Open Access

Strong convergence theorems for fixed point problems of a nonexpansive semigroup in a Banach space

Fixed Point Theory and Applications20132013:248

https://doi.org/10.1186/1687-1812-2013-248

Received: 3 June 2013

Accepted: 17 September 2013

Published: 7 November 2013

Abstract

In this paper, we study the implicit and explicit viscosity iteration schemes for a nonexpansive semigroup in a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition. Our results improve and generalize the corresponding results given by Yao et al. (Fixed Point Theory Appl. 2013, doi:10.1186/1687-1812-2013-31) and many others.

MSC:47H05, 47H10, 47H17.

Keywords

nonexpansive semigroup fixed point reflexive and strictly convex Banach space uniformly smooth Opial’s condition sunny nonexpansive retraction

1 Introduction

Let E be a real Banach space, and let K be a nonempty, closed and convex subset of E. A mapping T : K K is called nonexpansive if
T x T y x y , x , y K .
(1.1)
One parameter family S : = { T ( s ) : 0 s < } is said to be a nonexpansive semigroup from K into K if the following conditions are satisfied:
  1. (1)

    T ( 0 ) x = x for all x K ;

     
  2. (2)

    T ( s + t ) = T ( s ) T ( t ) for all s , t 0 ;

     
  3. (3)

    T ( t ) x T ( t ) y x y , x , y K and t 0 ;

     
  4. (4)

    for each x K , the mapping T ( ) x from [ 0 , ) into K is continuous.

     

Let F ( S ) denote the common fixed point set of the semigroup S, i.e., F ( S ) : = { x K : T ( s ) x = x , s > 0 } . It is known that F ( S ) is closed and convex.

A continuous operator of the semigroup S is said to be uniformly asymptotically regular (u.a.r.) on K if for all h 0 and any bounded subset C of K, lim s sup x C T ( h ) T ( s ) x T ( s ) x = 0 (see [1]).

Approximation of fixed points of nonexpansive mappings by a sequence of finite means has been considered by many authors (see [26]). In 2013, Yao et al. [7] introduced two new algorithms for finding a common fixed point of a nonexpansive semigroup { T ( s ) } s 0 in Hilbert spaces and proved that both approaches converge strongly to a common fixed point of { T ( s ) } s 0 .

Theorem 1.1 [7]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S = { T ( s ) } s 0 : C C be a nonexpansive semigroup with Fix ( S ) . Let { γ t } 0 < t < 1 and { λ t } 0 < t < 1 be two continuous nets of positive real numbers such that γ t ( 0 , 1 ) , lim t 0 γ t = 1 and lim t 0 λ t = + . Let { x t } be the net defined in the following implicit manner:
x t = P C [ t ( γ t x t ) + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s ] , t ( 0 , 1 ) .
(1.2)

Then, as t 0 + , the net { x t } strongly converges to x Fix ( s ) .

Theorem 1.2 [7]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S = { T ( s ) } s 0 : C C be a nonexpansive semigroup with Fix ( S ) . Let { x n } be the sequence generated iteratively by the following explicit algorithm:
x n + 1 = ( 1 β n ) x n + β n P C [ α n ( γ n x n ) + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s ] , n 0 ,
(1.3)
where { α n } , { β n } and { γ n } are sequences of real numbers in [ 0 , 1 ] and { λ n } is a sequence of positive real numbers. Suppose that the following conditions are satisfied:
  1. (i)

    lim n α n = 0 , n = 0 α n = and lim n γ n = 1 ;

     
  2. (ii)

    0 < lim inf n β n lim sup n β n < 1 ;

     
  3. (iii)

    lim n λ n = and lim n λ n 1 λ n = 1 .

     

Then the sequence { x n } generated by (1.3) strongly converges to a point x Fix ( s ) .

In this paper, we study the convergence of the following iterative schemes in a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition:
x t = Q K [ t ( γ t x t ) + ( 1 t ) T ( s t ) x t ] , t ( 0 , 1 ) , x n + 1 = ( 1 β n ) x n + β n Q K [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] , n 0 .

Our work improves and generalizes many others. In particular, our results extend the main results of Yao et al. [7].

2 Preliminaries

Let E be a real Banach space and E be the dual space of E. The duality mapping J : E 2 E is defined by
( x ) = { f E : x , f = x 2 = f 2 } .
(2.1)

By the Hahn-Banach theorem, J ( x ) is nonempty.

Let dim E 2 . The modulus of convexity of E is the function δ E : ( 0 , 2 ] [ 0 , 1 ] defined by
δ E ( ϵ ) : = inf { 1 x y 2 : x = y = 1 ; ϵ = x y } .
(2.2)

E is uniformly convex if ϵ ( 0 , 2 ] , there exists δ = δ ( ϵ ) > 0 such that if x , y E with x 1 , y 1 and x y ϵ , then x + y 2 1 δ . Equivalently, E is uniformly convex if and only if δ E ( ϵ ) > 0 , ϵ ( 0 , 2 ] . E is strictly convex if for all x , y E , x y , x = y = 1 , we have λ x + ( 1 λ ) y < 1 , λ ( 0 , 1 ) .

Let S ( E ) = { x E : x = 1 } . The space E is said to be smooth if
lim t 0 ( x + t y x ) / t
(2.3)

exists for all x , y S ( E ) . The norm of E is said to be Fréchet differentiable if for all x S ( E ) , the limit (2.3) exists uniformly for all y S ( E ) . E is said to have a uniformly Gâteaux differentiable norm if for all y S ( E ) , the limit (2.3) is attained uniformly for all x S ( E ) . The norm of E is said to be uniformly Fréchet differentiable (or uniformly smooth) if the limit (2.3) is attained uniformly for x , y S ( E ) × S ( E ) .

It is well known that if E is smooth, then J is single-valued, which is denoted by j. And if E has a uniformly Gâteaux differentiable norm, then J is norm-to-weak uniformly continuous on each bounded subset of E. The duality mapping J is said to be weakly sequentially continuous if J is single-valued and for any { x n } E with x n x , J ( x n ) J ( x ) . Gossez and Lami Dozo [8] proved that a space with a weakly continuous duality mappings satisfies Opial’s condition. Conversely, if a space satisfies Opial’s condition and has a uniformly Gâteaux differentiable norm, then it has a weakly continuous duality mapping.

Recall that if C and D are nonempty subsets of a Banach space E such that C is nonempty closed convex and D C , the mapping Q : C D is said to be sunny if
Q ( Q x + t ( x Q x ) ) = Q x ,

where Q x + t ( x Q x ) C for all x C and t 0 .

A mapping Q : C D is called a retraction if Q x = x for all x D .

A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C into D (see [9, 10]). It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from E onto C.

Proposition 2.1 [11]

Let C be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C. Let Q : C D be a retraction, and let J be the normalized duality mapping on E. Then the following are equivalent:
  1. (1)

    Q is sunny and nonexpansive.

     
  2. (2)

    Q x Q y 2 x y , J ( Q x Q y ) , x , y C .

     
  3. (3)

    x Q x , J ( y Q x ) 0 , x C , y D .

     

Proposition 2.2 [12]

Let C be a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with F ( T ) . Then the set F ( S ) is a sunny nonexpansive retraction of C.

Lemma 2.3 [13]

Let K be a nonempty closed convex subset of a reflexive Banach space E which satisfies Opial’s condition, and suppose that T : K E is nonexpansive. Then the mapping I T is demiclosed at zero, that is, x n x , x n T x n 0 implies x = T x .

Lemma 2.4 [14]

Let { x n } , { y n } be two bounded sequences in a Banach space E and β n ( 0 , 1 ) with 0 < lim inf n β n lim sup n β n < 1 . Suppose x n + 1 = β n y n + ( 1 β n ) x n for all integers n 0 and lim sup n ( y n + 1 y n x n + 1 x n ) 0 . Then lim n x n y n = 0 .

Lemma 2.5 [15]

Let { a n } be a sequence of nonnegative real numbers satisfying the following relation:
a n + 1 ( 1 ρ n ) a n + ρ n σ n , n 0 ,
where { ρ n } and { σ n } are sequences of real numbers such that
  1. (i)

    0 < ρ n < 1 ;

     
  2. (ii)

    n = 1 ρ n = ;

     
  3. (iii)

    lim sup n σ n 0 or n = 1 | ρ n σ n | is convergent.

     

Then lim n a n = 0 .

3 Main result

Theorem 3.1 Let E be a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition, and let K be a nonempty closed convex subset of E. Let S = { T ( s ) : s 0 } : K K be a uniformly asymptotically regular nonexpansive semigroup such that F ( S ) . Let { γ t } 0 < t < 1 and { s t } 0 < t < 1 be two continuous nets of positive real numbers such that γ t ( 0 , 1 ) , lim t 0 γ t = 1 and lim t 0 s t = + . Let { x t } be the net defined by
x t = Q K [ t ( γ t x t ) + ( 1 t ) T ( s t ) x t ] , t ( 0 , 1 ) .
(3.1)

Then, as t 0 + , the net { x t } converges strongly to a point x F ( S ) .

Proof Consider a mapping W on K defined by
W x : = Q K [ t ( γ t x ) + ( 1 t ) T ( s t ) x ] , t ( 0 , 1 ) .
x , y K , we have
W x W y t γ t ( x y ) + ( 1 t ) ( T ( s t ) x T ( s t ) y ) t γ t x y + ( 1 t ) x y = [ 1 ( 1 γ t ) t ] x y .
Hence, W is a contraction. So, it has a unique fixed point, denoted by x t . That is,
x t = Q K [ t ( γ t x t ) + ( 1 t ) T ( s t ) x t ] .

Therefore, the sequence { x t } defined by (3.1) is well defined.

Let p F ( S ) , then
x t p = Q K [ t ( γ t x t ) + ( 1 t ) T ( s t ) x t ] p t γ t ( x t p ) t ( 1 γ t ) p + ( 1 t ) ( T ( s t ) x t p ) t γ t x t p + t ( 1 γ t ) p + ( 1 t ) x t p = [ 1 ( 1 γ t ) t ] x t p + t ( 1 γ t ) p .
It follows that
x t p p .

Thus, { x t } is bounded, so is { T ( s t ) u n } .

Let R = p . It is clear that { x t } B ( p , R ) . Then B ( p , R ) K is a nonempty bounded closed convex subset of K and T ( s ) -invariant. Since { T ( s ) } is u.a.r. nonexpansive semigroup and lim t 0 s t = , then for all s > 0 ,
lim t 0 T ( s ) ( T ( s t ) x t ) T ( s t ) x t lim n sup x D T ( s ) ( T ( s t ) x ) T ( s t ) x = 0 ,
where D is any bounded subset of K containing { u n } . Since
x t T ( s t ) x t t γ t x t T ( s t ) x t 0 ,
and
x t T ( s ) x t x t T ( s t ) x t + T ( s t ) x t T ( s ) ( T ( s t ) x t ) + T ( s ) ( T ( s t ) x t ) T ( s ) x t 2 x t T ( s t ) x t + T ( s t ) x t T ( s ) ( T ( s t ) x t ) .
Thus, for all s > 0 , we have
lim t 0 x t T ( s ) x t = 0 .
(3.2)
Set y t = t ( γ t x t ) + ( 1 t ) T ( s t ) x t . Then x t = Q K y t . By Proposition 2.1(2), we can get that
x t p 2 = Q K y t Q K p 2 y t p , j ( x t p ) = t γ t x t p , j ( x t p ) t ( 1 γ t ) p , j ( x t p ) + ( 1 t ) T ( s t ) x t p , j ( x t p ) [ 1 ( 1 γ t ) t ] x t p 2 t ( 1 γ t ) p , j ( x t p ) .
Thus
x t p 2 p , j ( x t p ) , p F ( S ) .
(3.3)
Since { x t } is bounded and E is reflexive, there exists a subsequence { x t n } of { x t } such that x t n x . From (3.2), we have x t n T ( s ) x t n 0 as n . Since E satisfies Opial’s condition, it follows from Lemma 2.3 that x F ( S ) . From (3.3), we have
x t n p 2 p , j ( x t n p ) , p F ( S ) .
(3.4)
In particular, if we substitute x for p in (3.4), then we have
x t n x 2 x , j ( x t n x ) .
(3.5)
Since j is weakly sequentially continuous from E to E , it follows from (3.5) that
lim n x t n x 2 lim n x , j ( x t n x ) = 0 .
Suppose that there exists a subsequence { x t m } of { x t } such that x t m x ˜ . Then we have x ˜ F ( S ) and
x t m p 2 p , j ( x t m p ) , p F ( S ) .
(3.6)
Since x , x ˜ F ( S ) , from (3.4) and (3.6), we have
x t n x ˜ 2 x ˜ , j ( x t n x ˜ ) ,
(3.7)
and
x t m x 2 x , j ( x t m x ) .
(3.8)
Now, in (3.7) and (3.8), taking n and m , respectively. We get
x x ˜ 2 x ˜ , j ( x x ˜ ) ,
(3.9)
and
x ˜ x 2 x , j ( x ˜ x ) .
(3.10)
Adding up (3.9) and (3.10), we have
x x ˜ 2 0 .

We have proved that each cluster point of { x t } (as t 0 ) equals x . Thus x t x as t 0 . □

Remark 3.2 Theorem 3.1 improves and extends Theorem 3.1 of Yao et al. [7] in the following aspects.
  1. (1)

    From a real Hilbert space to a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition.

     
  2. (2)

    1 λ t 0 λ t T ( s ) x t d s is replaced by T ( s t ) x t .

     
Theorem 3.3 Let E be a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition, and let K be a nonempty closed convex subset of E. Let S = { T ( s ) : s 0 } : K K be a uniformly asymptotically regular nonexpansive semigroup such that F ( S ) . Let { x n } be a sequence generated in the following iterative process:
x n + 1 = ( 1 β n ) x n + β n Q K [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] , n 0 ,
(3.11)
where { α n } , { β n } and { γ n } are sequences of real numbers in [ 0 , 1 ] satisfying the following conditions:
  1. (1)

    lim n γ n = 1 , n = 1 ( 1 γ n ) α n = , lim n α n = 0 .

     
  2. (2)

    0 < lim inf n β n lim sup n β n < 1 .

     
  3. (3)

    h , s n 0 such that s n + 1 = h + s n and lim n s n = .

     

Then { x n } converges strongly to x F ( S ) .

Proof Let p F ( S ) , we can get
x n + 1 p = ( 1 β n ) x n + β n Q K [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] p ( 1 β n ) x n p + β n Q K [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] p ( 1 β n ) x n p + β n α n γ n ( x n p ) α n ( 1 γ n ) p + ( 1 α n ) ( T ( s n ) x n p ) ( 1 β n ) x n p + β n ( α n γ n x n p α n ( 1 γ n ) p + ( 1 α n ) x n p ) = [ 1 ( 1 γ n ) α n β n ] x n p + ( 1 γ n ) α n β n p max { x n p , p } max { x 0 p , p } .

Hence, { x n } is bounded, so is { T ( s n ) x n } .

Set y n = Q K [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] for all n 0 . Then x n + 1 = ( 1 β n ) x n + β n y n .
y n + 1 y n = Q K [ α n + 1 ( γ n + 1 x n + 1 ) + ( 1 α n + 1 ) T ( s n + 1 ) x n + 1 ] Q K [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] [ α n + 1 ( γ n + 1 x n + 1 ) + ( 1 α n + 1 ) T ( s n + 1 ) x n + 1 ] [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] = α n + 1 γ n + 1 ( x n + 1 x n ) + ( α n + 1 γ n + 1 α n γ n ) x n + ( 1 α n + 1 ) × ( T ( s n + 1 ) x n + 1 T ( s n + 1 ) x n + T ( s n + 1 ) x n T ( s n ) x n ) + ( α n + 1 α n ) T ( s n ) x n α n + 1 γ n + 1 x n + 1 x n + | α n + 1 γ n + 1 α n γ n | x n + ( 1 α n + 1 ) ( x n + 1 x n + T ( h ) T ( s n ) x n T ( s n ) x n ) + | α n + 1 α n | T ( s n ) x n = [ 1 ( 1 γ n + 1 ) α n + 1 ] x n + 1 x n + | α n + 1 γ n + 1 α n γ n | x n + ( 1 α n + 1 ) T ( h ) ( s n ) x n T ( s n ) x n + | α n + 1 α n | T ( s n ) x n .
So,
y n + 1 y n x n + 1 x n ( 1 γ n + 1 ) α n + 1 x n + 1 x n + | α n + 1 γ n + 1 α n γ n | x n + ( 1 α n + 1 ) T ( h ) ( s n ) x n T ( s n ) x n + | α n + 1 α n | T ( s n ) x n .
(3.12)
Since { T ( s ) : s 0 } is uniformly asymptotically regular and lim n s n = , it follows that
lim n T ( h ) T ( s n ) x n T ( s n ) x n lim n sup x B T ( h ) T ( s n ) x T ( s n ) x = 0 ,
(3.13)
where B is any bounded set containing { x n } . Moreover, since { x n } , { T ( s n ) x n } are bounded, and α n 0 as n , (3.12) implies that
lim sup n ( y n + 1 y n x n + 1 x n ) 0 .

Hence, by Lemma 2.4 we have lim n y n x n = 0 since x n + 1 x n = β n ( y n x n ) . Consequently, lim n x n + 1 x n = 0 .

It follows from (3.11) that
x n T ( s n ) x n x n x n + 1 + x n + 1 T ( s n ) x n x n x n + 1 + ( 1 β n ) ( x n T ( s n ) x n ) + β n ( Q K [ α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n ] T ( s n ) x n ) x n x n + 1 + ( 1 β n ) x n T ( s n ) x n + α n γ n x n T ( s n ) x n + α n ( 1 γ n ) T ( s n ) x n = x n x n + 1 + ( 1 β n + α n γ n ) x n T ( s n ) x n + α n ( 1 γ n ) T ( s n ) x n .
So,
x n T ( s n ) x n 1 β n α n γ n ( x n x n + 1 + α n ( 1 γ n ) T ( s n ) x n ) 0 .
(3.14)
Since
x n T ( h ) x n x n T ( s n ) x n + T ( s n ) x n T ( h ) T ( s n ) x n + T ( h ) T ( s n ) x n T ( h ) x n 2 x n T ( s n ) x n + T ( s n ) x n T ( h ) T ( s n ) x n ,
from (3.13) and (3.14), we have
lim n x n T ( h ) x n = 0 .
(3.15)
Notice that { x n } is bounded. Put x = Q F ( S ) ( 0 ) . Then there exists a positive number R such that B ( x , R ) K contains { x n } . Moreover, B ( x , R ) K is T ( s ) -invariant for all s 0 and so, without loss of generality, we can assume that { T ( s ) : s 0 } is a nonexpansive semigroup on B ( x , R ) K . We take a subsequence { x n k } of { x n } such that
lim sup n x , j ( x n x ) = lim k x , j ( x n k x ) .
We may also assume that x n k x ˜ . It follows from Lemma 2.3 and (3.15) that x ˜ F ( S ) and hence
x , j ( x ˜ x ) 0 .
Since j is weakly sequentially continuous, we have
lim sup n x , j ( x n x ) = lim k x , j ( x n k x ) = x , j ( x ˜ x ) 0 .
Since lim n y n x n = 0 , we have y n x x n x , so
lim sup n x , j ( y n x ) = lim sup n x , j ( x n x ) 0 .
Set u n = α n ( γ n x n ) + ( 1 α n ) T ( s n ) x n . It follows that y n = Q K u n for all n 0 . By Proposition 2.1(3), we have
y n u n , j ( y n x ) 0 ,
and so
y n x 2 = y n x , j ( y n x ) = y n u n , j ( y n x ) + u n x , j ( y n x ) u n x , j ( y n x ) = α n γ n x n x , j ( y n x ) α n ( 1 γ n ) x , j ( y n x ) + ( 1 α n ) T ( s n ) x n x , j ( y n x ) α n γ n x n x j ( y n x ) α n ( 1 γ n ) x , j ( y n x ) + ( 1 α n ) T ( s n ) x n x j ( y n x ) [ 1 ( 1 γ n ) α n ] x n x y n x α n ( 1 γ n ) x , j ( y n x ) 1 ( 1 γ n ) α n 2 x n x 2 + 1 2 y n x 2 α n ( 1 γ n ) x , j ( y n x ) ,
that is,
y n x 2 [ 1 ( 1 γ n ) α n ] x n x 2 2 α n ( 1 γ n ) x , j ( y n x ) .
By the convexity of 2 , we have
x n + 1 x 2 ( 1 β n ) x n x 2 + β n y n x 2 [ 1 ( 1 γ n ) α n β n ] x n x 2 2 ( 1 γ n ) α n β n x , j ( y n x ) .

By Lemma 2.5, we conclude that x n x . □

Remark 3.4 Theorem 3.3 improves and extends Theorem 3.3 of Yao et al. [7] in the following aspects.
  1. (1)

    From a real Hilbert space to a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition.

     
  2. (2)

    1 λ n 0 λ n T ( s ) x n d s is replaced by T ( s n ) x n .

     

Declarations

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Hubei Normal University

References

  1. Aleyner A, Censor Y: Best approximation to common fixed points of a semigroup of nonexpansive operators. J. Nonlinear Convex Anal. 2005, 6: 137–151.MathSciNetGoogle Scholar
  2. Sunthrayuth P, Kumam P: A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces. J. Nonlinear Anal. Optim. 2010, 1: 139–150.MathSciNetGoogle Scholar
  3. Yao, Y, Liou, Y-C, Yao, J-C: Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces. J. Nonlinear Convex Anal. (in press)Google Scholar
  4. Yang P, Yao Y, Liou Y-C, Chen R: Hybrid algorithms of nonexpansive semigroups for variational inequalities. J. Appl. Math. 2012. 10.1155/2012/634927Google Scholar
  5. Yao Y, Cho YJ, Liou Y-C: Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequalities. Fixed Point Theory Appl. 2011., 2011: Article ID 101 10.1186/1687-1812-2011-101Google Scholar
  6. Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. Appl. Math. Lett. 2007, 20: 751–757. 10.1016/j.aml.2006.09.003MathSciNetView ArticleGoogle Scholar
  7. Yao Y, Kang JI, Cho YJ, Liou Y-C: Approximation of fixed points for nonexpansive semigroup in Hilbert spaces. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-31Google Scholar
  8. Gossez J-P, Lami Dozo E: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pac. J. Math. 1972, 40: 565–573. 10.2140/pjm.1972.40.565MathSciNetView ArticleGoogle Scholar
  9. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.Google Scholar
  10. Kopecka E, Reich S: Nonexpansive retracts in Banach spaces. Banach Cent. Publ. 2007, 77: 161–174.MathSciNetView ArticleGoogle Scholar
  11. Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3MathSciNetView ArticleGoogle Scholar
  12. Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 1993, 2: 333–342.MathSciNetGoogle Scholar
  13. Jung JS: Iterative approach to common fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2005, 302: 509–520. 10.1016/j.jmaa.2004.08.022MathSciNetView ArticleGoogle Scholar
  14. Suzuki T: Strong convergence of Krasnoselskii and Mann’s sequences for one-parameter nonexpansive semigroup without Bochner integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleGoogle Scholar
  15. Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar

Copyright

© Wang et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.