Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces

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.

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

R E S E A R C H Open Access

Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces

Shih-sen Chang1, Jong Kyu Kim2*, Yeol Je Cho3 and Jae Yull Sim4

*Correspondence: mailto:[email protected]

Web End [email protected]

2Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, KoreaFull list of author information is available at the end of the article

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, Mouda, 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; Opials 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 xn x and xn x the strong and weak convergence of {xn}, respectively. Denote by N the set of all positive integers and by F(T) the set of xed points of a mapping T : D D.

Denition . Let T : D D be a mapping.

() T : D D is said to be nonexpansive if

Tx Ty x y , x, y D.

() T is said to be quasi-nonexpansive if F(T) is nonempty and

Tx p x p , x D, p F(T). (.)

() T is said to be nonspreading if

Tx Ty Tx y + Ty x , x, y D. (.)

It is easy to prove that equation (.) is equivalent to

Tx Ty x y + x Tx, y Ty , x, y D. (.)

2014 Chang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 2 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

() T is said to be k-strictly pseudo-nonspreading [], if there exists a constant k [, )

such that

TxTy xy +k x

Remark . It follows from Denition . that() if T is nonspreading and F(T) = , then T is quasi-nonexpansive;() if T is nonspreading, then it is k-strictly pseudo-nonspreading with k = . 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 . [] Let R denote the set of real numbers with the usual norm. Let T : R R be a mapping dened by

Tx =

Then T is a k-strictly pseudo-nonspreading mapping, but it is not nonspreading.

In , Kurokawa and Takahashi [] obtained a weak mean ergodic theorem of Baillons type [] for nonspreading mappings in Hilbert spaces. They further proved a strong-convergence theorem somewhat related to Halperns type [] for this class of mappings using the idea of mean convergence in Hilbert spaces.

In , Osilike and Isiogugu [] rst introduced the concept of k-strictly pseudo-nonspreading mappings and proved a weak mean convergence theorem of Baillons type similar to the ones obtained in []. Furthermore, using the idea of mean convergence, a strong-convergence theorem similar to the one obtained in [] is proved which extends and improves the main theorems of [] and an armative answer given to an open problem posed by Kurokawa and Takahashi [] for the case where the mapping T is averaged.

On the other hand, the split feasibility problem (SFP) in nitely dimensional spaces was rst introduced by Censor and Elfving [] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction []. 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 [].

The split feasibility problem in an innitely dimensional Hilbert space can be found in [, , ].

The purpose of this paper is to introduce the following multiple-set split feasibility problem (MSSFP) for an innite family of k-strictly pseudo-nonspreading mappings and a nite family of -strictly pseudo-nonspreading mappings in innitely dimensional Hilbert spaces, i.e., to nd x C such that

Ax Q, (.)

where H, H are two real Hilbert spaces, A : H H is a bounded linear operator, {Si}i= : H H is an innite family of ki-strictly pseudo-nonspreading mappings and

Tx(yTy)

+ xTx, yTy , x, y D. (.)

x, x (, ), x, x [, ).

(.)

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 3 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

{Ti}Ni= : H H is a nite family of i-strictly pseudo-nonspreading mappings, C :=

i= F(Si) and Q :=

Ni= F(Ti). Also we wish to study the weak and strong convergence of the following iterative sequence to a solution of problem (.):

i= i,nSi,yn,yn = xn + A(Tn(modN) I)Axn, n ,

where Si, := I + ( )Si, (, ) is a constant.In the sequel we denote the set of solutions of (MSSFP) equation (.), i.e.,

= {x C, Ax Q} = C A(Q). (.)

2 Preliminaries

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

Denition . Let E be a real Banach space, and T : E E be a mapping.() I T is said to be demiclosed at , if, for any sequence {xn} H with xn x,

(I T)xn , then x = Tx.() T is said to be semicompact, if, for any bounded sequence {xn} E,

limn xn Txn = , then there exists a subsequence {xni} {xn} such that {xni} converges strongly to some point x E.

Lemma . [] Let H be a real Hilbert space, D be a nonempty and closed convex subset of H, and T : D D be a k-strictly pseudo-nonspreading mapping.

() If F(T) = , then F(T) is closed and convex; () I T is demiclosed at zero.

Lemma . Let H be a real Hilbert space. Then the following statements hold:() For all x, y H and for all t [, ],

tx

() For all x, y H,

x + y x + y, x + y .

Lemma . [] Let E be a uniformly convex Banach space, Br() := {x E : x r} be a closed ball with center and radius r > . Then for any given sequence {x, x, . . . , xn, . . .}

Br() and any given number sequence {, , . . . , n, . . .} with i ,

i= i = , there exists a strictly increasing continuous and convex function g : [, r) [, ) with g() = such that for any i, j N , i < j,

n= nxn

x H chosen arbitrarily, xn+ = ,nyn +

+ ( t)y

= t x + ( t) y t( t) x y . (.)

n=n xn ijg

xi xj

. (.)

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 4 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

Lemma . [] Let {an}, {bn} and {n} be sequences of nonnegative real numbers satisfying

an+ ( + n)an + bn, n . (.)

If

i= n < and

i= bn < , then the limit limn an exists.

Lemma . Let D be a nonempty and closed convex subset of H and T : D D be a k-strictly pseudo-nonspreading mapping with F(T) = . Let T = I + ( )T, [k, ).

Then the following conclusions hold:() F(T) = F(T);

() I T is demiclosed at zero; () Tx Ty x y +

x Tx, y Ty ; () T is a quasi-nonexpansive mapping.

Proof Since (I T) = ( )(I T), the conclusions (), () are obvious.

Now we prove the conclusion (). In fact, since T is a k-strictly pseudo-nonspreading mapping, it follows from Lemma . that

Tx Ty = (x

y) + ( )(Tx Ty)

= x y + ( ) Tx Ty ( )

x

Tx (y Ty)

x y + ( )

x y + k

Tx (y Ty)

+ x Tx, y Ty

x

x

( )

Tx (y Ty)

= x y + ( ) x Tx, y Ty

( )( k)

x

Tx (y Ty)

x y + ( ) x Tx, y Ty

= x y +

( ) x Tx, y Ty , x, y D. (.)

If y F(T), then y F(T). Hence from equation (.),

Tx y = Tx Ty x y , x D. (.)

This completes the proof of Lemma ..

Lemma . [] Let H be a Hilbert space and {un} be a sequence in H such that there exists a nonempty set W H satisfying:

() for every w W, limn un w exists;() each weak-cluster point of the sequence {wn} is in W.

Then there exists w W such that {un} weakly converges to w.

3 Weak- and strong-convergence theorems

For solving the multiple-set split feasibility problem (MSSFP) equation (.), we assume that the following conditions are satised:

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 5 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

() H and H are two real Hilbert spaces, A : H H is a bounded linear operator and A : H H is the adjoint of A;

() {Si}i= : H H is an innite family of ki-strictly pseudo-nonspreading mappings with k := supi ki (, );

() {Ti}Ni= : H H is a nite family of i-strictly pseudo-nonspreading mappings with = max{i : i = , , . . . , N} (, );

() C :=

i= F(Si) = and Q :=

Ni= F(Ti) = .

Now we are in a position to give the following main theorem.

Theorem . Let H, H, A, A, {Si}i=, {Ti}Ni=, C, Q, k, be the same as above. Let {xn} be a sequence generated by

x H chosen arbitrarily, xn+ = ,nyn +

i= i,nSi,yn,yn = xn + A(Tn(modN) I)Axn,n , (.)

where Si, := I + ( )Si, i , [k, ) is a constant, {i,n} (, ) and > satisfy the following conditions:(a)

i= i,n = , for each n ;(b) for each i , lim infn ,ni,n > ;

(c) (, A ).

Let = {x C, Ax Q} = (the set of solutions of (MSSFP) equation (.) dened by equation (.)). Then we have the following:(I) both {xn} and {yn} converge weakly to some point x ;(II) in addition, if there exists some positive integer m such that Sm is semicompact, then both {xn} and {yn} converge strongly to x .

Proof First we prove the conclusion (I).

Step . We prove that the sequences {xn}, {yn} and {Si,yn} are bounded and, for each p , the following limits exist and

lim

n xn p =

lim

n yn p .

In fact, for given p , by the denition of ,

p C =

i=

F(Si) =

i=

F(Si,)

and

Ap Q :=

N

i=

F(Ti).

Therefore, we have

Ap = Tn(modN)Ap. (.)

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 6 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

Since {Si}i= is a family of k-strictly pseudo-nonspreading mappings, by Lemma ., C =

i= F(Si) is closed and convex. It follows from Lemma . that, for each n and p ,

xn+ p =

,n(yn p) +

i=i,n(Si,yn p)

i=i,n Si,yn p

= yn p , (.)

yn p =

,n yn p +

x

n p + A(Tn(modN) I)Axn

= xn p +

xn p, A(Tn(modN) I)Axn

+

A(T

n(mod N) I)Axn

(.)

and

A(T

n(mod N) I)Axn

=

A(Tn(modN) I)Axn, A(Tn(modN) I)Axn

=

AA(Tn(modN) I)Axn, (Tn(modN) I)Axn

A

(T

n(mod N) I)Axn

.

(.)

Further, since {Ti}Ni= is a nite family of -strictly pseudo-nonspreading mappings, we have

xn p, A(Tn(modN) I)Axn

= A(xn p), (Tn(modN) I)Axn

= A(xn p) + (Tn(modN) I)Axn (Tn(modN) I)Axn, (Tn(modN) I)Axn

= Tn(modN)Axn Ap, (Tn(modN) I)Axn

(T

n(mod N) I)Axn

=

Tn(modN)Axn Ap +

(T

n(mod N) I)Axn

Axn Ap

(T

n(mod N) I)Axn

=

Tn(modN)Axn Tn(modN)Ap +

(T

n(mod N) I)Axn

Axn Ap

(T

n(mod N) I)Axn

Axn Ap +

(T

n(mod N) I)Axn

+

(T

n(mod N) I)Axn

Axn Ap

(T

n(mod N) I)Axn

=

(T

n(mod N) I)Axn

.

(.)

Substituting equations (.) and (.) into equation (.) and simplifying, we have

yn p xn p

A

(Tn(modN)

I)Axn

.

(.)

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 7 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

By condition (c), ( A ) > , therefore we have

yn p xn p . (.)

Substituting equation (.) into equation (.), we have

xn+ p xn p , n .

This implies that the limit limn xn p exists. It follows from equations (.) and (.) that the limit limn yn p exists also, and

lim

n xn p =

lim

n yn p , p

. (.)

Therefore, {xn} and {yn} are bounded. Since for each i , Si, is quasi-nonexpansive, we have

Si,yn p yn p .

Hence {Si,yn} is also bounded.

Step . Now we prove that for any given positive integer l , the following conclusions hold:

lim

n yn Sl,yn = ;

lim

n Tn(modN)Axn Axn = . (.)

In fact, for any given p , it follows from equation (.), Lemma ., and equation (.) that

xn+ p =

,n(yn p) +

i=i,n(Si,yn p)

,n yn p +

i=i,n Si,yn p ,nl,ng

yn Si,yn

,n yn p +

i=i,n yn p ,nl,ng

yn Si,yn

= yn p ,nl,ng

yn Si,yn

xn p

A

(T

n(mod N) I)Axn

,nl,ng

yn Si,yn

, n . (.)

Therefore, we have

A

(T

n(mod N) I)Axn

+ ,nl,ng

yn Sl,yn

xn p xn+ p

(as n ).

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 8 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

By conditions (b) and (c) we have

lim

n (T

n(mod N) I)Axn

= ; lim

n g y

n Sl,yn

= . (.)

Since g is continuous and strictly increasing with g() = , from equation (.) we have

lim

n yn Sl,yn = for each l . (.)

Hence conclusion (.) is proved.Step . Now, we prove that

lim

n xn+ xn = ;

lim

n yn+ yn = . (.)

In fact, it follows from equation (.) that

xn+ xn =

,n(yn xn) +

i=i,n(Si,yn xn)

=

,n

A(Tn(modN) I)Axn +

i=i,n(Si,yn xn)

,n

A(Tn(modN) I)Axn

+

i=i,n Si,yn xn

,n

A(Tn(modN) I)Axn

+

i=i,n

Si,yn yn + yn xn

. (.)

By virtue of equations (.) and (.), one has

yn xn = A(T

n(mod N) I)Axn

(as n ). (.)

This together with equations (.) and (.) shows that

xn+ xn (as n ).

Similarly, we have

yn+ yn = x

n+ + A(T(n+)(modN) I)Axn+

xn + A(Tn(modN) I)Axn

xn+ xn +

A(T

(n+)(mod N) I)Axn+

+

A(T

n(mod N) I)Axn

(as n ).

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 9 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

Step . Now we show that every weak-cluster point x of the sequence {xn} is in . Indeed, since {yn} is a bounded sequence in H, there exists a subsequence {yni} {yn}

such that yni x H. It follows from equation (.) that

lim

n yni Sl,yni = for each l .

By Lemma ., (I Si) is demiclosed at zero. Since (I Sl,) = ( )(I Si), this implies that (I Sl,) is also demiclosed at zero. Hence x F(Sl,) = F(Sl). By the arbitrariness of l , we have

x

On the other hand, it follows from equations (.) and (.) that

xni = yni A(Tni(modN) I)Axni x. (.)

Since A is a bounded linear operator, this implies that Axni Ax. Also, by equation (.)

lim

ni Tni(modN)Axni Axni = . (.)

Hence for any given positive integer j = , , . . . , N, there exists a subsequence {nik} {ni} with nik(mod N) = j such that

lim

nik TjAxnik Axnik = .

Since Axnik Ax, and by Lemma ., I Tj is demiclosed at . This implies that Ax F(Tj). By the arbitrariness of j = , , . . . , N,

Ax

These show that x .

Step . Summing up the above arguments, we have proved that: (i) for each p , the limits limn xn p and limn yn p exist (see equation (.)); (ii) every weak-cluster point x of the sequence {xn} (or {yn}) is in . Taking W = and {un} = {xn} (or {yn})

in Lemma ., therefore all conditions in Lemma . are satised. By using Lemma ., xn x, yn x and x . This completes the proof of the conclusion (I).

Next we prove the conclusion (II).

Without loss of generality, we may assume that S is semicompact. Since (I S,) = ( )(I S), this implies that S, is also semicompact. In view of equation (.), we have

yn S,yn (as n ). (.)

i=

F(Si) = C.

N

j=

F(Tj) = Q.

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 10 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

Therefore, there exists a subsequence of {yni} {yn} such that yni u H. Since yni x, we have x = u and so yni x . By virtue of equation (.), we have

lim

n y

= ,

i.e., {yn} and {xn} both converge strongly to the point x . This completes the proof of Theorem ..

Remark . Theorem . improves and extends the corresponding results of Censor et al. [, , ], Byrne [], Yang [], Mouda [], Xu [], Censor and Segal [], Masad and Reich [], Deepho and Kumam [, ] 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 dierent from the ones given in [, , , , , ]. Under suitable conditions, some weak- and strong-convergence results for the algorithms are proved.

If we put = in Theorem ., we immediately get the following.

Corollary . Let H, {Si}i=, k be the same as above. Let {xn} be a sequence generated by

x H chosen arbitrarily, xn+ = ,nxn +

i= i,nSi,xn, n , (.)

where Si, := I + ( )Si, i , [k, ) is a constant, {i,n} (, ) satisfy the following conditions:(a)

i= i,n = , for each n ;(b) for each i , lim infn ,ni,n > . Let

i=

F(Si) = .

n x

n x

= , lim

n

x

i=

F(Si) = .

Then we have the following:(I) the sequence {xn} converges weakly to some point x F ;(II) in addition, if there exists some positive integer m such that Sm is semicompact, then the sequence {xn} converges strongly to x F .

4 Applications

In this section we utilize the results presented in Section to study the hierarchical variational inequality problem.

Let H be a real Hilbert space, {Si} : H H, i = , , . . . be a countable family of ki-strictly pseudo-nonspreading mappings with k = supi ki (, ), and

F :=

F :=

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 11 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

Let T : H H be a nonspreading mapping. The so-called hierarchical variational inequality problem for a countable family of mappings {Si} with respect to mapping T is to nd an x F such that

x Tx, x x

, x F . (.)

It is easy to see that equation (.) is equivalent to the following xed point problem: to nd x F such that

x = PF Tx, (.)

where PF is the metric projection from H onto F . Letting C = F and Q = F(PF T) (the xed point set of PF T) and A = I (the identity mapping on H), then the problem (.) is equivalent to the following multi-set split feasibility problem: to nd x C such that

x Q. (.)

Hence from Theorem . we have the following theorem.

Theorem . Let H, {Si}, T, C, Q, k be the same as above. Let {xn}, {yn} be the sequences dened by

where Si, := I + ( )Si, i , [k, ), {i,n} (, ) and > satisfy the following conditions:(a)

i= i,n = , for each n ;(b) for each i , lim infn ,ni,n > ;

(c) (, ).

If C Q = , then {xn} converges weakly to a solution of the hierarchical variational inequality problem (.). In addition, if one of the mappings Si is semicompact, then both {xn} and {yn} converge strongly to a solution of the hierarchical variational inequality problem (.).

Proof In fact, by the assumption that T is a nonspreading mapping, hence by Remark ., T is a -strictly pseudo-nonspreading with = . Taking N = and A = I in Theorem ., all conditions in Theorem . are satised. The conclusions of Theorem . can immediately be obtained from Theorem ..

Remark . If T = I (the identity mapping), then we can get the results of Corollary ..

Competing interests

The authors declare that they have no competing interests.

Authors contributions

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 nal manuscript.

x H chosen arbitrarily, xn+ = ,nyn +

i= i,nSi,yn, yn = xn + (T I)xn, n ,

(.)

Chang et al. Fixed Point Theory and Applications 2014, 2014:11 Page 12 of 12 http://www.fixedpointtheoryandapplications.com/content/2014/1/11

Web End =http://www.xedpointtheoryandapplications.com/content/2014/1/11

Author details

1College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.

2Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, Korea. 3Department of Mathematics Education and the RlNS, Gyeongsang National University, Chinju, 660-701, Korea. 4Department of Mathematics, Kyungnam University, Changwon, Gyeongnam 631-701, Korea.

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).

Received: 1 October 2013 Accepted: 17 December 2013 Published: 9 January 2014

References

1. Osilike, MO, Isiogugu, FO: Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. Nonlinear Anal. 74, 1814-1822 (2011)

2. Kurokawa, Y, Takahashi, W: Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces. Nonlinear Anal. 73, 1562-1568 (2010)

3. Baillon, J: Un theorem de type ergodique pour les contractions nonlineaires dans un espace de Hilbert. C. R. Acad. Sci., Ser. A-B 280(Aii), A1511-A1514 (1975)

4. Halpern, B: Fixed points of nonexpanding mappings. Bull. Am. Math. Soc. 73, 957-961 (1967)5. Censor, Y, Elfving, T: A multi-projection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221-239 (1994)

6. Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18, 441-453 (2002)

7. Censor, Y, Bortfeld, T, Martin, B, Tromov, A: A unied approach for inversion problem in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353-2365 (2006)

8. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications. Inverse Probl. 21, 2071-2084 (2005)

9. Censor, Y, Motova, A, Segal, A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244-1256 (2007)

10. Xu, HK: A variable Krasnoselskii-Mann algorithm and the multiple-sets split feasibility problem. Inverse Probl. 22,

2021-2034 (2006)

11. Yang, Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 20, 1261-1266 (2004)12. Zhao, J, Yang, Q: Several solution methods for the split feasibility problem. Inverse Probl. 21, 1791-1799 (2005)13. Chang, SS, Kim, JK, Wang, XR: Modied block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010, Article ID 869684 (2010). doi:10.1155/2010/869684

14. Mouda, A, Al-Shemas, E: Simultaneous iterative methods for split equality problem. Trans. Math. Program. Appl. 1, 1-11 (2013)

15. Mouda, A: The split common xed point problem for demi-contractive mappings. Inverse Probl. 26, 055007 (2010)16. Xu, HK: Iterative methods for split feasibility problem in innite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)

17. Censor, Y, Segal, A: The split common xed point problem for directed operators. J. Convex Anal. 16, 587-600 (2009)18. Masad, E, Reich, S: A note on the multiple-set split feasibility problem in Hilbert spaces. J. Nonlinear Convex Anal. 8, 367-371 (2007)

19. Deepho, J, Kumam, P: Split feasibility and xed-point problems for asymptotically quasi-nonexpansive mappings.J. Inequal. Appl. 2013, 322 (2013). doi:10.1186/1029-242X-2013-32220. Deepho, J, Kumam, P: A modied Halperns iterative scheme for solving split feasibility problems. Abstr. Appl. Anal. 2012, Article ID 876069 (2012)

21. Yang, L, Chang, SS, Cho, YJ, Kim, JK: Multiple-set split feasibility problems for total asymptotically strict pseudocontraction mappings. Fixed Point Theory Appl. 2011, 77 (2011). doi:10.1186/1687-1812-2011-77

22. Chang, SS, Cho, YJ, Kim, JK, Zhang, WB, Yang, L: Multiple-set split feasibility problems for asymptotically strict pseudocontractions. Abstr. Appl. Anal. 2012, Article ID 491760 (2012). doi:10.1155/2012/491760

doi:10.1186/1687-1812-2014-11Cite this article as: Chang et al.: Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces. Fixed Point Theory and Applications 2014 2014:11.