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Approximation of fixed points for nonexpansive semigroups in Hilbert spaces

Fixed Point Theory and Applications20132013:31

https://doi.org/10.1186/1687-1812-2013-31

  • Received: 30 July 2012
  • Accepted: 30 January 2013
  • Published:

Abstract

In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others.

MSC:47H05, 47H10, 47H17.

Keywords

  • nonexpansive semigroup
  • common fixed point
  • algorithms
  • projection

1 Introduction

Let H be a real Hilbert space with the inner product , and the norm . Let C be a nonempty closed convex subset of H. A mapping T : C C is said to be nonexpansive if
T x T y x y , x , y C .
Recall that a family S : = { T ( s ) } s 0 of mappings of C into itself is called a nonexpansive semigroup if it satisfies the following conditions:
  1. (S1)

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

     
  2. (S2)

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

     
  3. (S3)

    T ( s ) x T ( s ) y x y for all x , y C and s 0 ;

     
  4. (S4)

    for each x H , s T ( s ) x is continuous.

     

We denote by Fix ( T ( s ) ) the set of fixed points of T ( s ) and by Fix ( S ) the set of all common fixed points of S, i.e., Fix ( S ) = s 0 Fix ( T ( s ) ) . It is known that Fix ( S ) is closed and convex [[1], Lemma 1].

Approximation of fixed points of nonexpansive mappings by a sequence of finite means has been considered by many authors; see, for instance, [127]. This work was originated with the beautiful work of Baillon [5] in 1975 (see also [6] and [7] for a generalization): If C is a closed convex subset of a Hilbert space and T is a nonexpansive mapping from C into itself such that the set Fix ( T ) of fixed points of T is nonempty, then for each x C , the Cesàro mean
1 n k = 1 n T k x

converges weakly to x Fix ( T ) . In this case, if we put x = P Fix ( T ) x for each x C , then P Fix ( T ) is a nonexpansive retraction from C onto F ( T ) . In [18], Takahashi proved the existence of such a retraction for an amenable semigroup of nonexpansive mappings on a Hilbert space. In [19], Rodé also found a sequence of means on a semigroup generalizing the Cesàro means and extended Baillon’s theorem. In [28], Lau, Shioji and Takahashi extended Takahashi’s result and Rode’s result to a closed convex subset of a uniformly convex Banach space.

In the literature, a nonlinear ergodic theorem for nonexpansive semigroups has been considered by many authors (see [2946]). Especially, Shioji and Takahashi [17] introduced an implicit iteration { x n } in a Hilbert space defined by
x n = α n x + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s , n 0 ,
(1.1)
where { α n } is a sequence in ( 0 , 1 ) and { λ n } is a sequence of positive real numbers divergent to ∞. Under certain restrictions on the sequence { α n } , Shioji and Takahashi [17] proved strong convergence of { x n } generated by (1.1) to a member of Fix ( T ( s ) ) . In [16], Shimizu and Takahashi studied the strong convergence of the iterative sequence { x n } defined by
x n + 1 = α n x + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s , n 0 .
(1.2)

The corresponding viscosity approximations of (1.1) and (1.2) have been extended in [29]. Lau et al. [37] studied the iterative schemes of Browder and Halpern types for a nonexpansive semigroup { T ( s ) } s 0 on a compact convex subset C of a smooth (and strictly convex) Banach space with respect to a sequence { μ n } of strongly asymptotically invariant means defined on an appropriate invariant subspace of l ( S ) , the space of bounded real-valued functions on a semigroup S.

Motivated and inspired by the works in the literature, in this paper, we introduce two new algorithms for finding a common fixed point of a nonexpansive semigroup { T ( s ) } s 0 in Hilbert spaces and prove that both approaches converge strongly to a common fixed point of { T ( s ) } s 0 .

2 Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H. The metric (or nearest point) projection from H onto C is the mapping P C : H C which assigns to each point x C the unique point P C x C satisfying the property
x P C x = inf y C x y = : d ( x , C ) .
It is well known that P C is a nonexpansive mapping and satisfies
x y , P C x P C y P C x P C y 2 , x , y H .
Moreover, P C is characterized by the following properties:
x P C x , y P C x 0
(2.1)
and
x y 2 x P C x 2 + y P C x 2 , x H , y C .

We need the following lemmas for proving our main results.

Lemma 2.1 [16]

Let C be a nonempty bounded closed convex subset of a Hilbert space H and { T ( s ) } s 0 be a nonexpansive semigroup on C. Then, for any h 0 ,
lim t sup x C 1 t 0 t T ( s ) x d s T ( h ) 1 t 0 t T ( s ) x d s = 0 .

Lemma 2.2 [8]

Let C be a closed convex subset of a real Hilbert space H and S : C C be a nonexpansive mapping. Then the mapping I S is demiclosed. That is, if { x n } is a sequence in C such that x n x weakly and ( I S ) x n y strongly, then ( I S ) x = y .

Lemma 2.3 [13]

Let { x n } and { y n } be bounded sequences in a Banach space X and { γ n } be a sequence in [ 0 , 1 ] with 0 < lim inf n β n lim sup n β n < 1 . Suppose that
x n + 1 = ( 1 γ n ) x n + γ n y n , n 0 ,
and
lim sup n ( y n y n 1 x n x n 1 ) 0 .

Then lim n y n x n = 0 .

Lemma 2.4 [12]

Assume that { a n } is a sequence of nonnegative real numbers such that
a n + 1 ( 1 γ n ) a n + δ n γ n , n 1 ,
where { γ n } is a sequence in ( 0 , 1 ) and { δ n } is a sequence such that
  1. (a)

    n = 1 γ n = ;

     
  2. (b)

    lim sup n δ n 0 or n = 1 | δ n γ n | < .

     

Then lim n a n = 0 .

3 Main results

In this section, we show our main results.

Theorem 3.1 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 ) .
(3.1)

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

Proof First, we note that the net { x t } defined by (3.1) is well defined. We define the mapping
W x : = P C [ t ( γ t x ) + ( 1 t ) 1 λ t 0 λ t T ( s ) x d s ] , t ( 0 , 1 ) .
It follows that
W x W y t γ t ( x y ) + ( 1 t ) 1 λ t 0 λ t ( T ( s ) x T ( s ) y ) d s t γ t x y + ( 1 t ) 1 λ t 0 λ t ( T ( s ) x T ( s ) y ) d s t γ t x y + ( 1 t ) x y = [ 1 ( 1 γ t ) t ] x y .

This implies that the mapping W is a contraction and so it has a unique fixed point. Therefore, the net { x t } defined by (3.1) is well defined.

Take p Fix ( S ) . By (3.1), we have
x t p = P C [ t ( γ t x t ) + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s ] p t γ t ( x t p ) t ( 1 γ t ) p + ( 1 t ) ( 1 λ t 0 λ t T ( s ) x t d s p ) t γ t x t p + t ( 1 γ t ) p + ( 1 t ) 1 λ t 0 λ t T ( s ) x t T ( s ) p d s 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 ,
which implies that the net { x t } is bounded. Set R : = p . It is clear that { x t } B ( p , R ) . Notice that
1 λ t 0 λ t T ( s ) x t d s p x t p R .
Moreover, we observe that if x B ( p , R ) , then
T ( s ) x p T ( s ) x T ( s ) p x p R ,
i.e., B ( p , R ) is T ( s ) -invariant for all s. Set y t = t ( γ t x t ) + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s . Then x t = P C [ y t ] . It follows that
By Lemma 2.1, we deduce that for all 0 τ < ,
lim t 0 T ( τ ) x t x t = 0 .
(3.2)
Note that x t = P C [ y t ] . By using the property of the metric projection (2.1), we have
x t p 2 = x t y t , x t p + y t p , x t p y t p , x t p = t γ t x t p , x t p t ( 1 γ t ) p , x t p + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s p , x t p [ 1 ( 1 γ t ) t ] x t p 2 t ( 1 γ t ) p , x t p .
Therefore, we have
x t p 2 p , p x t , p Fix ( S ) .
(3.3)

From this inequality, immediately it follows that ω w ( x t ) = ω s ( x t ) , where ω w ( x t ) and ω s ( x t ) denote the sets of weak and strong cluster points of { x t } , respectively.

Let { t n } ( 0 , 1 ) be a sequence such that t n 0 as n . Put x n : = x t n , y n : = y t n and λ n : = λ t n . Since { x n } is bounded, without loss of generality, we may assume that the sequence { x n } converges weakly to a point x C . Also, y n x weakly. Noticing (3.2), we can use Lemma 2.2 to get x Fix ( S ) . From (3.3), we have
x n p 2 p , p x n , p Fix ( S ) .
(3.4)
In particular, if we substitute x for p in (3.4), then we have
x n x 2 x , x x n .
(3.5)

However, x n x . This together with (3.5) guarantees that x n x and so the net { x t } is relatively compact, as t 0 + , in the norm topology.

Now, in (3.4), taking n , we get
x p 2 p , p x , p Fix ( S ) .
This is equivalent to the following:
0 x , p x , p Fix ( S ) .

Therefore, x = P Fix ( T ) ( 0 ) , which is obviously unique. This is sufficient to conclude that the entire net { x t } converges in norm to x . This completes the proof. □

Remark 3.2

It is known that the algorithm
x t = P C [ t x t + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s ] , t ( 0 , 1 ) ,

has only weak convergence. However, our similar algorithm (3.1) (with γ t 1 ) has strong convergence.

Next, we introduce an explicit algorithm for the nonexpansive semigroup S = { T ( s ) } s 0 : C C and prove the strong convergence theorems of this algorithm.

Theorem 3.3 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 ,
(3.6)
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 (3.6) strongly converges to a point x Fix ( S ) .

Proof Take p Fix ( S ) . From (3.6), we have
It follows that, by induction,
x n p max { x 0 p , p } .
Set y n = P C [ α n ( γ n x n ) + ( 1 α n ) z n ] for all n 0 , where z n = 1 λ n 0 λ n T ( s ) x n d s . We have
and
Therefore, we have
where M > 0 is a constant such that
sup n 1 { x n 1 , z n 1 , 2 x n 1 p } M .
Hence we get
lim sup n ( y n y n 1 x n x n 1 ) 0 .
This together with Lemma 2.3 implies that
lim n y n x n = 0 .
Therefore, it follows that
lim n x n + 1 x n = lim n β n y n x n = 0 .
Note that
T ( τ ) x n x n T ( τ ) x n T ( τ ) 1 λ n 0 λ n T ( s ) x n d s + T ( τ ) 1 λ n 0 λ n T ( s ) x n d s 1 λ n 0 λ n T ( s ) x n d s + 1 λ n 0 λ n T ( s ) x n d s x n T ( τ ) 1 λ n 0 λ n T ( s ) x n d s 1 λ n 0 λ n T ( s ) x n d s + 2 x n 1 λ n 0 λ n T ( s ) x n d s .
(3.7)
From (3.6), we have
It follows that
(3.8)
From (3.7), (3.8) and Lemma 2.1, we have
lim n T ( τ ) x n x n = 0 , τ 0 .
(3.9)
Notice that { x n } is a bounded sequence and x ˜ is a weak limit of { x n } . Putting x = P Fix ( S ) ( 0 ) . Then there exists a positive number R such that B ( x , R ) contains { x n } . Moreover, B ( x , R ) 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 ) . By the demiclosedness principle (Lemma 2.2) and (3.9), we have x ˜ Fix ( S ) and hence
lim sup n x , x n + 1 x = lim n x , x ˜ x 0 .
Finally, we prove that x n x . Set u n = α n ( γ n x n ) + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s . It follows that y n = P C [ u n ] for all n 0 . By using the property of the metric projection (2.1), we have
y n u n , y n x 0
and so
y n x 2 = y n x , y n x = y n u n , y n x + u n x , y n x u n x , y n x = α n γ n x n x , y n x α n ( 1 γ n ) x , y n x + ( 1 α n ) z n x , y n x α n γ n x n x y n x α n ( 1 γ n ) x , y n x + ( 1 α n ) z n x y n x [ 1 ( 1 γ n ) α n ] x n x y n x α n ( 1 γ n ) x , y n x 1 ( 1 γ n ) α n 2 x n x 2 + 1 2 y n x α n ( 1 γ n ) x , 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 , y n x .
By the convexity of the norm, 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 , y n x .

Hence all the conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that x n x . This completes the proof. □

In Theorem 3.3, if we put β n = 1 for each n 1 , we have the following corollary.

Corollary 3.4 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 the sequence { x n } be generated iteratively by the following explicit algorithm:
x n + 1 = 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 ,
(3.10)
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)

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

     

Then the sequence { x n } generated by (3.10) strongly converges to a point x Fix ( S ) .

Remark 3.5

It is known that the algorithm
x n + 1 = ( 1 β n ) x n + β n P C [ α n x n + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s ] , n 0 ,

has only weak convergence. However, our similar algorithm (3.6) (with γ n 1 ) has strong convergence.

Declarations

Acknowledgements

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China
(2)
National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 463-1 Jeonmin-dong, Yusung-gu, Daejeon, 305-390, Korea
(3)
Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea
(4)
Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan

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