# 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}

**2014**:11

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

*k*-strictly pseudo-nonspreading mappingdemiclosenessOpial’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 $T:D\to D$.

**Definition 1.1**Let $T:D\to D$ be a mapping.

- (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)

**Remark 1.2**It follows from Definition 1.1 that

- (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].

*k*-strictly pseudo-nonspreading mappings and a finite family of

*ρ*-strictly pseudo-nonspreading mappings in infinitely dimensional Hilbert spaces,

*i.e.*, to find ${x}^{\ast}\in C$ such that

where ${S}_{i,\beta}:=\beta I+(1-\beta ){S}_{i}$, $\beta \in (0,1)$ is a constant.

*i.e.*,

## 2 Preliminaries

For this purpose, we first recall some definitions, notations and conclusions which will be needed in proving our main results.

**Definition 2.1**Let

*E*be a real Banach space, and $T:E\to E$ be a mapping.

- (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]

*Let*

*H*

*be a real Hilbert space*,

*D*

*be a nonempty and closed convex subset of*

*H*,

*and*$T:D\to D$

*be a*

*k*-

*strictly pseudo*-

*nonspreading mapping*.

- (1)
*If*$F(T)\ne \mathrm{\varnothing}$,*then*$F(T)$*is closed and convex*; - (2)
$I-T$

*is demiclosed at zero*.

**Lemma 2.3**

*Let*

*H*

*be a real Hilbert space*.

*Then the following statements hold*:

- (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]

*Let*

*E*

*be a uniformly convex Banach space*, ${B}_{r}(0):=\{x\in E:\parallel x\parallel \le r\}$

*be a closed ball with center*0

*and radius*$r>0$.

*Then for any given sequence*$\{{x}_{1},{x}_{2},\dots ,{x}_{n},\dots \}\subset {B}_{r}(0)$

*and any given number sequence*$\{{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n},\dots \}$

*with*${\lambda}_{i}\ge 0$, ${\sum}_{i=1}^{\mathrm{\infty}}{\lambda}_{i}=1$,

*there exists a strictly increasing continuous and convex function*$g:[0,2r)\to [0,\mathrm{\infty})$

*with*$g(0)=0$

*such that for any*$i,j\in \mathcal{N}$, $i<j$,

**Lemma 2.5** [10]

*Let*$\{{a}_{n}\}$, $\{{b}_{n}\}$

*and*$\{{\delta}_{n}\}$

*be sequences of nonnegative real numbers satisfying*

*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*.

**Lemma 2.6**

*Let*

*D*

*be a nonempty and closed convex subset of*

*H*

*and*$T:D\to D$

*be a*

*k*-

*strictly pseudo*-

*nonspreading mapping with*$F(T)\ne \mathrm{\varnothing}$.

*Let*${T}_{\beta}=\beta I+(1-\beta )T$, $\beta \in [k,1)$.

*Then the following conclusions hold*:

- (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.

*T*is a

*k*-strictly pseudo-nonspreading mapping, it follows from Lemma 2.3 that

This completes the proof of Lemma 2.6. □

**Lemma 2.7** [14]

*Let*

*H*

*be a Hilbert space and*$\{{u}_{n}\}$

*be a sequence in*

*H*

*such that there exists a nonempty set*$W\subset H$

*satisfying*:

- (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.

**Theorem 3.1**

*Let*${H}_{1}$, ${H}_{2}$,

*A*, ${A}^{\ast}$, ${\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}}$, ${\{{T}_{i}\}}_{i=1}^{N}$,

*C*,

*Q*,

*k*,

*ρ*

*be the same as above*.

*Let*$\{{x}_{n}\}$

*be a sequence generated by*

*where*${S}_{i,\beta}:=\beta I+(1-\beta ){S}_{i}$, $i\ge 1$, $\beta \in [k,1)$

*is a constant*, $\{{\alpha}_{i,n}\}\subset (0,1)$

*and*$\gamma >0$

*satisfy the following conditions*:

- (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)
$\gamma \in (0,\frac{1-\rho}{{\parallel A\parallel}^{2}})$.

*Let*$\mathrm{\Gamma}=\{x\in C,Ax\in Q\}\ne \mathrm{\varnothing}$ (

*the set of solutions of*(

*MSSFP*)

*equation*(1.6)

*defined by equation*(1.7)).

*Then we have the following*:

- (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).

*k*-strictly pseudo-nonspreading mappings, by Lemma 2.2, $C={\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i})$ is closed and convex. It follows from Lemma 2.6 that, for each $n\ge 1$ and $p\in \mathrm{\Gamma}$,

*ρ*-strictly pseudo-nonspreading mappings, we have

Hence $\{{S}_{i,\beta}{y}_{n}\}$ is also bounded.

*g*is continuous and strictly increasing with $g(0)=0$, from equation (3.12) we have

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.

*A*is a bounded linear operator, this implies that $A{x}_{{n}_{i}}\rightharpoonup A{x}^{\ast}$. Also, by equation (3.10)

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. □

**Remark 3.2**Theorem 3.1 improves and extends the corresponding results of Censor

*et al.*[5, 8, 9], Byrne [6], Yang [11], Moudafi [15], Xu [16], Censor and Segal [17], Masad and Reich [18], Deepho and Kumam [19, 20] and others in the following aspects:

- (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.

**Corollary 3.3**

*Let*

*H*, ${\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}}$,

*k*

*be the same as above*.

*Let*$\{{x}_{n}\}$

*be a sequence generated by*

*where*${S}_{i,\beta}:=\beta I+(1-\beta ){S}_{i}$, $i\ge 1$, $\beta \in [k,1)$

*is a constant*, $\{{\alpha}_{i,n}\}\subset (0,1)$

*satisfy the following conditions*:

- (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*$\mathcal{F}:=\bigcap _{i=1}^{\mathrm{\infty}}F({S}_{i})\ne \mathrm{\varnothing}.$

*Then we have the following*:

- (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.

*H*be a real Hilbert space, $\{{S}_{i}\}:H\to H$, $i=1,2,\dots $ be a countable family of ${k}_{i}$-strictly pseudo-nonspreading mappings with $k={sup}_{i\ge 1}{k}_{i}\in (0,1)$, and

*T*is to find an ${x}^{\ast}\in \mathcal{F}$ such that

*H*onto ℱ. Letting $C=\mathcal{F}$ and $Q=F({P}_{\mathcal{F}}T)$ (the fixed point set of ${P}_{\mathcal{F}}T$) and $A=I$ (the identity mapping on

*H*), then the problem (4.2) is equivalent to the following multi-set split feasibility problem: to find ${x}^{\ast}\in C$ such that

Hence from Theorem 3.1 we have the following theorem.

**Theorem 4.1**

*Let*

*H*, $\{{S}_{i}\}$,

*T*,

*C*,

*Q*,

*k*

*be the same as above*.

*Let*$\{{x}_{n}\}$, $\{{y}_{n}\}$

*be the sequences defined by*

*where*${S}_{i,\beta}:=\beta I+(1-\beta ){S}_{i}$, $i\ge 1$, $\beta \in [k,1)$, $\{{\alpha}_{i,n}\}\subset (0,1)$

*and*$\gamma >0$

*satisfy the following conditions*:

- (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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