Open Access

Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces

Fixed Point Theory and Applications20122012:142

https://doi.org/10.1186/1687-1812-2012-142

Received: 8 May 2012

Accepted: 22 August 2012

Published: 5 September 2012

Abstract

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces. The main results presented in this paper extend and improve the corresponding results of many authors.

MSC:47H05, 47H09, 49M05.

Keywords

strong convergence total quasi-ϕ-asymptotically nonexpansive semigroups generalized projection

1 Introduction

Throughout this article, we assume that E is a real Banach space with norm , E is the dual space of E; , is the duality pairing between E and E ; C is a nonempty closed convex subset of E; N and R denote the natural number set and the set of nonnegative real numbers respectively. The mapping J : E 2 E defined by
J ( x ) = { f E : x , f = x 2 ; f = x , x E }

is called the normalized duality mapping. Let T : C C be a nonlinear mapping; F ( T ) denotes the set of fixed points of mapping T.

Alber et al.[1] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied the methods of approximation of fixed points. They are defined as follows.

Definition 1.1 Let T : C C be a mapping. T is said to be total asymptotically nonexpansive if there exist sequences { μ n } , { ν n } with μ n , ν n 0 as n and a strictly increasing continuous function ψ : R R with ψ ( 0 ) = 0 such that T n x T n y x y + μ n ψ ( x y ) + ν n holds for all x , y C and all n N .

T is said to be total asymptotically quasi-nonexpansive if F ( T ) , there exist sequences { μ n } , { ν n } with μ n , ν n 0 as n and a strictly increasing continuous function ψ : R R with ψ ( 0 ) = 0 such that T n x p x p + μ n ψ ( x p ) + ν n holds for all x C , p F ( T ) and all n N .

Chidume and Ofoedu [2] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings and total asymptotically quasi-nonexpansive mappings in Banach spaces. Chidume et al.[3] gave a new iterative sequence and necessary and sufficient conditions for this sequence to converge to a common fixed point of finite total asymptotically nonexpansive mappings. Chang [4] established some new approximation theorems of common fixed points for a countable family of total asymptotically nonexpansive mappings in Banach spaces.

Recently, many researchers have focused on studying the convergence of iterative algorithms for quasi-ϕ-asymptotically nonexpansive (see [59]) and total quasi-ϕ-asymptotically nonexpansive (see [1012]) mappings. Ye et al.[13] used a new hybrid projection algorithm to obtain strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. Kim [14] used hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings to prove the strong convergence theorems. Saewan [15] used the shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings.

A Banach space E is said to be strictly convex if x + y 2 < 1 for x = y = 1 and x y ; it is also said to be uniformly convex if lim n x n y n = 0 for any two sequences { x n } , { y n } in E such that x n = y n = 1 and lim n x n + y n 2 = 1 . Let U = { x E : x = 1 } be the unit sphere of E, then the Banach space E is said to be smooth provided lim t 0 x + t y y t exists for each x , y U . It is also said to be uniformly smooth if the limit is attained uniformly for each x , y U . It is well known that if E is reflexive and smooth, then the duality mapping J is single valued. A Banach space E is said to have the Kadec-Klee property if a sequence { x n } of E satisfies that x n x E and x n x , then x n x . It is known that if E is uniformly convex, then E has the Kadec-Klee property.

In the sequel, we assume that E is a smooth, strictly convex and reflexive Banach space and C is a nonempty closed convex subset of E. We use ϕ : E × E R + to denote the Lyapunov functional defined by
ϕ ( x , y ) = x 2 2 x , J y + y 2 , x , y E .
It is obvious that
( x y ) 2 ϕ ( x , y ) ( x + y ) 2 , x , y E ,
(1)
and
ϕ ( x , J 1 ( λ J y + ( 1 λ ) J z ) ) λ ϕ ( x , y ) + ( 1 λ ) ϕ ( x , z ) .
(2)
Following Alber [16], the generalized projection Π C x : E C is defined by
Π C x = arg inf y C ϕ ( y , x ) , x E .

The quasi-ϕ-asymptotically nonexpansive and total quasi-ϕ-asymptotically nonexpansive mappings are defined as follows.

Definition 1.2 A mapping T : C C is said to be quasi-ϕ-asymptotically nonexpansive, if F ( T ) , there exist sequences { k n } [ 1 , + ) with k n 1 as n such that
ϕ ( p , T n x ) k n ϕ ( p , x )

holds for all x C , p F ( T ) and all n N .

A mapping T : C C is said to be total quasi-ϕ-asymptotically nonexpansive, if F ( T ) , there exist sequences { μ n } , { ν n } with μ n , ν n 0 as n and a strictly increasing continuous function ψ : R R with ψ ( 0 ) = 0 such that
ϕ ( p , T n x ) ϕ ( p , x ) + μ n ψ ( ϕ ( p , x ) ) + ν n

holds for all x C , p F ( T ) and all n N .

In recent years, many researchers have considered the convergence of asymptotically nonexpansive semigroups [17, 18]. The asymptotically nonexpansive semigroups are defined as follows.

Definition 1.3[17]

One-parameter family T : = { T ( t ) : t 0 } of mappings from C into itself is said to be an asymptotically nonexpansive semigroup on C, if the following conditions are satisfied:
  1. (a)

    T ( 0 ) x = x for each x C ;

     
  2. (b)

    T ( t + s ) x = T ( s ) T ( t ) for any t , s R + and x C ;

     
  3. (c)

    For any x C , the mapping t T ( t ) x is continuous;

     
  4. (d)
    There exist sequences { k n } [ 1 , + ) with k n 1 as n such that
    T n ( t ) x T n ( t ) y k n x y
     

holds for all x , y C , n N .

We use F ( T ) to denote the common fixed point set of the semigroup T, i.e., F ( T ) = t 0 F ( T ( t ) ) .

Chang [19] used the modified Halpern-Mann type iteration algorithm for quasi-ϕ-asymptotically nonexpansive semigroups to prove the strong convergence in the Banach space. The quasi-ϕ-asymptotically nonexpansive semigroups are defined as follows.

Definition 1.4[19]

One-parameter family T : = { T ( t ) : t 0 } of mappings from C into itself is said to be a quasi-ϕ-asymptotically nonexpansive semigroup on C if the conditions (a), (b), (c) in Definition 1.3 and following condition (e) are satisfied:
  1. (e)
    For all x , y C , p F ( ( T ) ) , t 0 , there exist sequences { k n } [ 1 , + ) with k n 1 as n , such that
    ϕ ( p , T n ( t ) x ) k n ϕ ( p , x )
     

holds for all n N .

2 Preliminaries

This section contains some definitions and lemmas which will be used in the proofs of our main results in the following section.

Definition 2.1 One-parameter family T : = { T ( t ) : t 0 } of mappings from C into itself is said to be a total quasi-ϕ-asymptotically nonexpansive semigroup on C if conditions (a), (b), (c) in Definition 1.3 and following condition (f) are satisfied:
  1. (f)
    If F ( T ) , there exist sequences { μ n } , { ν n } with μ n , ν n 0 as n and a strictly increasing continuous function ψ : R R with ψ ( 0 ) = 0 such that
    ϕ ( p , T n ( t ) x ) ϕ ( p , x ) + μ n ψ ( ϕ ( p , x ) ) + ν n
     

holds for all x C , p F ( T ) and all n N .

A total quasi-ϕ-asymptotically nonexpansive semigroup T is said to be uniformly Lipschitzian if there exists a bounded measurable function L : [ 0 , ) ( 0 , + ) such that
T ( n ) ( t ) x T ( n ) ( t ) y L ( t ) x y , x , y C , t 0 , n N .

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-nonexpansive asymptotically semigroups to prove the strong convergence in Banach spaces. The results presented in the article improve and extend the corresponding results of [5, 6, 912, 14, 15, 19] and many others.

In order to prove the results of this paper, we shall need the following lemmas:

Lemma 2.1 (See [16])

Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:
  1. (i)

    ϕ ( x , Π C y ) + ϕ ( Π C y , y ) ϕ ( x , y ) for all x C , y E ;

     
  2. (ii)

    If x E and z C , then z = Π C x z y , J x J z 0 , y C ;

     
  3. (iii)

    For x , y E , ϕ ( x , y ) = 0 if and only if x = y .

     

Lemma 2.2[19]

Let E be a uniformly convex and smooth Banach space and let { x n } and { y n } be two sequences of E. If ϕ ( x n , y n ) 0 and either { x n } or { y n } is bounded, then x n y n 0 .

Lemma 2.3[10]

Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and C be a nonempty closed convex subset of E. Let T : C C be a closed and total quasi-ϕ-asymptotically nonexpansive mapping defined by Definition  1.2. If ν 1 = 0 , then the fixed point set F ( T ) of T is a closed and convex subset of C.

3 Main results

Theorem 3.1 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty closed convex subset of E. Let T : = { T ( t ) : t 0 } be a total quasi-ϕ-asymptotically nonexpansive semigroup from C into itself defined by Definition  2.1. Suppose T : = { T ( t ) : t 0 } is closed, uniformly L-Lipschitz and F ( T ) : = t 0 F ( T ( t ) ) . Suppose there exists M > 0 such that ψ ( η n ) M η n . Let α n be a sequence in [ 0 , 1 ] and β n be a sequence in ( 0 , 1 ) satisfying the following conditions: lim n α n = 0 , 0 < lim inf n β n < lim sup n β n < 1 . Let x n be a sequence generated by
{ x 1 E , chosen arbitrarily ; C 1 = C , l n , t = β n J x n + ( 1 β n ) J T n ( t ) x n , y n , t = J 1 [ α n J x 1 + ( 1 α n ) l n , t ] , t 0 , C n + 1 = { z C n : sup t 0 ϕ ( z , y n , t ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x 1 , n 1 ,
(3)

where ξ n = μ n M sup p F ( T ) ϕ ( p , x n ) . If ν 1 = 0 and F ( T ) is bounded in C, then the iterative sequence { x n } converges strongly to a common fixed point x F ( T ) in C.

Proof (I) We prove F ( T ) and C n ( n N ) all are closed and convex subsets in C.

It follows from Lemma 2.3 that F ( T ( t ) ) , t 0 is a closed and convex subset of C. So F ( T ) is closed and convex in C. By the assumption we know that C 1 = C is closed and convex. We suppose that C n is closed and convex for some n 2 . By the definition of ϕ, we have that
C n + 1 = { z C n : sup t 0 ϕ ( z , y n , t ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } = t 0 { z C : ϕ ( z , y n , t ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } C n = t 0 { z C : 2 α n z , J x 1 + 2 ( 1 α n ) z , J x n 2 z , J y n , t α n x 1 2 + ( 1 α n ) x n 2 y n , t 2 } C n .
This shows that C n + 1 is closed and convex.
  1. (II)

    We prove that F ( T ) C n .

     
In fact F ( T ) C 1 = C . Suppose that F ( T ) C n , n 2 . Let
ω n , t = J 1 ( β n J x n + ( 1 β n ) J T n ( t ) x n ) , t 0 .
It follows from (2) that for any u F ( T ) C n , we have
ϕ ( u , y n , t ) = ϕ ( u , J 1 ( α n J x 1 + ( 1 α n ) J ω n , t ) ) α n ϕ ( u , x 1 ) + ( 1 α n ) ϕ ( u , ω n , t ) ,
and
ϕ ( u , ω n , t ) = ϕ ( u , J 1 ( β n J x n + ( 1 β n ) J T n ( t ) x n ) ) β n ϕ ( u , x n ) + ( 1 β n ) ϕ ( u , T n ( t ) x n ) β n ϕ ( u , x n ) + ( 1 β n ) [ ϕ ( u , x n ) + μ n ψ ( ϕ ( u , x n ) ) + ν n ] ϕ ( u , x n ) + ( 1 β n ) ( μ n M ϕ ( u , x n ) + ν n ) .
Therefore, we have
sup t 0 ϕ ( u , y n , t ) α n ϕ ( u , x 1 ) + ( 1 α n ) [ ϕ ( u , x n ) + ( 1 β n ) ( μ n M ϕ ( u , x n ) + ν n ) ] α n ϕ ( u , x 1 ) + ( 1 α n ) ϕ ( u , x n ) + μ n M sup p F ( T ) ϕ ( p , x n ) + ν n = α n ϕ ( u , x 1 ) + ( 1 α n ) ϕ ( u , x n ) + ξ n .
Where ξ n = μ n M sup p F ( T ) ϕ ( p , x n ) + ν n . This shows that u C n + 1 , so F ( T ) C n + 1 .
  1. (III)

    We prove that { x n } is a Cauchy sequence in C.

     
Since x n = Π C n x 1 , from Lemma 2.1(ii), we have
x n y , J x 1 J x n 0 , y C n .
Again since F ( T ) C n , n 1 , we have
x n u , J x 1 J x n 0 , u F ( T ) .
It follows from Lemma 2.1(i) that for each u F ( T ) , n 1 ,
ϕ ( x n , x 1 ) = ϕ ( Π C n x 1 , x 1 ) ϕ ( u , x 1 ) ϕ ( u , x n ) ϕ ( u , x 1 ) .
Therefore, ϕ ( x n , x 1 ) is bounded. By virtue of (1), x n is also bounded. Since x n = Π C n x 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , we have ϕ ( x n , x 1 ) ϕ ( x n + 1 , x 1 ) . This implies that { ϕ ( x n , x 1 ) } is nondecreasing. Hence, the limit lim n ϕ ( x n , x 1 ) exists. By the construction of C n , for any positive integer m n , we have C m C n and x m = Π C 1 x 1 C n . This shows that
ϕ ( x m , x n ) = ϕ ( x m , Π C n x 1 ) ϕ ( x m , x 1 ) ϕ ( x n , x 1 ) 0 , as  m , n .
It follows from Lemma 2.2 that lim n , m x m x n = 0 . Hence x n is a Cauchy sequence in C. Since C is complete, without loss of generality, we can assume that x n p (some point in C). By the assumption, we have that
lim n ξ n = lim n [ μ n M sup p F ( T ) ϕ ( p , x n ) + ν n ] = 0 .
(4)
  1. (IV)

    Now we prove p F ( T ) .

     
Since x n + 1 C n + 1 and α n 0 , it follows from (3) and (4) that
sup t 0 ϕ ( x n + 1 , y n , t ) α n ϕ ( x n + 1 , x 1 ) + ( 1 α n ) ϕ ( x n + 1 , x n ) + ξ n 0 as  n .
Since x n p , by Lemma 2.2, for each t 0 , we have
lim n y n , t = p .
(5)
Since x n is bounded, and T = { T ( t ) , t 0 } is a total quasi-ϕ-asymptotically nonexpansive semigroup with sequence μ n , ν n , p F ( T ) , we have
ϕ ( p , T n ( t ) x ) ϕ ( p , x ) + μ n ψ ( ϕ ( p , x ) ) + ν n ϕ ( p , x ) + μ n M ϕ ( p , x ) + ν n .
This implies that { T n ( t ) x n } t 0 is uniformly bounded. Since for each t 0 ,
ω n , t = J 1 ( β n J x n + ( 1 β n ) J T n ( t ) x n ) β n x n + ( 1 β n ) T n ( t ) x n max { x n , T n ( t ) x n } .
This implies that { ω n , t } , t 0 is also uniformly bounded. Since α n 0 , from (3) we have
lim n J y n , t J ω n , t = lim n α n J x 1 J ω n , t = 0 , t 0 .
(6)
Since E is uniformly smooth, J 1 is uniformly continuous on each bounded subset of E , it follows from (5) and (6) that
lim n ω n , t = p , t 0 .
Since x n p and J is uniformly continuous on each bounded subset of E, we have J x n J p , and for each t 0 ,
0 = lim n J ω n , t J p = lim n β n J x n + ( 1 β n ) J T n ( t ) x n J p = lim n β n ( J x n J p ) + ( 1 β n ) ( J T n ( t ) x n J p ) = lim n ( 1 β n ) ( J T n ( t ) x n J p ) .
By condition 0 < lim inf n β n < lim sup n β n < 1 , we have that
lim n J T n ( t ) x n J p = 0 , uniformly for  t 0 .
Since J is uniformly continuous, this shows that lim n T n ( t ) x n = p = 0 uniformly for t 0 . Again by the assumptions that the semigroup T : = { T ( t ) : t 0 } is closed and uniformly L-Lipschitzian, we have
(7)
By lim n T n ( t ) x n = p uniformly for t 0 , x n p and L ( t ) is a bounded and measurable function, and from (7) we have that
lim n T n + 1 ( t ) x n T n ( t ) x n = 0 uniformly for  t 0 ,
and
lim n T n + 1 ( t ) x n = p uniformly for  t 0 ,
so we get
lim n T ( t ) T n ( t ) x n = p uniformly for  t 0 .
By virtue of the closeness of semigroup T, we have that T ( t ) p = p , i.e., p F ( T ( t ) ) . By the arbitrariness of t 0 , we have p F ( T ) = t 0 F ( T ( t ) ) .
  1. (V)

    Finally, we prove x n p = Π F ( T ) x 1 .

     
Let ω = Π F ( T ) x 1 . Since ω F ( T ) C n and x n = Π C n x 1 , we get ϕ ( x n , x 1 ) ϕ ( ω , x 1 ) , n 1 . This implies that
ϕ ( p , x 1 ) = lim n ϕ ( x n , x 1 ) ϕ ( ω , x 1 ) .
(8)

In view of the definition of Π F ( T ) x 1 , from (8), we have p = ω . Therefore, x n p = Π F ( T ) x 1 . This completes the proof of Theorem 3.1. □

Declarations

Acknowledgements

This work was supported by National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of Sichuan Provincial Education Department (No. 12ZB345 and No. 11ZA172).

Authors’ Affiliations

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

References

  1. Alber YI, Chidume CE, Zegeye H: Approximating fixed point of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006., 2006: Article ID 10673Google Scholar
  2. Chidume CE, Ofoedu EU: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2007, 333(1):128–141. 10.1016/j.jmaa.2006.09.023MathSciNetView ArticleGoogle Scholar
  3. Chidume CE, Ofoedu EU: A new iteration process for approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. Int. J. Math. Math. Sci. 2009., 2009: Article ID 615107. doi:10.1155/2009/615107Google Scholar
  4. Chang S, Kim JK, Joseph Lee HW, Chan CK: A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 37. doi:10.1186/1029–242X-2012–37Google Scholar
  5. Chang S, Chan CK, Joseph Lee HW: Modified block iterative algorithm for Quasi- ϕ -asymptotically nonexpansive mappings and equilibrium problem in Banach spaces. Appl. Math. Comput. 2011, 217: 7520–7530. 10.1016/j.amc.2011.02.060MathSciNetView ArticleGoogle Scholar
  6. Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215(11):3874–3883. 10.1016/j.amc.2009.11.031MathSciNetView ArticleGoogle Scholar
  7. Saewan S, Kumam P: Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi- ϕ -asymptotically nonexpansive mappings. Abstr. Appl. Anal. 2010., 2010: Article ID 22. doi:10.1155/2010/357120Google Scholar
  8. Chang S, Wanga L, Tang Y-K, Wanga B, Qin L-J: Strong convergence theorems for a countable family of quasi- ϕ -asymptotically nonexpansive nonself mappings. Appl. Math. Comput. 2012, 218: 7864–7870. 10.1016/j.amc.2012.02.002MathSciNetView ArticleGoogle Scholar
  9. Qin X, Huang S, Wang T: On the convergence of hybrid projection algorithms for asymptotically quasi- ϕ -nonexpansive mappings. Comput. Math. Appl. 2011, 61(4):851–859. 10.1016/j.camwa.2010.12.033MathSciNetView ArticleGoogle Scholar
  10. Chang S, Joseph Lee HW, Chan CK, Zhang WB: A modified Halpern type iterative algorithm for total quasi- ϕ -asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 2012, 218(11):6489–6497. 10.1016/j.amc.2011.12.019MathSciNetView ArticleGoogle Scholar
  11. Wang X, Chang S, Wang L, Tang Y-K, Xu YG: Strong convergence theorems for nonlinear operator equations with total quasi- ϕ -asymptotically nonexpansive mappings and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 34. doi:10.1186/1687–1812–2012–34Google Scholar
  12. Zuo P, Chang S, Liu M: On a hybrid algorithm for a family of total quasi- ϕ -asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 70. doi:10.1186/1687–1812–2012–70Google Scholar
  13. Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.MathSciNetView ArticleGoogle Scholar
  14. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10Google Scholar
  15. Saewan S, Kumam P: The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 9Google Scholar
  16. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartosator AG. Dekker, New York; 1996:15–50.Google Scholar
  17. Chang SS, Joseph Lee HW, Chan CK: Convergence theorem of common fixed point for asymptotically nonexpansive semigroups in Banach spaces. Appl. Math. Comput. 2009, 212: 60–65. 10.1016/j.amc.2009.01.086MathSciNetView ArticleGoogle Scholar
  18. Zegeye H, Shahzadb N, Damana OA: Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings. Math. Comput. Model. 2011, 54: 2077–2086. 10.1016/j.mcm.2011.05.016View ArticleGoogle Scholar
  19. Chang S, Wang L, Tang Y-K, Zao Y-H, Wang B: Strong convergence theorems of quasi- ϕ -asymptotically nonexpansive semi-groups in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 15. doi:10.1186/1687–1812–2012–15Google Scholar

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© Quan et al.; licensee Springer. 2012

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