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Weak and strongconvergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces
Fixed Point Theory and Applications volume 2014, Article number: 11 (2014)
Abstract
The purpose of this article is to study the weak and strongconvergence theorems of solutions to split a feasibility problem for a family of nonspreadingtype 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.
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
$$\parallel TxTy\parallel \le \parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D.$$ 
(2)
T is said to be quasinonexpansive if $F(T)$ is nonempty and
$$\parallel Txp\parallel \le \parallel xp\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T).$$(1.1) 
(3)
T is said to be nonspreading if
$$2{\parallel TxTy\parallel}^{2}\le {\parallel Txy\parallel}^{2}+{\parallel Tyx\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D.$$(1.2)
It is easy to prove that equation (1.2) is equivalent to

(4)
T is said to be kstrictly pseudononspreading [1], if there exists a constant $k\in [0,1)$ such that
$${\parallel TxTy\parallel}^{2}\le {\parallel xy\parallel}^{2}+k{\parallel xTx(yTy)\parallel}^{2}+2\u3008xTx,yTy\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D.$$(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 quasinonexpansive;

(2)
if T is nonspreading, then it is kstrictly pseudononspreading with $k=0$. But the converse is not true from the following example. Thus, we know that the class of kstrictly pseudononspreading mappings is more general than the class of nonspreading mappings.
Example 1.3 [1]
Let ℛ denote the set of real numbers with the usual norm. Let $T:\mathcal{R}\to \mathcal{R}$ be a mapping defined by
Then T is a kstrictly pseudononspreading 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 strongconvergence 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 kstrictly pseudononspreading 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 strongconvergence 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].
The purpose of this paper is to introduce the following multipleset split feasibility problem (MSSFP) for an infinite family of kstrictly pseudononspreading mappings and a finite family of ρstrictly pseudononspreading mappings in infinitely dimensional Hilbert spaces, i.e., to find ${x}^{\ast}\in C$ such that
where ${H}_{1}$, ${H}_{2}$ are two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator, ${\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}}:{H}_{1}\to {H}_{1}$ is an infinite family of ${k}_{i}$strictly pseudononspreading mappings and ${\{{T}_{i}\}}_{i=1}^{N}:{H}_{2}\to {H}_{2}$ is a finite family of ${\rho}_{i}$strictly pseudononspreading mappings, $C:={\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i})$ and $Q:={\bigcap}_{i=1}^{N}F({T}_{i})$. Also we wish to study the weak and strong convergence of the following iterative sequence to a solution of problem (1.6):
where ${S}_{i,\beta}:=\beta I+(1\beta ){S}_{i}$, $\beta \in (0,1)$ is a constant.
In the sequel we denote Γ the set of solutions of (MSSFP) equation (1.6), 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)
$IT$ is said to be demiclosed at 0, if, for any sequence $\{{x}_{n}\}\subset H$ with ${x}_{n}\rightharpoonup {x}^{\ast}$, $\parallel (IT){x}_{n}\parallel \to 0$, 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 kstrictly pseudononspreading mapping.

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

(2)
$IT$ 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 $t\in [0,1]$,
$${\parallel tx+(1t)y\parallel}^{2}=t{\parallel x\parallel}^{2}+(1t){\parallel y\parallel}^{2}t(1t){\parallel xy\parallel}^{2}.$$(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 kstrictly pseudononspreading 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)
${\parallel {T}_{\beta}x{T}_{\beta}y\parallel}^{2}\le {\parallel xy\parallel}^{2}+\frac{2}{1\beta}\u3008x{T}_{\beta}x,y{T}_{\beta}y\u3009$;

(4)
${T}_{\beta}$ is a quasinonexpansive mapping.
Proof Since $(I{T}_{\beta})=(1\beta )(IT)$, the conclusions (1), (2) are obvious.
Now we prove the conclusion (3). In fact, since T is a kstrictly pseudononspreading mapping, it follows from Lemma 2.3 that
If $y\in F(T)$, then $y\in F({T}_{\beta})$. Hence from equation (2.4),
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 weakcluster 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 strongconvergence theorems
For solving the multipleset split feasibility problem (MSSFP) equation (1.6), we assume that the following conditions are satisfied:

(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 pseudononspreading 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 pseudononspreading 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).
Step 1. We prove that the sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ and $\{{S}_{i,\beta}{y}_{n}\}$ are bounded and, for each $p\in \mathrm{\Gamma}$, the following limits exist and
In fact, for given $p\in \mathrm{\Gamma}$, by the definition of Γ,
and
Therefore, we have
Since ${\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}}$ is a family of kstrictly pseudononspreading 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 $p\in \mathrm{\Gamma}$,
and
Further, since ${\{{T}_{i}\}}_{i=1}^{N}$ is a finite family of ρstrictly pseudononspreading mappings, we have
Substituting equations (3.5) and (3.6) into equation (3.4) and simplifying, we have
By condition (c), $(1\rho \gamma {\parallel A\parallel}^{2})>0$, therefore we have
Substituting equation (3.8) into equation (3.3), we have
This implies that the limit ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel $ exists. It follows from equations (3.8) and (3.3) that the limit ${lim}_{n\to \mathrm{\infty}}\parallel {y}_{n}p\parallel $ exists also, and
Therefore, $\{{x}_{n}\}$ and $\{{y}_{n}\}$ are bounded. Since for each $i\ge 1$, ${S}_{i,\beta}$ is quasinonexpansive, we have
Hence $\{{S}_{i,\beta}{y}_{n}\}$ is also bounded.
Step 2. Now we prove that for any given positive integer $l\ge 1$, the following conclusions hold:
In fact, for any given $p\in \mathrm{\Gamma}$, it follows from equation (3.1), Lemma 2.4, and equation (3.7) that
Therefore, we have
By conditions (b) and (c) we have
Since g is continuous and strictly increasing with $g(0)=0$, from equation (3.12) we have
Hence conclusion (3.10) is proved.
Step 3. Now, we prove that
In fact, it follows from equation (3.1) that
By virtue of equations (3.1) and (3.10), one has
This together with equations (3.10) and (3.15) shows that
Similarly, we have
Step 4. Now we show that every weakcluster point ${x}^{\ast}$ of the sequence $\{{x}_{n}\}$ is in Γ.
Indeed, since $\{{y}_{n}\}$ is a bounded sequence in ${H}_{1}$, there exists a subsequence $\{{y}_{{n}_{i}}\}\subset \{{y}_{n}\}$ such that ${y}_{{n}_{i}}\rightharpoonup {x}^{\ast}\in {H}_{1}$. It follows from equation (3.10) that
By Lemma 2.2, $(I{S}_{i})$ is demiclosed at zero. Since $(I{S}_{l,\beta})=(1\beta )(I{S}_{i})$, this implies that $(I{S}_{l,\beta})$ is also demiclosed at zero. Hence ${x}^{\ast}\in F({S}_{l,\beta})=F({S}_{l})$. By the arbitrariness of $l\ge 1$, we have
On the other hand, it follows from equations (3.1) and (3.10) that
Since A is a bounded linear operator, this implies that $A{x}_{{n}_{i}}\rightharpoonup A{x}^{\ast}$. Also, by equation (3.10)
Hence for any given positive integer $j=1,2,\dots ,N$, there exists a subsequence $\{{n}_{{i}_{k}}\}\subset \{{n}_{i}\}$ with ${n}_{{i}_{k}}(modN)=j$ such that
Since $A{x}_{{n}_{{i}_{k}}}\rightharpoonup A{x}^{\ast}$, and by Lemma 2.2, $I{T}_{j}$ is demiclosed at 0. This implies that $A{x}^{\ast}\in F({T}_{j})$. By the arbitrariness of $j=1,2,\dots ,N$,
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 weakcluster 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).
Without loss of generality, we may assume that ${S}_{1}$ is semicompact. Since $(I{S}_{1,\beta})=(1\beta )(I{S}_{1})$, this implies that ${S}_{1,\beta}$ is also semicompact. In view of equation (3.10), we have
Therefore, there exists a subsequence of $\{{y}_{{n}_{i}}\}\subset \{{y}_{n}\}$ such that ${y}_{{n}_{i}}\to {u}^{\ast}\in {H}_{1}$. Since ${y}_{{n}_{i}}\rightharpoonup {x}^{\ast}$, we have ${x}^{\ast}={u}^{\ast}$ and so ${y}_{{n}_{i}}\to {x}^{\ast}\in \mathrm{\Gamma}$. By virtue of equation (3.9), we have
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 demicontractive mappings, to the more general family of kstrictly pseudononspreading 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 strongconvergence 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.
Let H be a real Hilbert space, $\{{S}_{i}\}:H\to H$, $i=1,2,\dots $ be a countable family of ${k}_{i}$strictly pseudononspreading mappings with $k={sup}_{i\ge 1}{k}_{i}\in (0,1)$, and
Let $T:H\to H$ be a nonspreading mapping. The socalled hierarchical variational inequality problem for a countable family of mappings $\{{S}_{i}\}$ with respect to mapping T is to find an ${x}^{\ast}\in \mathcal{F}$ such that
It is easy to see that equation (4.1) is equivalent to the following fixed point problem: to find ${x}^{\ast}\in \mathcal{F}$ such that
where ${P}_{\mathcal{F}}$ is the metric projection from 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 multiset 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 ρstrictly pseudononspreading 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.
References
 1.
Osilike MO, Isiogugu FO: Weak and strong convergence theorems for nonspreadingtype mappings in Hilbert spaces. Nonlinear Anal. 2011, 74: 1814–1822. 10.1016/j.na.2010.10.054
 2.
Kurokawa Y, Takahashi W: Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 1562–1568. 10.1016/j.na.2010.04.060
 3.
Baillon J: Un theorem de type ergodique pour les contractions nonlineaires dans un espace de Hilbert. C. R. Acad. Sci., Ser. AB 1975, 280(Aii):A1511A1514.
 4.
Halpern B: Fixed points of nonexpanding mappings. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S000299041967118640
 5.
Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692
 6.
Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 2002, 18: 441–453. 10.1088/02665611/18/2/310
 7.
Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problem in intensitymodulated radiation therapy. Phys. Med. Biol. 2006, 51: 2353–2365. 10.1088/00319155/51/10/001
 8.
Censor Y, Elfving T, Kopf N, Bortfeld T: The multiplesets split feasibility problem and its applications. Inverse Probl. 2005, 21: 2071–2084. 10.1088/02665611/21/6/017
 9.
Censor Y, Motova A, Segal A: Perturbed projections and subgradient projections for the multiplesets split feasibility problem. J. Math. Anal. Appl. 2007, 327: 1244–1256. 10.1016/j.jmaa.2006.05.010
 10.
Xu HK: A variable Krasnosel’skiiMann algorithm and the multiplesets split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/02665611/22/6/007
 11.
Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/02665611/20/4/014
 12.
Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/02665611/21/5/017
 13.
Chang SS, Kim JK, Wang XR: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010., 2010: Article ID 869684 10.1155/2010/869684
 14.
Moudafi A, AlShemas E: Simultaneous iterative methods for split equality problem. Trans. Math. Program. Appl. 2013, 1: 1–11.
 15.
Moudafi A: The split common fixed point problem for demicontractive mappings. Inverse Probl. 2010., 26: Article ID 055007
 16.
Xu HK: Iterative methods for split feasibility problem in infinitedimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018
 17.
Censor Y, Segal A: The split common fixed point problem for directed operators. J. Convex Anal. 2009, 16: 587–600.
 18.
Masad E, Reich S: A note on the multipleset split feasibility problem in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 367–371.
 19.
Deepho J, Kumam P: Split feasibility and fixedpoint problems for asymptotically quasinonexpansive mappings. J. Inequal. Appl. 2013., 2013: Article ID 322 10.1186/1029242X2013322
 20.
Deepho J, Kumam P: A modified Halpern’s iterative scheme for solving split feasibility problems. Abstr. Appl. Anal. 2012., 2012: Article ID 876069
 21.
Yang L, Chang SS, Cho YJ, Kim JK: Multipleset split feasibility problems for total asymptotically strict pseudocontraction mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 77 10.1186/16871812201177
 22.
Chang SS, Cho YJ, Kim JK, Zhang WB, Yang L: Multipleset split feasibility problems for asymptotically strict pseudocontractions. Abstr. Appl. Anal. 2012., 2012: Article ID 491760 10.1155/2012/491760
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).
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The main idea of this paper is proposed by JKK and SSC. JKK and SSC prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Chang, S., Kim, J.K., Cho, Y.J. et al. Weak and strongconvergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces. Fixed Point Theory Appl 2014, 11 (2014). https://doi.org/10.1186/16871812201411
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Keywords
 split feasibility problem
 convex feasibility problem
 kstrictly pseudononspreading mapping
 demicloseness
 Opial’s condition