact, by the assumption that T is a nonspreading mapping, hence by Remark 1.2, T is a ρ-strictly pseudo-nonspreading with $\rho =0$. Taking $N=1$ and $A=I$ in Theorem 3.1, all conditions in Theorem 3.1 are satisfied. The conclusions of Theorem 4.1 can immediately be obtained from Theorem 3.1. □
Remark 4.2 If $T=I$ (the identity mapping), then we can get the results of Corollary 3.3.
Declarations
Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the republic of Korea (2013R1A1A2054617).
Authors’ Affiliations
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