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

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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.

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 x’ and ‘ x n 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:DD.

Definition 1.1 Let T:DD be a mapping.

  1. (1)

    T:DD is said to be nonexpansive if

    TxTyxy,x,yD.
  2. (2)

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

    Txpxp,xD,pF(T).
    (1.1)
  3. (3)

    T is said to be nonspreading if

    2 T x T y 2 T x y 2 + T y x 2 ,x,yD.
    (1.2)

It is easy to prove that equation (1.2) is equivalent to

T x T y 2 x y 2 +2xTx,yTy,x,yD.
(1.3)
  1. (4)

    T is said to be k-strictly pseudo-nonspreading [1], if there exists a constant k[0,1) such that

    T x T y 2 x y 2 +k x T x ( y T y ) 2 +2xTx,yTy,x,yD.
    (1.4)

Remark 1.2 It follows from Definition 1.1 that

  1. (1)

    if T is nonspreading and F(T), then T is quasi-nonexpansive;

  2. (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]

Let denote the set of real numbers with the usual norm. Let T:RR be a mapping defined by

Tx= { x , x ( , 0 ) , 2 x , x [ 0 , ) .
(1.5)

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 [79].

The split feasibility problem in an infinitely dimensional Hilbert space can be found in [6, 8, 1012].

The purpose of this paper is to introduce the following multiple-set split feasibility problem (MSSFP) for an infinite family of 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 C such that

A x Q,
(1.6)

where H 1 , H 2 are two real Hilbert spaces, A: H 1 H 2 is a bounded linear operator, { S i } i = 1 : H 1 H 1 is an infinite family of k i -strictly pseudo-nonspreading mappings and { T i } i = 1 N : H 2 H 2 is a finite family of ρ i -strictly pseudo-nonspreading mappings, C:= i = 1 F( S i ) and Q:= 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):

{ x 1 H 1 chosen arbitrarily , x n + 1 = α 0 , n y n + i = 1 α i , n S i , β y n , y n = x n + γ A ( T n ( mod N ) I ) A x n , n 1 ,

where S i , β :=βI+(1β) S i , β(0,1) is a constant.

In the sequel we denote Γ the set of solutions of (MSSFP) equation (1.6), i.e.,

Γ={xC,AxQ}=C A 1 (Q).
(1.7)

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:EE be a mapping.

  1. (1)

    IT is said to be demiclosed at 0, if, for any sequence { x n }H with x n x , (IT) x n 0, then x =T x .

  2. (2)

    T is said to be semicompact, if, for any bounded sequence { x n }E, lim n x n T x n =0, then there exists a subsequence { x n i }{ x n } such that { x n i } converges strongly to some point x E.

Lemma 2.2 [1]

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

  1. (1)

    If F(T), then F(T) is closed and convex;

  2. (2)

    IT is demiclosed at zero.

Lemma 2.3 Let H be a real Hilbert space. Then the following statements hold:

  1. (1)

    For all x,yH and for all t[0,1],

    t x + ( 1 t ) y 2 =t x 2 +(1t) y 2 t(1t) x y 2 .
    (2.1)
  2. (2)

    For all x,yH,

    x + y 2 x 2 +2y,x+y.

Lemma 2.4 [13]

Let E be a uniformly convex Banach space, B r (0):={xE:xr} be a closed ball with center 0 and radius r>0. Then for any given sequence { x 1 , x 2 ,, x n ,} B r (0) and any given number sequence { λ 1 , λ 2 ,, λ n ,} with λ i 0, i = 1 λ i =1, there exists a strictly increasing continuous and convex function g:[0,2r)[0,) with g(0)=0 such that for any i,jN, i<j,

n = 1 λ n x n 2 n = 1 λ n x n 2 λ i λ j g ( x i x j ) .
(2.2)

Lemma 2.5 [10]

Let { a n }, { b n } and { δ n } be sequences of nonnegative real numbers satisfying

a n + 1 (1+ δ n ) a n + b n ,n1.
(2.3)

If i = 1 δ n < and i = 1 b n <, then the limit lim n a n exists.

Lemma 2.6 Let D be a nonempty and closed convex subset of H and T:DD be a k-strictly pseudo-nonspreading mapping with F(T). Let T β =βI+(1β)T, β[k,1). Then the following conclusions hold:

  1. (1)

    F(T)=F( T β );

  2. (2)

    I T β is demiclosed at zero;

  3. (3)

    T β x T β y 2 x y 2 + 2 1 β x T β x,y T β y;

  4. (4)

    T β is a quasi-nonexpansive mapping.

Proof Since (I T β )=(1β)(IT), the conclusions (1), (2) are obvious.

Now we prove the conclusion (3). In fact, since T is a k-strictly pseudo-nonspreading mapping, it follows from Lemma 2.3 that

T β x T β y 2 = β ( x y ) + ( 1 β ) ( T x T y ) 2 = β x y 2 + ( 1 β ) T x T y 2 β ( 1 β ) x T x ( y T y ) 2 β x y 2 + ( 1 β ) { x y 2 + k x T x ( y T y ) 2 + 2 x T x , y T y } β ( 1 β ) x T x ( y T y ) 2 = x y 2 + 2 ( 1 β ) x T x , y T y ( 1 β ) ( β k ) x T x ( y T y ) 2 x y 2 + 2 ( 1 β ) x T x , y T y = x y 2 + 2 ( 1 β ) x T β x , y T β y , x , y D .
(2.4)

If yF(T), then yF( T β ). Hence from equation (2.4),

T β xy= T β x T β yxy,xD.
(2.5)

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 WH satisfying:

  1. (1)

    for every wW, lim n u n w exists;

  2. (2)

    each weak-cluster point of the sequence { w n } is in W.

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

3 Weak- and strong-convergence theorems

For solving the multiple-set split feasibility problem (MSSFP) equation (1.6), we assume that the following conditions are satisfied:

  1. (1)

    H 1 and H 2 are two real Hilbert spaces, A: H 1 H 2 is a bounded linear operator and A : H 2 H 1 is the adjoint of A;

  2. (2)

    { S i } i = 1 : H 1 H 1 is an infinite family of k i -strictly pseudo-nonspreading mappings with k:= sup i 1 k i (0,1);

  3. (3)

    { T i } i = 1 N : H 2 H 2 is a finite family of ρ i -strictly pseudo-nonspreading mappings with ρ=max{ ρ i :i=1,2,,N}(0,1);

  4. (4)

    C:= i = 1 F( S i ) and Q:= i = 1 N F( T i ).

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

Theorem 3.1 Let H 1 , H 2 , A, A , { S i } i = 1 , { T i } i = 1 N , C, Q, k, ρ be the same as above. Let { x n } be a sequence generated by

{ x 1 H 1 chosen arbitrarily , x n + 1 = α 0 , n y n + i = 1 α i , n S i , β y n , y n = x n + γ A ( T n ( mod N ) I ) A x n , n1,
(3.1)

where S i , β :=βI+(1β) S i , i1, β[k,1) is a constant, { α i , n }(0,1) and γ>0 satisfy the following conditions:

  1. (a)

    i = 0 α i , n =1, for each n1;

  2. (b)

    for each i1, lim inf n α 0 , n α i , n >0;

  3. (c)

    γ(0, 1 ρ A 2 ).

Let Γ={xC,AxQ} (the set of solutions of (MSSFP) equation (1.6) defined by equation (1.7)). Then we have the following:

  1. (I)

    both { x n } and { y n } converge weakly to some point x Γ;

  2. (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 Γ.

Proof First we prove the conclusion (I).

Step 1. We prove that the sequences { x n }, { y n } and { S i , β y n } are bounded and, for each pΓ, the following limits exist and

lim n x n p= lim n y n p.

In fact, for given pΓ, by the definition of Γ,

pC= i = 1 F( S i )= i = 1 F( S i , β )

and

ApQ:= i = 1 N F( T i ).

Therefore, we have

Ap= T n ( mod N ) Ap.
(3.2)

Since { S i } i = 1 is a family of k-strictly pseudo-nonspreading mappings, by Lemma 2.2, C= i = 1 F( S i ) is closed and convex. It follows from Lemma 2.6 that, for each n1 and pΓ,

x n + 1 p = α 0 , n ( y n p ) + i = 1 α i , n ( S i , β y n p ) x n + 1 p α 0 , n y n p + i = 1 α i , n S i , β y n p x n + 1 p = y n p ,
(3.3)
y n p 2 = x n p + γ A ( T n ( mod N ) I ) A x n 2 y n p 2 = x n p 2 + 2 γ x n p , A ( T n ( mod N ) I ) A x n y n p 2 + γ 2 A ( T n ( mod N ) I ) A x n 2
(3.4)

and

γ 2 A ( T n ( mod N ) I ) A x n 2 = γ 2 A ( T n ( mod N ) I ) A x n , A ( T n ( mod N ) I ) A x n = γ 2 A A ( T n ( mod N ) I ) A x n , ( T n ( mod N ) I ) A x n γ 2 A 2 ( T n ( mod N ) I ) A x n 2 .
(3.5)

Further, since { T i } i = 1 N is a finite family of ρ-strictly pseudo-nonspreading mappings, we have

x n p , A ( T n ( mod N ) I ) A x n = A ( x n p ) , ( T n ( mod N ) I ) A x n = A ( x n p ) + ( T n ( mod N ) I ) A x n ( T n ( mod N ) I ) A x n , ( T n ( mod N ) I ) A x n = T n ( mod N ) A x n A p , ( T n ( mod N ) I ) A x n ( T n ( mod N ) I ) A x n 2 = 1 2 { T n ( mod N ) A x n A p 2 + ( T n ( mod N ) I ) A x n 2 A x n A p 2 } ( T n ( mod N ) I ) A x n 2 = 1 2 { T n ( mod N ) A x n T n ( mod N ) A p 2 + ( T n ( mod N ) I ) A x n 2 A x n A p 2 } ( T n ( mod N ) I ) A x n 2 1 2 { A x n A p 2 + ρ ( T n ( mod N ) I ) A x n 2 } + 1 2 { ( T n ( mod N ) I ) A x n 2 A x n A p 2 } ( T n ( mod N ) I ) A x n 2 = ρ 1 2 ( T n ( mod N ) I ) A x n 2 .
(3.6)

Substituting equations (3.5) and (3.6) into equation (3.4) and simplifying, we have

y n p 2 x n p 2 γ ( 1 ρ γ A 2 ) ( T n ( mod N ) I ) A x n 2 .
(3.7)

By condition (c), (1ργ A 2 )>0, therefore we have

y n p 2 x n p 2 .
(3.8)

Substituting equation (3.8) into equation (3.3), we have

x n + 1 p x n p,n1.

This implies that the limit lim n x n p exists. It follows from equations (3.8) and (3.3) that the limit lim n y n p exists also, and

lim n x n p= lim n y n p,pΓ.
(3.9)

Therefore, { x n } and { y n } are bounded. Since for each i1, S i , β is quasi-nonexpansive, we have

S i , β y n p y n p.

Hence { S i , β y n } is also bounded.

Step 2. Now we prove that for any given positive integer l1, the following conclusions hold:

lim n y n S l , β y n =0; lim n T n ( mod N ) A x n A x n =0.
(3.10)

In fact, for any given pΓ, it follows from equation (3.1), Lemma 2.4, and equation (3.7) that

x n + 1 p 2 = α 0 , n ( y n p ) + i = 1 α i , n ( S i , β y n p ) 2 α 0 , n y n p 2 + i = 1 α i , n S i , β y n p 2 α 0 , n α l , n g ( y n S i , β y n ) α 0 , n y n p 2 + i = 1 α i , n y n p 2 α 0 , n α l , n g ( y n S i , β y n ) = y n p 2 α 0 , n α l , n g ( y n S i , β y n ) x n p 2 γ ( 1 ρ γ A 2 ) ( T n ( mod N ) I ) A x n 2 α 0 , n α l , n g ( y n S i , β y n ) , n 1 .
(3.11)

Therefore, we have

γ ( 1 ρ γ A 2 ) ( T n ( mod N ) I ) A x n 2 + α 0 , n α l , n g ( y n S l , β y n ) x n p 2 x n + 1 p 2 0 ( as  n ) .

By conditions (b) and (c) we have

lim n ( T n ( mod N ) I ) A x n =0; lim n g ( y n S l , β y n ) =0.
(3.12)

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

lim n y n S l , β y n =0for each l1.
(3.13)

Hence conclusion (3.10) is proved.

Step 3. Now, we prove that

lim n x n + 1 x n =0; lim n y n + 1 y n =0.
(3.14)

In fact, it follows from equation (3.1) that

x n + 1 x n 2 = α 0 , n ( y n x n ) + i = 1 α i , n ( S i , β y n x n ) 2 = α 0 , n ( γ A ( T n ( mod N ) I ) A x n ) + i = 1 α i , n ( S i , β y n x n ) 2 α 0 , n γ A ( T n ( mod N ) I ) A x n 2 + i = 1 α i , n S i , β y n x n 2 α 0 , n γ A ( T n ( mod N ) I ) A x n 2 + i = 1 α i , n ( S i , β y n y n + y n x n ) 2 .
(3.15)

By virtue of equations (3.1) and (3.10), one has

y n x n = γ A ( T n ( mod N ) I ) A x n 0 ( as  n ) .
(3.16)

This together with equations (3.10) and (3.15) shows that

x n + 1 x n 0(as n).

Similarly, we have

y n + 1 y n = x n + 1 + γ A ( T ( n + 1 ) ( mod N ) I ) A x n + 1 [ x n + γ A ( T n ( mod N ) I ) A x n ] x n + 1 x n + γ A ( T ( n + 1 ) ( mod N ) I ) A x n + 1 + γ A ( T n ( mod N ) I ) A x n 0 ( as  n ) .

Step 4. Now we show that every weak-cluster point x 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 }{ y n } such that y n i x H 1 . It follows from equation (3.10) that

lim n y n i S l , β y n i =0for each l1.

By Lemma 2.2, (I S i ) is demiclosed at zero. Since (I S l , β )=(1β)(I S i ), this implies that (I S l , β ) is also demiclosed at zero. Hence x F( S l , β )=F( S l ). By the arbitrariness of l1, we have

x i = 1 F( S i )=C.

On the other hand, it follows from equations (3.1) and (3.10) that

x n i = y n i γ A ( T n i ( mod N ) I)A x n i x .
(3.17)

Since A is a bounded linear operator, this implies that A x n i A x . Also, by equation (3.10)

lim n i T n i ( mod N ) A x n i A x n i =0.
(3.18)

Hence for any given positive integer j=1,2,,N, there exists a subsequence { n i k }{ n i } with n i k (modN)=j such that

lim n i k T j A x n i k A x n i k =0.

Since A x n i k A x , and by Lemma 2.2, I T j is demiclosed at 0. This implies that A x F( T j ). By the arbitrariness of j=1,2,,N,

A x j = 1 N F( T j )=Q.

These show that x Γ.

Step 5. Summing up the above arguments, we have proved that: (i) for each pΓ, the limits lim n x n p and lim n y n p exist (see equation (3.2)); (ii) every weak-cluster point x of the sequence { x n } (or { y n }) is in Γ. Taking W=Γ 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 x , y n x and x Γ. 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 , β )=(1β)(I S 1 ), this implies that S 1 , β is also semicompact. In view of equation (3.10), we have

y n S 1 , β y n 0(as n).
(3.19)

Therefore, there exists a subsequence of { y n i }{ y n } such that y n i u H 1 . Since y n i x , we have x = u and so y n i x Γ. By virtue of equation (3.9), we have

lim n y n x =0, lim n x n x =0,

i.e., { y n } and { x n } both converge strongly to the point x Γ. 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:

  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;

  2. (b)

    for the algorithms, we propose some new hybrid iterative algorithms which are different from the ones given in [57, 9, 17, 18, 21, 22]. Under suitable conditions, some weak- and strong-convergence results for the algorithms are proved.

If we put γ=0 in Theorem 3.1, we immediately get the following.

Corollary 3.3 Let H, { S i } i = 1 , k be the same as above. Let { x n } be a sequence generated by

{ x 1 H chosen arbitrarily , x n + 1 = α 0 , n x n + i = 1 α i , n S i , β x n , n1,
(3.20)

where S i , β :=βI+(1β) S i , i1, β[k,1) is a constant, { α i , n }(0,1) satisfy the following conditions:

  1. (a)

    i = 0 α i , n =1, for each n1;

  2. (b)

    for each i1, lim inf n α 0 , n α i , n >0. Let

    F:= i = 1 F( S i ).

Then we have the following:

  1. (I)

    the sequence { x n } converges weakly to some point x F;

  2. (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 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 }:HH, i=1,2, be a countable family of k i -strictly pseudo-nonspreading mappings with k= sup i 1 k i (0,1), and

F:= i = 1 F( S i ).

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

x T x , x x 0,xF.
(4.1)

It is easy to see that equation (4.1) is equivalent to the following fixed point problem: to find x F such that

x = P F T x ,
(4.2)

where P F is the metric projection from H onto . Letting C=F and Q=F( P F T) (the fixed point set of P 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 C such that

x Q.
(4.3)

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

{ x 1 H 1 chosen arbitrarily , x n + 1 = α 0 , n y n + i = 1 α i , n S i , β y n , y n = x n + γ ( T I ) x n , n 1 ,
(4.4)

where S i , β :=βI+(1β) S i , i1, β[k,1), { α i , n }(0,1) and γ>0 satisfy the following conditions:

  1. (a)

    i = 0 α i , n =1, for each n1;

  2. (b)

    for each i1, lim inf n α 0 , n α i , n >0;

  3. (c)

    γ(0,1).

If CQ, 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 ρ=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.

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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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Correspondence to Jong Kyu Kim.

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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 final manuscript.

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Chang, S., Kim, J.K., Cho, Y.J. et al. Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces. Fixed Point Theory Appl 2014, 11 (2014) doi:10.1186/1687-1812-2014-11

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Keywords

  • split feasibility problem
  • convex feasibility problem
  • k-strictly pseudo-nonspreading mapping
  • demicloseness
  • Opial’s condition