Open Access

Multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in Hilbert spaces

Journal of Inequalities and Applications20142014:69

https://doi.org/10.1186/1029-242X-2014-69

Received: 19 October 2013

Accepted: 30 January 2014

Published: 13 February 2014

Abstract

Some weak and strong convergence theorems for solving multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in infinite-dimensional Hilbert spaces are proved. The results presented in the paper extend and improve the corresponding results of Xu (Inverse Probl. 22(6):2021-2034, 2006), Osilike and Isiogugu (Nonlinear Anal. 74:1814-1822, 2011), Chang et al. (Abstr. Appl. Anal. 2012:491760, 2012), and others.

MSC:47H05, 47H09, 49M05.

Keywords

weak and strong convergencemultiple-set split feasibilityκ-asymptotically strictly pseudo-nonspreading mapping

1 Introduction

Throughout this article, we always assume that H 1 , H 2 are real Hilbert spaces; ‘→’ and ‘’ denote strong and weak convergence, respectively.

The split feasibility problem ( S F P ) in finite dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems. The ( S F P ) can be used in various disciplines such as medical image reconstruction [2], image restoration, computer tomography, and radiation therapy treatment planning [35]. The multiple-set split feasibility problem ( M S S F P ) was studied in [47].

Let A : H 1 H 2 be a bounded linear operator, S i : H 1 H 1 and T i : H 2 H 2 , i = 1 , 2 , , N , be two finite families of mappings, C : = i = 1 N F ( S i ) and Q : = i = 1 N F ( T i ) , where F ( S i ) and F ( T i ) are the sets of fixed points of S i and T i , respectively.

The so-called multiple set split feasibility problem is
to find  x C  such that  A x Q .
(1.1)
In the sequel, we use Γ to denote the set of solutions of the problem ( M S S F P ) (1.1), that is,
Γ = { x C : A x Q } .
(1.2)
Let H be a real Hilbert space and K be a nonempty closed convex subset of H. Following Kohsaka and Takahashi [811], a mapping T : K K is said to be nonspreading if
2 T x T y 2 T x y 2 + T y x 2 for all  x , y K .
It is to see that the above inequality is equivalent to
T x T y 2 x y 2 + 2 x T x , y T y for all  x , y K .

In 1967, Browder and Petryshyn [12] introduced the concept of κ-strictly pseudo-nonspreading mapping.

Definition 1.1 [12]

Let H be a real Hilbert space. A mapping T : D ( T ) H H is said to be κ-strictly pseudo-nonspreading if there exists κ [ 0 , 1 ) such that
T x T y 2 x y 2 + κ x T x ( y T y ) 2 + 2 x T x , y T y , x , y D ( T ) .

Clearly, every nonspreading mapping is κ-strictly pseudo-nonspreading.

The class of asymptotically strict pseudo-contractions was introduced by Qihou [13] in 1996. Kim and Xu [14], Inchan and Nammanee [15], Zhou [16] Cho [17], and Ge [18] proved that the class of asymptotically strict pseudo-contractions is demiclosed at the origin and also obtained some weak convergence theorems for the class of mappings. In 2012, Osilike and Isiogugu [19] introduced a class of nonspreading type mappings which is more general than the class studied in [11] in Hilbert spaces and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, Chang et al. [7] studied the multiple-set split feasibility problem for an asymptotically strict pseudo-contraction in the framework of infinite-dimensional Hilbert spaces.

Definition 1.2 [7]

Let H be a real Hilbert space, we say that the mapping T : D ( T ) H H is a κ-asymptotically strict pseudo-contraction if there exists a constant κ [ 0 , 1 ) and a sequence { k n } [ 1 , ) with k n 1 ( n ) such that
T n x T n y 2 k n x y 2 + κ x T n x ( y T n y ) 2

holds for all x , y D ( T ) .

In this article we introduce the following class of κ-asymptotically strictly pseudo-nonspreading mappings which is more general than that of κ-strictly pseudo-nonspreading mappings and κ-asymptotically strict pseudo-contractions.

Definition 1.3 Let H be a real Hilbert space. A mapping T : D ( T ) H H is said to be κ-asymptotically strictly pseudo-nonspreading if there exists a constant κ [ 0 , 1 ) and a sequence { k n } [ 1 , ) with k n 1 ( n ) such that
T n x T n y 2 k n x y 2 + κ x T n x ( y T n y ) 2 + 2 x T n x , y T n y , x , y D ( T ) .
(1.3)

Example 1.4 Now, we give an example of κ-asymptotically strict pseudo-contractive mapping.

Let C be a unit ball in a real Hilbert l 2 , and let T : C C be a mapping defined by
T : ( x 1 , x 2 , ) ( 0 , x 1 2 , a 2 x 2 , a 3 x 3 , ) ,
(1.4)

where { a i } is a sequence in ( 0 , 1 ) such that i = 2 α i = 1 2 .

It is proved in Goebel and Kirk [20] that
  1. (i)

    T x T y 2 x y , x , y C ;

     
  2. (ii)

    T n x T n y 2 i = 2 n a j x y , n 2 and x , y C .

     
Define k 1 1 2 = 2 , k n 1 2 = 2 i = 2 n a j , n 2 , then
lim n k n = lim n ( 2 i = 2 n a j ) 2 = 1 .
Letting κ = 0 , then x , y C , n 1 , we have
T n x T n y 2 k n x y 2 + κ x y ( T n x T n y ) 2 ̲ .

This implies that T is a κ-asymptotically strict pseudo-contractive mapping.

Example 1.5 Now, we give an example of κ-asymptotically strictly pseudo-nonspreading mapping.

Let X = l 2 with the norm be defined by
x = i = 1 x i 2 , x = ( x 1 , x 2 , , x n , ) X ,
and let C = { x = ( x 1 , x 2 , , x n , ) | x i R 1 , i = 1 , 2 , } be an orthogonal subspace of X (i.e., x , y C , we have x , y = 0 ). It is obvious that C is a nonempty closed convex subset of X. For each x = ( x 1 , x 2 , , x n , ) C , we define a mapping T : C C by
T x = { ( x 1 , x 2 , , x n , ) if  i = 1 x i < 0 ; ( x 1 , x 2 , , x n , ) if  i = 1 x i 0 .
(1.5)

Next we prove that T is a κ-asymptotically strictly pseudo-nonspreading mapping.

In fact, for any x , y C , we have the following cases.

Case 1. If i = 1 x i < 0 and i = 1 y i < 0 , then we have T n x = x , T n y = y , and so then inequality (1.3) holds.

Case 2. If i = 1 x i < 0 and i = 1 y i 0 , then we have that T n x = x , T n y = ( 1 ) n y . This implies that
{ T n x T n y 2 = x ( 1 ) n y 2 = x 2 + y 2 ; k n x y 2 = k n ( x 2 + y 2 ) ; x T n x ( y T n y ) 2 = [ 1 ( 1 ) n ] 2 y 2 ; 2 x T n x , y T n y = 0 .

Therefore inequality (1.3) holds.

Case 3. If i = 1 x i 0 and i = 1 y i 0 , then we have T n x = ( 1 ) n x , T n y = ( 1 ) n y . Hence we have
{ T n x T n y 2 = ( 1 ) n x ( 1 ) n y 2 = x y 2 = x 2 + y 2 ; k n x y 2 = k n ( x 2 + y 2 ) ; x T n x ( y T n y ) 2 = [ 1 ( 1 ) n ] 2 x y 2 = [ 1 ( 1 ) n ] 2 ( x 2 + y 2 ) ; 2 x T n x , y T n y = 0 .

Thus inequality (1.3) still holds. Therefore the mapping defined by (1.5) is a κ-asymptotically strictly pseudo-nonspreading mapping.

The purpose of this article is under suitable conditions to prove some weak and strong convergence theorems for solving multiple-set split feasibility problem (1.1) for a κ-asymptotically strictly pseudo-nonspreading mapping in infinite-dimensional Hilbert spaces. The results presented in the paper extend and improve the corresponding results of Xu [6], Osilike and Isiogugu [19], Chang et al. [7], and many others.

2 Preliminaries

In the sequel, we first recall some definitions, notations, and conclusions which will be needed in proving our main results.

Let E be a real Banach space. A mapping T with domain D ( T ) and range R ( T ) in E is said to be demiclosed at origin if whenever { x n } is a sequence in D ( T ) converging weakly to a point x D ( T ) and ( I T ) x n converging strongly to 0, then T x = x .

A Banach space E is said to have the O p i a l property if, for any sequence { x n } with x n x , we have
lim inf n x n x < lim inf n x n y

for all y E with y x .

It is well known that each Hilbert space possesses the Opial property.

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

A mapping T : K K is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that
T n x T n y L x y , x , y K .
Let K be a nonempty closed convex subset of a real Hilbert space H. The metric projection P K : H K is a mapping such that for each x H , P K x is the unique point in K such that x P K x x y , y K . It is known that for each x H ,
x P K x , y P K x 0 , y K .
Lemma 2.1 Let H be a real Hilbert space, then the following results hold:
  1. (i)
    For all x , y H and for all t [ 0 , 1 ] ,
    t x + ( 1 t ) y 2 = t x 2 + ( 1 t ) y 2 t ( 1 t ) x y 2 .
     
  2. (ii)

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

     
  3. (iii)
    If { x n } n = 1 is a sequence in H which converges weakly to z H , then
    lim sup n x n y 2 = lim sup n x n z 2 + z y 2 , y H .
     

Lemma 2.2 Let K be a nonempty closed convex subset of a real Hilbert space H, and let T : K K be a continuous κ-asymptotically strictly pseudo-nonspreading mapping. If F ( T ) , then it is a closed and convex subset.

Proof Let { x n } F ( T ) be a sequence such that lim n x n = x K . Now we prove that x F ( T ) . In fact, since T is κ-asymptotically strictly pseudo-nonspreading, for each m 1 , we have
T m x x n 2 = T m x T m x n 2 k m x n x 2 + 2 x T m x , x n T m x n + κ x T m x ( x n T m x n ) 2 = k m x n x 2 + κ x T m x 2 .
Taking the limit as n in the above inequality, we have
T m x x 2 κ x T m x 2 .

Since κ ( 0 , 1 ) , we have T m x x = 0 for each m 1 . Hence T x = x . This shows that F ( T ) is closed.

Now we prove that F ( T ) is convex. In fact, let p 1 , p 2 F ( T ) , and z = λ p 1 + ( 1 λ ) p 2 , we prove that z F ( T ) . Since p 1 z = ( 1 λ ) ( p 1 p 2 ) and p 2 z = λ ( p 2 p 1 ) , by using Lemma 2.1(i), we have
z T m z 2 = λ ( p 1 T m z ) + ( 1 λ ) ( p 2 T m z ) 2 = λ p 1 T m z 2 + ( 1 λ ) p 2 T m z 2 λ ( 1 λ ) p 1 p 2 2 λ ( k m p 1 z 2 + κ p 1 T m p 1 ( z T m z ) 2 + 2 p 1 T m p 1 , z T m z ) + ( 1 λ ) ( k m p 2 z 2 + κ p 2 T m p 2 ( z T m z ) 2 + 2 p 2 T m p 2 , z T m z ) λ ( 1 λ ) p 1 p 2 2 = λ ( k m p 1 z 2 + κ z T m z 2 ) + ( 1 λ ) ( k m p 2 z 2 + κ z T m z 2 ) λ ( 1 λ ) p 1 p 2 2 .
Taking lim sup m on both sides of the above inequality, we have
lim sup m z T m z 2 lim sup m κ z T m z 2 .
Since κ < 1 , we have
lim sup m T m z z 2 = 0 ,

and so lim m T m z = z , i.e., T z = z . This completes the proof. □

Lemma 2.3 Let K be a nonempty closed convex subset of a real Hilbert space H, and let T : K K be a continuous κ-asymptotically strictly pseudo-nonspreading mapping. Then ( I T ) is demiclosed at 0, that is, if x n x and lim sup m lim sup n ( I T m ) x n = 0 , then ( I T ) x = 0 .

Proof Since { x n } is weak convergence, { x n } is bounded. For each x H , define f : H [ 0 , ) by
f ( x ) : = lim sup n x n x 2 , x H .
From Lemma 2.1(iii), we have
f ( x ) = lim sup n x n x 2 + x x 2 , x H .
Thus we have
f ( x ) = f ( x ) + x x 2 , x H .
In particular, for each m 1 ,
f ( T m x ) = f ( x ) + T m x x 2 .
(2.1)
On the other hand, we have
f ( T m x ) = lim sup n x n T m x 2 = lim sup n x n T m x n + T m x n T m x 2 = lim sup n ( x n T m x n 2 + 2 x n T m x n , T m x n T m x + T m x n T m x 2 ) .
Since lim sup m lim sup n ( I T m ) x n = 0 and T is a κ-asymptotically strictly pseudo-nonspreading mapping, taking lim sup m on both sides of the above equality, we get
lim sup m f ( T m x ) lim sup m T m x n T m x 2 lim sup m lim sup n ( k m x n x 2 + κ x n T m x n ( x T m x ) 2 + 2 x n T m x n , x T m x ) .
By virtue of lim sup m lim sup n ( I T m ) x n = 0 and k m 1 ( m ), we have
lim sup m f ( T m x ) f ( x ) + lim sup m κ x T m x 2 .
(2.2)
On the other hand, it follows from (2.1) that
lim sup m f ( T m x ) = f ( x ) + lim sup m T m x x 2 , x H .
(2.3)

Since κ < 1 , it follows from (2.2) and (2.3) that lim sup m T m x x 2 = 0 . So lim m T m x = x and T x = x . This completes the proof. □

3 Main results

Theorem 3.1 Let H 1 , H 2 , A, { S i } , { T i } , C, Q be the same as in multiple set split feasibility problem (1.1). For each i = 1 , 2 , , N , let T i be a uniformly L i ˜ -Lipschitzian and κ i -asymptotically strictly pseudo-nonspreading mapping, S i be a uniformly L i -Lipschitzian and ϱ i -asymptotically strictly pseudo-nonspreading mapping. Let { x n } be the sequence generated by
{ x 1 H 1  chosen arbitrarily , u n = x n + γ A ( T n ( mod N ) n I ) A x n , x n + 1 = ( 1 α n ) u n + α n S n ( mod N ) n u n ,
(3.1)

where γ is a constant and γ ( 0 , 1 κ λ ) , λ is the spectral of the operator A A , κ = max { κ 1 , κ 2 , , κ N } and { α n } is a sequence in ( 0 , 1 ϱ ] with ϱ = max { ϱ 1 , ϱ 2 , , ϱ N } . If Γ , then the sequence { x n } converges weakly to a point x Γ .

Proof The proof is divided into five steps.

(I) We first prove the limit lim n x n p exists for any p Γ .

Since p Γ , we have p C : = i = 1 N F ( S i ) and A p Q : = i = 1 N F ( T i ) . It follows from (3.1) that
x n + 1 p 2 = u n p + α n ( S n ( mod N ) n u n u n ) 2 = u n p 2 + 2 α n u n p , S n ( mod N ) n u n u n + α n 2 u n S n ( mod N ) n u n 2 .
(3.2)
Because S i is a ϱ i -asymptotically strictly pseudo-nonspreading mapping, for any v H 1 , we have
S n ( mod N ) n u n S n ( mod N ) n v 2 u n v 2 + ϱ u n S n ( mod N ) n u n ( v S n ( mod N ) n v ) 2 + 2 u n S n ( mod N ) n u n , v S n ( mod N ) n v .
Taking v = p , we have
S n ( mod N ) n u n p 2 u n p 2 + ϱ u n S n ( mod N ) n u n 2 .
Therefore we have
S n ( mod N ) n u n p 2 = S n ( mod N ) n u n u n + u n p 2 = S n ( mod N ) n u n u n 2 + 2 S n ( mod N ) n u n u n , u n p + u n p 2 u n p 2 + ϱ u n S n ( mod N ) n u n 2 .
Simplifying the above inequality, we have that
2 α n S n ( mod N ) n u n u n , u n p α n ( ϱ 1 ) u n S n ( mod N ) n u n 2 .
(3.3)
It follows from (3.2) and (3.3) that
x n + 1 p 2 u n p 2 + α n ( ϱ 1 ) u n S n ( mod N ) n u n 2 + α n 2 u n S n ( mod N ) n u n 2 = u n p 2 α n ( 1 ϱ α n ) u n S n ( mod N ) n u n 2 .
(3.4)
On the other hand,
u n p 2 = x n p + γ A ( T n ( mod N ) n I ) A x n 2 = x n p 2 + 2 γ x n p , A ( T n ( mod N ) n I ) A x n + γ 2 A ( T n ( mod N ) n I ) A x n 2 = x n p 2 + 2 γ x n p , A ( T n ( mod N ) n I ) A x n + γ 2 A ( T n ( mod N ) n I ) A x n , A ( T n ( mod N ) n I ) A x n = x n p 2 + 2 γ x n p , A ( T n ( mod N ) n I ) A x n + γ 2 A A ( T n ( mod N ) n I ) A x n , ( T n ( mod N ) n I ) A x n x n p 2 + 2 γ x n p , A ( T n ( mod N ) n I ) A x n + γ 2 A 2 ( T n ( mod N ) n I ) A x n 2 .
(3.5)
Since T i is a κ i -asymptotically strictly pseudo-nonspreading mapping and noting A p i 1 N F ( T i ) , we have
T n ( mod N ) n A x n A p 2 = T n ( mod N ) n A x n T n ( mod N ) n A p 2 A x n A p 2 + κ T n ( mod N ) n A x n A x n 2 .
(3.6)
Again since
T n ( mod N ) n A x n A p 2 = T n ( mod N ) n A x n A x n + A x n A p 2 = T n ( mod N ) n A x n A x n 2 + A x n A p 2 + 2 T n ( mod N ) n A x n A x n , A x n A p ,
(3.7)
hence from (3.6) and (3.7) we have that
2 T n ( mod N ) n A x n A x n , A x n A p ( κ 1 ) ( T n ( mod N ) n I ) A x n 2 .
(3.8)
By virtue of (3.8) we have
T n ( mod N ) n A x n A x n , T n ( mod N ) n A x n A p = T n ( mod N ) n A x n A x n , T n ( mod N ) n A x n A p + A x n A x n = ( T n ( mod N ) n I ) A x n 2 + T n ( mod N ) n A x n A x n , A x n A p ( T n ( mod N ) n I ) A x n 2 + κ 1 2 ( T n ( mod N ) n I ) A x n 2 = κ + 1 2 ( T n ( mod N ) n I ) A x n 2 .
(3.9)
It follows from (3.9) that
2 γ x n p , A ( T n ( mod N ) n I ) A x n = 2 γ A ( x n p ) , ( T n ( mod N ) n I ) A x n = 2 γ A ( x n p ) + ( T n ( mod N ) n I ) A x n ( T n ( mod N ) n I ) A x n , ( T n ( mod N ) n I ) A x n = 2 γ T n ( mod N ) n A x n A p , ( T n ( mod N ) n I ) A x n 2 γ ( T n ( mod N ) n I ) A x n 2 [ γ ( 1 + κ ) 2 γ ] ( T n ( mod N ) n I ) A x n 2 = γ ( κ 1 ) ( T n ( mod N ) n I ) A x n 2 .
(3.10)
Substituting (3.10) into (3.5) and then substituting the resulting inequality into (3.4), we have
x n + 1 p 2 x n p 2 + γ 2 A 2 ( T n ( mod N ) n I ) A x n 2 + [ γ ( κ 1 ) ] ( T n ( mod N ) n I ) A x n 2 α n ( 1 κ α n ) u n S n ( mod N ) n u n 2 x n p 2 γ ( 1 κ γ A 2 ) ( T n ( mod N ) n I ) A x n 2 α n ( 1 κ α n ) u n S n ( mod N ) n u n 2 x n p 2 .
(3.11)

This shows that the limit lim n x n p exists.

(II) Now we prove that the limit lim n u n p exists.

By (3.11) we have
γ ( 1 κ γ A 2 ) ( T n ( mod N ) n I ) A x n 2 + α n ( 1 κ α n ) u n S n ( mod N ) n u n 2 x n p 2 x n + 1 p 2 .
This implies that
lim n ( T n ( mod N ) n I ) A x n = 0 ,
(3.12)
and
lim n u n S n ( mod N ) n u n = 0 .
(3.13)
It follows from (3.5), (3.12), and (3.13) that the limit lim n u n p exists and
lim n x n p = lim n u n p .

(III) Now, we prove that lim n x n + 1 x n = 0 , lim n u n + 1 u n = 0 .

In fact, it follows from (3.1) that
x n + 1 x n = ( 1 α n ) u n + α n S n ( mod N ) n u n x n = ( 1 α n ) ( x n + γ A ( T n ( mod N ) n I ) A x n ) + α n S n ( mod N ) n u n x n = ( 1 α n ) ( γ A ( T n ( mod N ) n I ) A x n ) + α n ( S n ( mod N ) n u n x n ) = ( 1 α n ) ( γ A ( T n ( mod N ) n I ) A x n ) + α n ( S n ( mod N ) n u n u n ) + α n ( u n x n ) = ( 1 α n ) ( γ A ( T n ( mod N ) n I ) A x n ) + α n ( S n ( mod N ) n u n u n ) + α n γ A ( T n ( mod N ) n I ) A x n = γ A ( T n ( mod N ) n I ) A x n + α n ( S n ( mod N ) n u n u n ) .
(3.14)
This together with (3.12) and (3.13) shows that
lim n x n + 1 x n = 0 .
(3.15)
Similarly, it follows from (3.1), (3.12), and (3.15) that
u n + 1 u n = x n + 1 + γ A ( T n + 1 ( mod N ) n + 1 I ) A x n + 1 [ x n + γ A ( T n ( mod N ) n I ) A x n ] x n + 1 x n + γ A ( T n + 1 ( mod N ) n + 1 I ) A x n + 1 + γ A ( T n ( mod N ) n I ) A x n 0 ( as  n ) .
(3.16)
(IV) We prove that, for each j = 1 , 2 , , N ,
u i N + j S j u i N + j 0 , A x i N + j T j A x i N + j 0 ( i ) .
(3.17)
In fact, it follows from (3.13) that
u i N + j S j i N + j u i N + j 0 ( i ) .
(3.18)
Since S j is uniformly L j -Lipschitzian continuous, it follows from (3.16) and (3.18) that
u i N + j S j u i N + j u i N + j S j i N + j u i N + j + S j i N + j u i N + j S j u i N + j u i N + j S j i N + j u i N + j + L j S j i N + j 1 u i N + j u i N + j u i N + j S j i N + j u i N + j + L j [ S j i N + j 1 u i N + j S j i N + j 1 u i N + j 1 + S j i N + j 1 u i N + j 1 u i N + j ] u i N + j S j i N + j u i N + j + L j 2 u i N + j u i N + j 1 + L j [ S j i N + j 1 u i N + j 1 u i N + j 1 + u i N + j 1 u i N + j ] 0 ( as  n ) .
Similarly, we can prove that for each i = 1 , 2 , , N ,
A x i N + j T j i N + j A x i N + j 0 ( i ) .
(3.19)
Since T j is uniformly L j ˜ -Lipschitzian continuous, in the same way as above, we can also prove that
A x i N + j T j A x i N + j 0 ( as  i ) .

(V) Finally, we prove that x n x , u n x , and it is a solution of problem ( M S S F P ) (1.1).

In fact, since { u n } is bounded, there exists a subsequence { u n i } { u n } such that u n i x H 1 . Hence, for any positive integer j = 1 , 2 , , N , there exists a subsequence n i ( j ) n i with n i ( j ) mod N = j such that u n i ( j ) x . Again from (3.17) we have that
u n i ( j ) S j u n i ( j ) 0 , n i ( j ) .
(3.20)
Since S j is demiclosed at zero, it follows that x F ( S j ) . By the arbitrariness of j = 1 , 2 , , N , we have
x C : = i = 1 N F ( S i ) .

Moreover, from (3.1) and (3.13) we have x n i = u n i γ A ( T n i ( mod N ) n i I ) A x n i x . Since A is a linear bounded operator, it follows that A x n i A x . For any positive integer k = 1 , 2 , , N , there exists a subsequence x n i ( k ) x n i with n i ( k ) ( mod N ) = k such that A x n i ( k ) A x and A x n i ( k ) T k A x n i ( k ) 0 . Since T k is demiclosed at zero, we have A x F ( T k ) . By the arbitrariness of k, it follows that A x Q : = k = 1 N F ( T k ) . This together with x C shows that x Γ , that is, x is a solution to the problem ( M S S F P ) (1.1).

Next we prove that x n x and u n x .

In fact, assume that there exists another subsequence u n l u n such that u n l y Γ with y x . Consequently, by virtue of the existence of lim n x n p and the Opial property of a Hilbert space, we have
lim inf n i u n i x < lim inf n i u n i y = lim inf n u n y lim inf n j u n j y < lim inf n j u n j x = lim inf n u n x = lim inf n i u n i x .
This is a contradiction. Therefore, u n x . By (3.1) and (3.13), we have
x n = u n γ A ( T n ( mod N ) n I ) A x n x .

This completes the proof of Theorem 3.1. □

Theorem 3.2 Let H 1 , H 2 , A, { S i } , { T i } , C, Q be the same as in Theorem 3.1. For each i = 1 , 2 , , N , let T i be a uniformly L i ˜ -Lipschitzian and κ i -asymptotically strictly pseudo-nonspreading mapping, S i be a uniformly L i -Lipschitzian and ϱ i -asymptotically strictly pseudo-nonspreading mapping. Let { x n } be the sequence generated by
{ x 1 H 1  chosen arbitrarily , u n = x n + γ A ( T n ( mod N ) n I ) A x n , x n + 1 = ( 1 α n ) u n + α n S n ( mod N ) n u n ,

where γ is a constant and γ ( 0 , 1 κ λ ) , λ is the spectral of the operator A A , κ = max { κ 1 , κ 2 , , κ N } and { α n } is a sequence in ( 0 , 1 ϱ ] with ϱ = max { ϱ 1 , ϱ 2 , , ϱ N } . If Γ and if there exists a positive integer j such that S j is semicompact, then the sequence { x n } converges strongly to a point x Γ .

Proof Without loss of generality, we can assume that S 1 is semicompact. It follows from (3.17) that
u n i ( 1 ) S 1 u n i ( 1 ) 0 , n i ( 1 ) .
Therefore, there exists a subsequence of { u n i ( 1 ) } , which (for the sake of convenience) we still denote by { u n i ( 1 ) } , such that u n i ( 1 ) u H 1 . Since u n i ( 1 ) x , x = u , and so u n i ( 1 ) x Γ . By virtue of lim n x n p exists, we know that
lim n u n x = 0 , lim n x n x = 0 ,

that is, { u n } and { x n } both converge strongly to the point x Γ . This completes the proof of Theorem 3.2. □

4 Applications

In this section we shall utilize the results presented in Section 3 to study the hierarchical variational inequality problem.

Let H be a real Hilbert space, S i , i = 1 , 2 , , N , be uniformly L i -Lipschitzian and ϱ i -asymptotically strictly pseudo-nonspreading mappings with F : = i = 1 F ( S i ) . Let T : H H be a nonspreading mapping. The so-called hierarchical variational inequality problem for a finite family of mappings { S i } with respect to the mapping T is to find an x F such that
x T x , x x 0 , x F .
(4.1)
It is easy to see that (4.1) is equivalent to the following fixed point problem:
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), problem (4.2) is equivalent to the following multi-set split feasibility problem:
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 be the same as above. Let { x n } , { u n } be the sequences defined by
{ x 1 H 1  chosen arbitrarily , u n = x n + γ ( T I ) x n , n 1 , x n + 1 = ( 1 α n ) u n + α n S n ( mod N ) n u n ,
(4.4)

where γ is a constant and γ ( 0 , 1 ) , and { α n } is a sequence in ( 0 , 1 ϱ ] with ϱ = max { ϱ 1 , ϱ 2 , , ϱ N } . If Γ , then { x n } converges weakly to a solution of hierarchical variational inequality problem (4.1).

Proof In fact, by the assumption that T is a nonspreading mapping, T is κ-strictly pseudo-nonspreading with κ = 0 . Taking N = 1 and A = I in Theorem 3.1, by the same method as that given in Theorem 3.1, we can prove that { x n } converges weakly to a point x Γ , which is a solution of hierarchical variational inequality problem (4.1) immediately. □

Declarations

Acknowledgements

The authors would like to express their thanks to the referees and the editors for their helpful comments and advices. This work was supported by the National Research Foundation of Yibin University (No. 2011B07) and by the Scientific Research Fund Project of Sichuan Provincial Education Department (No. 12ZB345) and the National Natural Sciences Foundation of China (Grant No. 11361170).

Authors’ Affiliations

(1)
Institute for Mathematics, Yibin University
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics

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© Quan and Chang; licensee Springer. 2014

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