Open Access

Split feasibility problems for total quasi-asymptotically nonexpansive mappings

Fixed Point Theory and Applications20122012:151

https://doi.org/10.1186/1687-1812-2012-151

Received: 23 April 2012

Accepted: 30 August 2012

Published: 18 September 2012

Abstract

The purpose of this paper is to propose an algorithm for solving the split feasibility problems for total quasi-asymptotically nonexpansive mappings in infinite-dimensional Hilbert spaces. The results presented in the paper not only improve and extend some recent results of Moudafi [Nonlinear Anal. 74:4083-4087, 2011; Inverse Problem 26:055007, 2010], but also improve and extend some recent results of Xu [Inverse Problems 26:105018, 2010; 22:2021-2034, 2006], Censor and Segal [J. Convex Anal. 16:587-600, 2009], Censor et al. [Inverse Problems 21:2071-2084, 2005], Masad and Reich [J. Nonlinear Convex Anal. 8:367-371, 2007], Censor et al. [J. Math. Anal. Appl. 327:1244-1256, 2007], Yang [Inverse Problem 20:1261-1266, 2004] and others.

MSC:47J05, 47H09, 49J25.

Keywords

split feasibility problem convex feasibility problem total quasi-asymptotically nonexpansive mappings demi-closeness Opial condition

1 Introduction

Throughout this paper, we always assume that H 1 , H 2 are real Hilbert spaces, ‘→’, ‘’ denote strong and weak convergence, respectively, and F ( T ) is a fixed point set of a mapping T.

The split feasibility problem (SFP) in finite-dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning [35]. The split feasibility problem in an infinite-dimensional real Hilbert space can be found in [2, 4, 610].

The purpose of this paper is to introduce and study the following split feasibility problem for total quasi-asymptotically nonexpansive mappings in the framework of infinite-dimensional real Hilbert spaces:
find  x C  such that  A x Q ,
(1.1)
where A : H 1 H 2 is a bounded linear operator, S : H 1 H 1 and T : H 2 H 2 are mappings; C : = F ( S ) and Q : = F ( T ) . In the sequel, we use Γ to denote the set of solutions of (SFP)-(1.1), i.e.,
Γ = { x C , A x Q } .
(1.2)

2 Preliminaries

We first recall some definitions, notations and conclusions which will be needed in proving our main results.

Let E be a Banach space. A mapping T : E E is said to be demi-closed at origin if for any sequence { x n } E with x n x and ( I T ) x n 0 , x = T x .

A Banach space E is said to have the Opial property, if for any sequence { x n } with x n x ,
lim inf n x n x < lim inf n x n y , y E  with  y x .

Remark 2.1 It is well known that each Hilbert space possesses the Opial property.

Definition 2.2 Let H be a real Hilbert space.
  1. (1)
    A mapping G : H H is said to be a ( { ν n } , { μ n } , ζ ) -total quasi-asymptotically nonexpansive mapping if F ( G ) ; and there exist nonnegative real sequences { ν n } , { μ n } with ν n 0 and μ n 0 and a strictly increasing continuous function ζ : R + R + with ζ ( 0 ) = 0 such that for each n 1 ,
    p G n x 2 p x 2 + ν n ζ ( p x ) + μ n , p F ( G ) , x H .
    (2.1)
     

Now, we give an example of total quasi-asymptotically nonexpansive mapping.

Let C be a unit ball in a real Hilbert space 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 , ) , ( x 1 , x 2 , , ) l 2 ,

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

It is proved in Goebal and Kirk [17] that
  1. (i)

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

     
  2. (ii)

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

     
Denote by k 1 1 2 = 2 , k n 1 2 = 2 j = 2 n a j , n 2 , then
lim n k n = lim n ( 2 j = 2 n a j ) 2 = 1 .
Letting ν n = ( k n 1 ) , n 1 , ζ ( t ) = t , t 0 and { μ n } be a nonnegative real sequence with μ n 0 , from (i) and (ii), x , y C , n 1 , we have
T n x T n y 2 x y 2 + ν n ζ ( x y 2 ) + μ n .
(2.2)
Again, since 0 C and 0 F ( T ) , this implies that F ( T ) . From (2.2), we have
p T n y 2 p y 2 + ν n ζ ( p y 2 ) + μ n p F ( T ) , y C .
(2.3)
This shows that the mapping T defined as above is a total quasi-asymptotically nonexpansive mapping.
  1. (2)
    A mapping G : H H is said to be ( { k n } ) -quasi-asymptotically nonexpansive if F ( G ) ; and there exists a sequence { k n } [ 1 , ) with k n 1 such that for all n 1 ,
    p G n x 2 k n p x 2 , p F ( G ) , x H .
    (2.4)
     
  2. (3)
    A mapping G : H H is said to be quasi-nonexpansive if F ( G ) such that
    p G x p x , p F ( G ) , x H .
    (2.5)
     

Remark 2.3 It is easy to see that every quasi-nonexpansive mapping is a ( { 1 } ) -quasi-asymptotically nonexpansive mapping and every { k n } -quasi-asymptotically nonexpansive mapping is a ( { ν n } , { μ n } , ζ ) -total quasi-asymptotically nonexpansive mapping with { ν n = k n 1 } , { μ n = 0 } and ζ ( t ) = t 2 , t 0 .

Definition 2.4 (1) A mapping G : H H 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 H  and  n 1 .
  1. (2)

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

     
Proposition 2.5 Let G : H H be a ( { ν n } , { μ n } , ζ ) -total quasi-asymptotically nonexpansive mapping. Then for each q F ( G ) and for each x H , the following inequalities are equivalent: for each n 1
(2.6)
(2.7)
Proof (I) (2.1) (2.6) In fact, since
G n x q 2 = G n x x + x q 2 = G n x x 2 + x q 2 + 2 G n x x , x q , x H , q F ( G ) ,
from (2.1) we have that
G n x x 2 + x q 2 + 2 G n x x , x q x q 2 + ν n ζ ( q x ) + μ n .

Simplifying it, inequality (2.6) is obtained.

Conversely, from (2.6) the inequality (2.1) can be obtained immediately.
  1. (II)
    (2.6) (2.7) In fact, since
    x G n x , x q = x G n x , x G n x + G n x q = x G n x 2 + x G n x , G n x q
     
it follows from (2.6) that
2 ( x G n x 2 + x G n x , G n x q ) x G n x 2 ν n ζ ( q x ) μ n .

Simplifying it, the inequality (2.7) is obtained.

Conversely, from (2.7) the inequality (2.6) can be obtained immediately.

This completes the proof of Proposition 2.5. □

Lemma 2.6 [11]

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 , n 1 .

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

3 Split feasibility problem

For solving the split feasibility problem (1.1), let us 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;

     
  2. 2.

    S : H 1 H 1 and T : H 2 H 2 are two uniformly L-Lipschitzian and ( { ν n } , { μ n } , ζ )-total quasi-asymptotically nonexpansive mappings satisfying the following conditions:

     
  3. (i)

    T and S both are demi-closed at origin;

     
  4. (ii)

    n = 1 ( μ n + ν n ) < ;

     
  5. (iii)

    there exist positive constants M and M such that ζ ( t ) ζ ( M ) + M t 2 , t 0 .

     

We are now in a position to give the following result.

Theorem 3.1 Let H 1 , H 2 , A, S, T, L, { μ n } , { ν n } , ζ be the same as above. Let { x n } be the sequence generated by:
{ x 1 H 1 chosen arbitrarily , x n + 1 = ( 1 α n ) u n + α n S n ( u n ) , u n = x n + γ A ( T n I ) A x n , n 1 ,
(3.1)
where { α n } is a sequence in [ 0 , 1 ] , and γ > 0 is a constant satisfying the following conditions:
  1. (iv)

    0 < lim inf n α n lim sup n α n < 1 ; and γ ( 0 , 1 A 2 ) ,

     
  2. (I)

    If Γ (where Γ is the set of solutions to ((SFP)-(1.1)), then { x n } converges weakly to a point x Γ .

     
  3. (II)

    In addition, if S is also semi-compact, then { x n } and { u n } both converge strongly to x Γ .

     
The proof of conclusion (I) (1) First, we prove that for each p Γ , the following limits exist:
lim n x n p = lim n u n p .
(3.2)
In fact, since p Γ , we have p C : = F ( S ) and A p Q : = F ( T ) . It follows from (3.1) and (2.4) that
x n + 1 p 2 = u n p α n ( u n S n u n ) 2 = u n p 2 2 α n u n p , u n S n u n + α n 2 u n S n u n 2 u n p 2 α n { u n S n u n 2 ν n ζ ( u n p ) μ n } + α n 2 u n S n u n 2 ( by (2.6) ) = u n p 2 α n ( 1 α n ) u n S n u n 2 + α n ( ν n ζ ( u n p ) + μ n ) .
(3.3)
On the other hand, by condition (iii), we have
ζ ( u n p ) ζ ( M ) + M u n p 2 .
(3.4)
Substituting (3.4) into (3.3) and simplifying, we have
x n + 1 p 2 ( 1 + α n ν n M ) u n p 2 α n ( 1 α n ) u n S n u n 2 + α n ( ν n ζ ( M ) + μ n ) ( 1 + ν n M ) u n p 2 α n ( 1 α n ) u n S n u n 2 + ν n ζ ( M ) + μ n .
(3.5)
On the other hand,
u n p 2 = x n p + γ A ( T n I ) A x n 2 = x n p 2 + γ 2 A ( T n I ) A x n 2 + 2 γ x n p , A ( T n I ) A x n ,
(3.6)
and
γ 2 A ( T n I ) A x n 2 = γ 2 A ( T n I ) A x n , A ( T n I ) A x n = γ 2 A A ( T n I ) A x n , ( T n I ) A x n γ 2 A 2 T n A x n A x n 2 ,
(3.7)
and
(3.8)
In (2.5), taking x = A x n , G n = T n , q = A p , and noting A p F ( T ) , from (2.7) and condition (iii), we have
(3.9)
Substituting (3.9) into (3.8) and simplifying it, we have
(3.10)
Substituting (3.7) and (3.10) into (3.6) after simplifying, we have
u n p 2 ( 1 + γ ν n M A 2 ) x n p 2 + γ ( ν n ζ ( M ) + μ n ) γ ( 1 γ A 2 ) ( T n I ) A x n 2 .
(3.11)
Substituting (3.11) into (3.5) and simplifying it, we have
x n + 1 p 2 ( 1 + ν n M ) { ( 1 + γ ν n M A 2 ) x n p 2 + γ ( ν n ζ ( M ) + μ n ) γ ( 1 γ A 2 ) ( T n I ) A x n 2 } α n ( 1 α n ) u n S n u n 2 + ν n ζ ( M ) + μ n ( 1 + ξ n ) x n p 2 + η n γ ( 1 γ A 2 ) ( T n I ) A x n 2 α n ( 1 α n ) u n S n u n 2 ,
(3.12)
where
ξ n = ν n ( M + γ M A 2 + γ ν n M A 2 ) , η n = [ ( 1 + ν n M ) γ + 1 ] ( ν n ζ ( M ) + μ n ) .
By condition (iii), we have
n = 1 ξ n < , and n = 1 η n < .
By condition (iv), ( 1 γ A 2 ) > 0 . Hence, from (3.12), we have
x n + 1 p 2 ( 1 + ξ n ) x n p 2 + η n , n 1 .
By Lemma 2.6, the following limit exists:
lim n x n p .
(3.13)
Now, we rewrite (3.12) as follows:
γ ( 1 γ A 2 ) ( T n I ) A x n 2 + α n ( 1 α n ) u n S n u n 2 x n p 2 x n + 1 p 2 + ξ n x n p 2 + η n 0 ( as  n ) .
This together with the condition (iv) implies that
lim n u n S n u n = 0 ;
(3.14)
and
lim n ( T n I ) A x n = 0 .
(3.15)
It follows from (3.6), (3.14) and (3.15) that the limit lim n u n p exists and
lim n u n p = lim n x n p .
The conclusion (3.2) is proved.
  1. (2)
    Next, we prove that
    lim n x n + 1 x n = 0 and lim n u n + 1 u n = 0 .
    (3.16)
     
In fact, it follows from (3.1) that
x n + 1 x n = ( 1 α n ) u n + α n S n ( u n ) x n = ( 1 α n ) ( x n + γ A ( T n I ) A x n ) + α n S n ( u n ) x n = ( 1 α n ) γ A ( T n I ) A x n + α n ( S n ( u n ) x n ) = ( 1 α n ) γ A ( T n I ) A x n + α n ( S n ( u n ) u n ) + α n ( u n x n ) = ( 1 α n ) γ A ( T n I ) A x n + α n ( S n ( u n ) u n ) + α n γ A ( T n I ) A x n = γ A ( T n I ) A x n + α n ( S n ( u n ) u n ) .
In view of (3.14) and (3.15), we have that
lim n x n + 1 x n = 0 .
(3.17)
Similarly, it follows from (3.1), (3.15) and (3.17) that
u n + 1 u n = x n + 1 + γ A ( T n + 1 I ) A x n + 1 ( x n + γ A ( T n I ) A x n ) x n + 1 x n + γ A ( T n + 1 I ) A x n + 1 + γ A ( T n I ) A x n 0 ( as  n ) .
(3.18)
The conclusion (3.16) is proved.
  1. (3)
    Next, we prove that
    u n S u n 0 and A x n T A x n 0 ( as  n ) .
    (3.19)
     
In fact, from (3.14), we have
ζ n : = u n S n u n 0 ( as  n ) .
(3.20)
Since S is uniformly L-Lipschitzian continuous, it follows from (3.16) and (3.20) that
u n S u n u n S n u n + S n u n S u n ζ n + L S n 1 u n u n ζ n + L { S n 1 u n S n 1 u n 1 + S n 1 u n 1 u n } ζ n + L 2 u n u n 1 + L S n 1 u n 1 u n 1 + u n 1 u n ζ n + L ( 1 + L ) u n u n 1 + L ζ n 1 0 ( as  n ) .
Similarly, from (3.15), we have
A x n T n A x n 0 ( as  n ) .
(3.21)
Since T is uniformly L-Lipschitzian continuous, by the same way as above, from (3.16) and (3.21), we can also prove that
A x n T A x n 0 ( as  n ) .
(3.22)
  1. (4)

    Finally, we prove that x n x and u n x , which is a solution of (SFP)-(1.1).

     
Since { u n } is bounded, there exists a subsequence { u n i } { u n } such that u n i x (some point in H 1 ). From (3.19), we have
u n i S u n i 0 ( as  n i ) .
(3.23)

By the assumption that S is demi-closed at zero, we get that x F ( S ) .

Moreover, from (3.1) and (3.15), we have
x n i = u n i γ A ( T n i I ) A x n i x .
Since A is a linear bounded operator, we get A x n i A x . In view of (3.19), we have
A x n i T A x n i 0 ( as  n i ) .

Since T is demi-closed at zero, we have A x F ( T ) . Summing up the above argument, it is clear that x Γ , i.e., x is a solution to the (SFP)-(1.1).

Now, we prove that x n x and u n x .

Suppose, to the contrary, that if there exists another subsequence { u n j } { u n } such that u n j y Γ with y x , then by virtue of (3.2) and the Opial property of Hilbert space, we have
lim inf n i u n i x < lim inf n i u n i y = lim n u n y = lim n j u n j y < lim inf n j u n j x = lim n u n x = lim inf n i u n i x .
This is a contradiction. Therefore, u n x . By using (3.1) and (3.15), we have
x n = u n γ A ( T n n I ) A x n x .

 □

The proof of conclusion (II) By the assumption that S is semi-compact, it follows from (3.23) that there exists a subsequence of { u n i } (without loss of generality, we still denote it by { u n i } ) such that u n i u H (some point in H). Since u n i x . This implies that x = u , and so u n i x Γ . By virtue of (3.2), we know that lim n u n x = 0 and lim n x n x = 0 , i.e., { u n } and { x n } both converge strongly to x Γ .

This completes the proof of Theorem 3.1. □

Theorem 3.2 Let H 1 , H 2 and A be the same as in Theorem  3.1. Let S : H 1 H 1 and T : H 2 H 2 be two ( { k n } ) -quasi-asymptotically nonexpansive mappings with { k n } [ 1 , ) , k n 1 satisfying the following conditions:
  1. (i)

    T and S both are demi-closed at origin;

     
  2. (ii)

    n = 1 ( k n 1 ) < .

     
Let { x n } be the sequence generated by
{ x 1 H 1 chosen arbitrarily , x n + 1 = ( 1 α n ) u n + α n S n ( u n ) , u n = x n + γ A ( T n I ) A x n , n 1 ,
(3.24)

where { α n } is a sequence in [ 0 , 1 ] and γ > 0 is a constant satisfying the condition (iv) in Theorem  3.1. Then the conclusions in Theorem  3.1 still hold.

Proof By assumptions, S : H 1 H 1 and T : H 2 H 2 both are ( { k n } ) -quasi-asymptotically nonexpansive mappings with { k n } [ 1 , ) , k n 1 ; by Remark 2.3, S and T both are uniformly L-Lipschitzian (where L = sup n 1 k n ) and ( { ν n } , { μ n } , ζ )-total quasi-asymptotically nonexpansive mapping with { ν n = k n 1 } , { μ n = 0 } and ζ ( t ) = t 2 , t 0 . Therefore, all conditions in Theorem 3.1 are satisfied. The conclusions of Theorem 3.2 can be obtained from Theorem 3.1 immediately. □

Theorem 3.3 Let H 1 , H 2 and A be the same as in Theorem  3.1. Let S : H 1 H 1 and T : H 2 H 2 be two quasi-nonexpansive mappings and demi-closed at origin. Let { x n } be the sequence generated by
{ x 1 H 1 chosen arbitrarily , x n + 1 = ( 1 α n ) u n + α n S n ( u n ) , u n = x n + γ A ( T n I ) A x n , n 1 ,
(3.25)

where { α n } is a sequence in [ 0 , 1 ] and γ > 0 is a constant satisfying the condition (iv) in Theorem  3.1. Then the conclusions in Theorem  3.1 still hold.

Proof By the assumptions, S : H 1 H 1 and T : H 2 H 2 are quasi-nonexpansive mappings. By Remark 2.3, S and T both are uniformly L-Lipschitzian (where L = 1 ) and ( { 1 } )- quasi-asymptotically nonexpansive mappings. Therefore, all conditions in Theorem 3.2 are satisfied. The conclusions of Theorem 3.3 can be obtained from Theorem 3.2 immediately. □

Remark 3.4 Theorems 3.1, 3.2 and 3.3 not only improve and extend the corresponding results of Moudafi [12, 13], but also improve and extend the corresponding results of Censor et al. [4, 5], Yang [7], Xu [14], Censor and Segal [15], Masad and Reich [16] and others.

Declarations

Acknowledgements

This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ010) and the Natural Science Foundation of Yunnan Province (Grant No.2011FB074).

Authors’ Affiliations

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

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