Open Access

Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces

Fixed Point Theory and Applications20122012:116

https://doi.org/10.1186/1687-1812-2012-116

Received: 22 February 2012

Accepted: 2 July 2012

Published: 20 July 2012

Abstract

In this paper, we provide the existence and convergence theorems of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in general Banach spaces, which extend and improve many recent results in this area.

MSC:47H09, 47H10, 47H20.

Keywords

asymptotically nonexpansive type mapping left amenable semigroup reversible semigroup fixed point

1 Introduction

Let E be a Banach space and C a nonempty bounded closed convex subset of E. A mapping T on C is said to be nonexpansive if T x T y x y for all x , y C . A well-known result of Browder [1] asserts that if E is uniformly convex, then every nonexpansive mapping on C has a fixed point. Kirk [2], Belluce and Kirk [3] extended this result to the case that X has a normal structure or Opial’s property. Goebel and Kirk [4] proved that if E is a uniformly convex Banach space, then every asymptotically nonexpansive mapping on C has a fixed point.

As is well known, not every semigroup of nonexpansive mappings on a subset of a Banach space has a fixed point [5]. The existence and convergence of fixed points for semigroups of various mappings have been studied extensively [610]. Recently, Suzuki and Takahashi [8], Takahashi and Zembayashi [9], Zhu and Li [10] proved the existence theorems of fixed points for semigroups = { T ( t ) : t 0 } of nonexpansive, asymptotically nonexpansive and asymptotically nonexpansive type mappings, respectively. For instance, in [9], Takahashi and Zembayashi proved the following theorem:

Theorem 1.1 [9]

Let C be a nonempty compact convex subset of a Banach space E and = { T ( t ) : t 0 } be a semigroup of asymptotically nonexpansive mappings on C, then the set of common fixed points F ( ) of is nonempty.

Many results are known in the case that the semigroup G is commutative, amenable or reversible [1124]. In the case of an amenable semigroup, the first result was established by Takahashi [21] where he proved:

Theorem 1.2 [21]

Let C be a nonempty compact convex subset of a Banach space E. Let = { T ( t ) : t G } be an amenable semigroup of nonexpansive mappings on C. Then C contains a common fixed point for .

Theorem 1.2 was proved for a commutative semigroup by DeMarr [11]. Later in [13], Lau showed that the fixed point property is equivalent to the existence of a left invariant mean on A P ( ) , the space of almost periodic functions on the semigroup . It should be pointed out that if is left reversible, then A P ( ) always has a left invariant mean [13], but the converse is false [14]. And in [16], Lau, Miyake and Takahashi gave the following existence theorem:

Theorem 1.3 [16]

Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup (with identity) and = { T ( t ) : t G } be a semigroup of nonexpansive mappings on C. Let X be a left invariant -stable subspace of l ( G ) containing 1, and μ be a left invariant mean on X. Then F ( ) = F ( T μ ) C a , where C a denotes the set of almost periodic elements in C, i.e., all x C such that { T ( s ) x : s G } is relatively compact in the norm topology of E. Further, if C is compact, then the set F ( ) is nonempty.

In [20], Saeidi extended Theorem 1.3 to the case for left reversible semigroups of asymptotically nonexpansive mappings. Inspired and motivated by [810, 16, 20, 21, 23], we investigate the existence and convergence of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. We first provide the existence theorem of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. Utilizing this result, we obtain a strong convergence theorem of iterative sequences for left amenable semigroups of asymptotically nonexpansive type mappings. The results obtained in this paper extend and improve many recent results in [810, 16, 20, 23].

2 Preliminaries

Let C be a nonempty bounded subset of a Banach space E. Let G be a semitopological semigroup, i.e., G is a semigroup with a Hausdorff topology such that for s G the mappings s s t and s t s from G to G are continuous. Let = { T ( t ) : t G } be a continuous representation of G on C, i.e., T ( t s ) x = T ( t ) T ( s ) x , t , s G , x C and the mapping ( t , x ) T ( t ) x from G × C into C is continuous when G × C has the product topology. Recall that is said to be
  1. (1)
    nonexpansive if for all x , y C and t G ,
    T ( t ) x T ( t ) y x y ;
     
  2. (2)
    asymptotically nonexpansive [2527] if there exists a function k : G [ 0 , + ) with inf s G sup t G k ( t s ) 1 such that for all x , y C and t G ,
    T ( t ) x T ( t ) y k ( t ) x y ;
     
  3. (3)
    asymptotically nonexpansive type [2527] if for each x C , there exists a function r ( , x ) : G [ 0 , + ) with inf s G sup t G r ( t s , x ) = 0 such that for all x , y C and t G ,
    T ( t ) x T ( t ) y x y + r ( t , x ) .
     

It is easily seen that ( 1 ) ( 2 ) ( 3 ) and that both inclusions are proper [2527].

Let l ( G ) be the Banach space of all bounded real valued functions on G with the supremum norm. Then, for each s G and f l ( G ) , we can define l s f in l ( G ) by ( l s f ) ( t ) = f ( s t ) for all t G . Let X be a subspace of l ( G ) containing 1 and X be its dual space. An element μ X is called a mean on X if μ = μ ( 1 ) = 1 . We always denote the value of μ at f X by μ t f ( t ) = μ ( f ) . Let X be left invariant, i.e., l s ( X ) X for all s G . A mean μ on X is said to be left invariant if μ ( l s f ) = μ ( f ) for all s G and f X . Further, X is called left amenable if X has a left invariant mean. In this case, we also say that G is a left amenable semigroup. Recall that a semigroup G is called left reversible if any two closed right ideals of G have nonvoid intersection. In this case, ( G , ) is a directed system when the binary relation ≤ on G is defined by s t if and only if { s } s G ¯ { t } t G ¯ , s , t G . As is well known, the class of left reversible semigroups includes all commutative semigroups and if a semigroup G is left amenable, then G is left reversible. But the converse is false [28].

Let = { T ( t ) : t G } be an asymptotically nonexpansive type semigroup on C. Let F ( ) denote the set of all fixed points of , i.e., F ( ) = { x C : T ( s ) x = x , s G } . A subspace X of l ( G ) is called -stable if functions s T ( s ) x , x and s T ( s ) x y on G are in X for all x , y C and x E . We know that if μ is a mean on X and if for each x E the function s T ( s ) x , x is contained in X and C is weakly compact, then there exists a unique point x 0 of E such that μ s T ( s ) x , x = x 0 , x for all x E . Such a point x 0 is always denoted by T μ x . Obviously, T μ x = x for each x F ( ) .

3 Main results

Lemma 3.1 Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup and = { T ( t ) : t G } be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r ( s , x ) = 0 for all x C . Let X be a left invariant -stable subspace of l ( G ) containing 1, and μ be a left invariant mean on X. Then F ( ) = F ( T μ ) C a .

Proof If F ( T μ ) C a is empty, then so is F ( ) as F ( ) F ( T μ ) C a . Let z F ( T μ ) C a and define d = μ s T ( s ) z z , then for all t G , we have
T ( t ) z z = T ( t ) z T μ z = sup { | T ( t ) z T μ z , x | : x E , x = 1 } = sup { | μ s T ( t ) z T ( s ) z , x | : x E , x = 1 } sup { μ s T ( t ) z T ( s ) z x : x E , x = 1 } = μ s T ( t ) z T ( s ) z = μ s T ( t ) z T ( t s ) z (by  μ -left invariant) μ s T ( s ) z z + r ( t , z ) = d + r ( t , z ) ,
i.e., for all t G ,
T ( t ) z z d + r ( t , z ) .
(3.1)
Next, we shall show d = 0 . In fact, if d > 0 , then for each t G ,
d = μ s T ( s ) z z = μ s T ( t s ) z z sup s G T ( t s ) z z ,
i.e.,
sup s G T ( t s ) z z d , t G .
(3.2)
By lim sup s G r ( s , z ) = 0 , then for any n N , there exists s n G such that
sup t s n r ( t , z ) < 1 4 n .
(3.3)
It follows from (3.2) that we can choose a cluster point u 1 of the net { T ( s ) z : s G } in the set C with u 1 z d and there exists t n ( 1 ) G satisfying t n ( 1 ) s n and T ( t n ( 1 ) ) z u 1 < 1 4 n . Combining it with (3.1) and (3.3), we get
u 1 z u 1 T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z z 1 4 n + d + r ( t n ( 1 ) , z ) d + 1 2 n (by  t n ( 1 ) s n ) .
Hence u 1 z d and so u 1 z = d . It follows from (3.1) and (3.3) that
T ( t n ( 1 ) s n s ) z u 1 T ( t n ( 1 ) s n s ) z T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z u 1 T ( s n s ) z z + r ( t n ( 1 ) , z ) + T ( t n ( 1 ) ) z u 1 d + r ( s n s , z ) + 1 2 n d + 3 4 n
(3.4)
for all s G . Noting
d = u 1 z = u 1 T μ z = sup { | u 1 T μ z , x | : x E , x = 1 } = sup { | μ s u 1 T ( s ) z , x | : x E , x = 1 } μ s u 1 T ( s ) z ,
(3.5)
we obtain
μ s ( T ( t n ( 1 ) s n s ) z z + T ( t n ( 1 ) s n s ) z u 1 ) = μ s T ( t n ( 1 ) s n s ) z z + μ s T ( t n ( 1 ) s n s ) z u 1 = μ s T ( s ) z z + μ s T ( s ) z u 1 2 d .
This implies that
sup s G [ T ( t n ( 1 ) s n s ) z z + T ( t n ( 1 ) s n s ) z u 1 ] 2 d .
Thus there exists s n ( 1 ) G such that
T ( t n ( 1 ) s n s n ( 1 ) ) z z + T ( t n ( 1 ) s n s n ( 1 ) ) z u 1 2 d 1 4 n .
(3.6)
Since { T ( s ) z : s G } is a relatively compact set, { T ( t n ( 1 ) s n s n ( 1 ) ) z } , as a subset of { T ( s ) z : s G } , has a strong convergent subsequence. Without loss of generality, we can assume that T ( t n ( 1 ) s n s n ( 1 ) ) z u 2 C . Setting t n ( 2 ) = t n ( 1 ) s n s n ( 1 ) , then t n ( 2 ) t n ( 1 ) s n , T ( t n ( 2 ) ) z u 2 and by (3.6),
u 2 z + u 2 u 1 2 d .
(3.7)
On the other hand,
u 2 z u 2 T ( t n ( 2 ) ) z + T ( t n ( 2 ) ) z z u 2 T ( t n ( 2 ) ) z + d + r ( t n ( 2 ) , z ) (by (3.1)) u 2 T ( t n ( 2 ) ) z + d + 1 4 n (by (3.3))
and
u 2 u 1 u 2 T ( t n ( 2 ) ) z + T ( t n ( 2 ) ) z u 1 = u 2 T ( t n ( 2 ) ) z + T ( t n ( 1 ) s n s n ( 1 ) ) z u 1 u 2 T ( t n ( 2 ) ) z + d + 3 4 n (by (3.4)) .
Thus we can conclude u 2 z d and u 2 u 1 d . So by (3.7),
u 2 z = u 2 u 1 = d .
Similar to the proof of (3.5), we can prove μ s u 2 T ( s ) z d and
μ s ( T ( t n ( 2 ) s n s ) z z + T ( t n ( 2 ) s n s ) z u 1 + T ( t n ( 2 ) s n s ) z u 2 ) = μ s T ( t n ( 2 ) s n s ) z z + μ s T ( t n ( 2 ) s n s ) z u 1 + μ s T ( t n ( 2 ) s n s ) z u 2 = μ s T ( s ) z z + μ s T ( s ) z u 1 + μ s T ( s ) z u 2 3 d .
This means
sup s G ( T ( t n ( 2 ) s n s ) z z + T ( t n ( 2 ) s n s ) z u 1 + T ( t n ( 2 ) s n s ) z u 2 ) 3 d .
Thus there exists s n ( 2 ) G such that
T ( t n ( 2 ) s n s n ( 2 ) ) z z + T ( t n ( 2 ) s n s n ( 2 ) ) z u 1 + T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 3 d 1 n .
Therefore,
T ( t n ( 2 ) s n s n ( 2 ) ) z z d + r ( t n ( 2 ) s n s n ( 2 ) , z ) d + 1 4 n , T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z T ( t n ( 2 ) ) z + T ( t n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 T ( s n s n ( 2 ) ) z z + r ( t n ( 2 ) , z ) + T ( t n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 d + r ( s n s n ( 2 ) , z ) + r ( t n ( 2 ) , z ) + T ( t n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 d + 1 2 n + T ( t n ( 2 ) ) z u 2
and
T ( t n ( 2 ) s n s n ( 2 ) ) z u 1 T ( t n ( 2 ) s n s n ( 2 ) ) z T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z u 1 T ( t n ( 1 ) s n s n ( 1 ) s n s n ( 2 ) ) z T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z u 1 T ( s n s n ( 1 ) s n s n ( 2 ) ) z z + r ( t n ( 1 ) , z ) + T ( t n ( 1 ) ) z u 1 d + r ( s n s n ( 1 ) s n s n ( 2 ) , z ) + r ( t n ( 1 ) , z ) + T ( t n ( 1 ) ) z u 1 d + 1 2 n + T ( t n ( 1 ) ) z u 1 .
Since { T ( t n ( 2 ) s n s n ( 2 ) ) z } has a strong convergent subsequence, without loss of generality, we can assume that T ( t n ( 2 ) s n s n ( 2 ) ) z u 3 C . Setting t n ( 3 ) = t n ( 2 ) s n s n ( 2 ) , then t n ( 3 ) t n ( 2 ) , T ( t n ( 3 ) ) z u 3 ,
u 3 z d , u 3 u 2 d , u 3 u 1 d
and
u 3 z + u 3 u 1 + u 3 u 2 3 d .
Thus we have found u 3 C such that
u 3 u 1 = u 3 u 2 = u 3 z = d .
Now, by mathematical induction, we can find a sequence { u i } C satisfying
u i z = d , u i u j = d ( i , j N , i j ) .
Since T ( t n ( i ) ) z u i , we can seek out t n i ( i ) G with T ( t n i ( i ) ) z u i d 4 . Thus
T ( t n i ( i ) ) z T ( t n j ( j ) ) z d 2 ( i , j N , i j ) ,

which is a contradiction with the relative compactness of { T ( t n i ( i ) ) z : i N } . Therefore, we can conclude d = 0 .

In the following, we shall show z F ( ) . Indeed, for any h G , T ( h ) : C C is continuous at z, then for all ε > 0 , there exists a δ > 0 ( δ < ε ) such that for all x C with x z < δ ,
T ( h ) x T ( h ) z < ε .
By (3.1) and the definition of r ( , z ) , we can get
inf s G sup t G T ( t s ) z z d + inf s G sup t G r ( t s , z ) = 0
and so we can find a s δ G such that sup t G T ( t s δ ) z z < δ , i.e., for all t G ,
T ( t s δ ) z z < δ .
Hence
T ( h ) z z T ( h ) z T ( h ) T ( t s δ ) z + T ( h ) T ( t s δ ) z z = T ( h ) z T ( h ) T ( t s δ ) z + T ( h t s δ ) z z < ε + δ < 2 ε .

Since ε > 0 is arbitrary, we get z F ( ) . This completes the proof. □

Now we can give the existence theorem of fixed points for left amenable semigroups of non-Lipschitzian mappings in Banach spaces.

Theorem 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and = { T ( t ) : t G } be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r ( s , x ) = 0 for all x C . Let X be a left invariant -stable subspace of l ( G ) containing 1, and μ be a left invariant mean on X. Then the set F ( ) is nonempty.

Proof For all x , y C and t G , we have
T μ x T μ y = sup { | T μ x T μ y , x | : x E , x = 1 } = sup { | μ s T ( s ) x T ( s ) y , x | : x E , x = 1 } sup { μ s T ( s ) x T ( s ) y x : x E , x = 1 } = μ s T ( s ) x T ( s ) y = μ s T ( t s ) x T ( t s ) y sup s G T ( t s ) x T ( t s ) y x y + sup s G r ( t s , z ) ,

and so by lim sup s G r ( s , z ) = 0 , we get T μ x T μ y x y , i.e., T μ is a nonexpansive mapping from C into itself. Since a nonexpansive mapping of a compact convex subset of a Banach space into itself has a fixed point [29], T μ has a fixed point z. By Lemma 3.1, z F ( ) . This completes the proof. □

Remark 3.1 Theorem 3.1 is an extension of the main results in [810, 16, 20, 23] to the case for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces.

Recall that for each s G , we define a point evaluation δ s on X by δ s ( f ) = f ( s ) for every f X . A convex combination of point evaluation is called a finite mean on G. If λ is a finite mean on G, say λ = Σ i = 1 n a i δ s i , where s i G , a i 0 , i = 1 , 2 , , n , and Σ i = 1 n a i = 1 , then λ ( t ) T ( t ) x , x = Σ i = 1 n a i T ( s i ) x , x = Σ i = 1 n a i T ( s i ) x , x for all x E . For convenience, we denote it by λ ( t ) T ( t ) x = Σ i = 1 n a i T ( s i ) x . A net { λ α : α I } of finite means on G is said to be strongly left regular if
lim α I λ α l s λ α = 0

for all s G , where A is a directed system and l s is the conjugate operator of l s .

Corollary 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and = { T ( t ) : t G } be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r ( s , x ) = 0 for all x C . Let X be a left invariant -stable subspace of l ( G ) containing 1 and { λ α : α I } be a net of strongly left regular finite means on G. If z C satisfies
lim inf α I λ α ( t ) T ( t ) z z = 0 ,

then z F ( ) .

Proof Since lim inf α I λ α ( t ) T ( t ) z z = 0 and { λ α : α I } D , we can find a subnet { λ α β : β A } of { λ α : α I } such that lim β A λ α β ( t ) T ( t ) z = z and ω lim β A λ α β = μ , where A is a directed system. Hence μ is a left invariant mean on X (see [30]) and T μ z = z , which implies z F ( ) . This completes the proof. □

Remark 3.2 Corollary 3.1 is an extension of the main results in [810, 23].

Next we shall prove the strong convergence theorem for the iterative sequences of left reversible semigroups of asymptotically nonexpansive type mappings. We need a lemma which plays a crucial role in the proof of Theorem 3.2.

Lemma 3.2 [30]

Let z n and w n be bounded sequences in a Banach space X and let α n be a sequence in ( 0 , 1 ) with 0 < lim inf n α n lim sup n α n < 1 . Suppose that z n + 1 = α n w n + ( 1 α n ) z n for all n N and
lim sup n ( w n w n + k z n z n + k ) 0

for all k N . Then lim inf n w n z n = 0 .

Theorem 3.2 Let C be a nonempty compact convex subset of a Banach space X and G be a left reversible semigroup. Let = { T ( t ) : t G } be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r ( s , x ) = 0 for all x C . Let X be a left invariant -stable subspace of l ( G ) containing 1, and μ be a left invariant mean on X. Let x 1 C and define a sequence { x n } in C by
x n + 1 = α n T μ x n + ( 1 α n ) x n ,

for all n N , where α n [ 0 , 1 ] satisfies 0 < lim inf n α n lim sup n α n < 1 . Then x n converges strongly to a fixed point z F ( ) .

Proof It follows from
T μ x n + 1 x n + 1 T μ x n + 1 T μ x n + T μ x n x n + 1 = T μ x n + 1 T μ x n + ( 1 α n ) T μ x n x n = T μ x n x n + T μ x n + 1 T μ x n x n + 1 x n T μ x n x n
that lim n T μ x n x n exists. By Lemma 3.1, we get lim inf n T μ x n x n = 0 and so lim n T μ x n x n = 0 . Since C is compact, there exists a subsequence { x n k } { x n } such that x n k z C . Hence, z is a fixed point of T μ . By Lemma 3.1, we have z F ( ) and
x n + 1 z = α n T μ x n + ( 1 α n ) x n z α n T μ x n z + ( 1 α n ) x n z x n z .

Hence lim n x n z exists. Thus lim n x n z = lim k x n k z = 0 , which implies that x n converges strongly to z F ( ) . This completes the proof. □

In the following, we shall give an example of a semigroup which is asymptotically nonexpansive type but not asymptotically nonexpansive on a compact set.

Example 3.1 [27]

Let Δ be the Cantor ternary set. Define the Cantor ternary function
τ ( x ) = { n = 1 + b n 2 n , x = n = 1 + 2 b n 3 n Δ ( b n = 0 , 1 ) , sup { τ ( y ) , y x , y Δ } , x [ 0 , 1 ] Δ
then τ : [ 0 , 1 ] [ 0 , 1 ] is a continuous and increasing but not absolutely continuous function with τ ( 0 ) = 0 , τ ( 1 2 ) = 1 2 (see [31]). Since a Lipschitzian function is absolutely continuous, τ is non-Lipschitzian. For all t > 0 , we define T ( t ) : [ 0 , 1 ] [ 0 , 1 ] by
T ( t ) x = { x 2 t , 0 x 1 2 , τ ( 1 x ) 2 t , 1 2 < x 1 .
Then T ( t ) is continuous but not Lipschitzian continuous (since τ is non-Lipschitzian) and for all x , y [ 0 , 1 ] , | T ( t ) x | 1 2 t + 1 ,
| T ( t ) x T ( t ) y | 1 2 t | x y | + 1 2 t .

Therefore, we can conclude that the semigroup = { T ( t ) : t > 0 } is asymptotically nonexpansive type but not an asymptotically nonexpansive on [ 0 , 1 ] . Also, 0 is a fixed point of .

Declarations

Acknowledgement

This manuscript has benefited greatly from the constructive comments and helpful suggestions of the anonymous referees, the authors would like to express their deep gratitude to them. This research is supported by the Natural Science Foundation of China (10971182, 11201410), the Natural Science Foundation of Jiangsu Province (BK2009179, BK2010309 and BK2012260), the Jiangsu Government Scholarship for Overseas Studies, the Natural Science Foundation of Jiangsu Education Committee (10KJB110012) and the Natural Science Foundation of Yangzhou University.

Authors’ Affiliations

(1)
College of Mathematics, Yangzhou University
(2)
School of Mathematical Sciences, Monash University

References

  1. Browder F: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleGoogle Scholar
  2. Kirk WA: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleGoogle Scholar
  3. Belluce LP, Kirk WA: Nonexpansive mappings and fixed points in Banach spaces. Ill. J. Math. 1967, 11: 474–479.MathSciNetGoogle Scholar
  4. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleGoogle Scholar
  5. Alspach DE: A fixed point free nonexpansive map. Proc. Am. Math. Soc. 1981, 82: 423–424. 10.1090/S0002-9939-1981-0612733-0MathSciNetView ArticleGoogle Scholar
  6. Suzuki T: Mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups. Nonlinear Anal. 2008, 68: 3870–3878. 10.1016/j.na.2007.04.026MathSciNetView ArticleGoogle Scholar
  7. Suzuki T: Browder’s type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces. Isr. J. Math. 2007, 157: 239–257. 10.1007/s11856-006-0010-6View ArticleGoogle Scholar
  8. Suzuki T, Takahashi W: Strong convergence of Mann’s type sequences for one-parameter nonexpansive semigroups in general Banach spaces. J. Nonlinear Convex Anal. 2004, 5: 209–216.MathSciNetGoogle Scholar
  9. Takahashi W, Zembayashi K: Fixed point theorems for one parameter asymptotically nonexpansive semigroups in Banach spaces. Nonlinear Anal. 2006, 65: 433–441. 10.1016/j.na.2005.08.020MathSciNetView ArticleGoogle Scholar
  10. Zhu LP, Li G: Fixed points theorems for asymptotically nonexpansive type semigroups in general Banach spaces. Acta Math. Sci. 2009, 29: 290–296.Google Scholar
  11. DeMarr R: Common fixed-points for commuting contraction mappings. Pac. J. Math. 1963, 13: 1139–1141.MathSciNetView ArticleGoogle Scholar
  12. Kada O, Lau ATM, Takahashi W: Asymptotically invariant net and fixed point set for semigroup of nonexpansive mappings. Nonlinear Anal. 1997, 29: 537–550.MathSciNetView ArticleGoogle Scholar
  13. Lau ATM: Invariant means on almost periodic functions and fixed point properties. Rocky Mt. J. Math. 1973, 3: 69–76. 10.1216/RMJ-1973-3-1-69View ArticleGoogle Scholar
  14. Lau ATM: Amenability of semigroups. de Gruyter Expositions in Mathematics 1. In Proceedings of 1989 Oberwolfach Conference on Analytical and Topological Properties of Semigroups. Edited by: Hofmann KH, Lawson JD, Pym JS. Berlin, New York; 1990:313–334.Google Scholar
  15. Lau ATM: Invariant means and fixed point properties of semigroup of nonexpansive mappings. Taiwan. J. Math. 2008, 12: 1525–1542.Google Scholar
  16. Lau ATM, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach space. Nonlinear Anal. 2007, 67: 1211–1225. 10.1016/j.na.2006.07.008MathSciNetView ArticleGoogle Scholar
  17. Lau ATM, Takahashi W: Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure. J. Funct. Anal. 1996, 25: 79–88.MathSciNetView ArticleGoogle Scholar
  18. Lau ATM, Takahashi W: Invariant means and fixed point properties for non-expansive representations of topological semigroups. Topol. Methods Nonlinear Anal. 1995, 5: 39–57.MathSciNetGoogle Scholar
  19. Lau ATM, Zhang Y: Fixed point properties of semigroup of nonexpansive mappings. J. Funct. Anal. 2008, 254: 2534–2554. 10.1016/j.jfa.2008.02.006MathSciNetView ArticleGoogle Scholar
  20. Saeidi S: Strong convergence of Browder’s type iterations for left amenable semigroups of Lipschitzian mappings in Banach spaces. Fixed Point Theory Appl. 2009, 5: 93–103. 10.1007/s11784-008-0092-3MathSciNetView ArticleGoogle Scholar
  21. Takahashi W: Fixed point theorem for amenable semigroups of nonexpansive mappings. Kodai Math. Semin. Rep. 1969, 21: 383–386. 10.2996/kmj/1138845984View ArticleGoogle Scholar
  22. Takahashi W: Fixed point theorems and nonlinear ergodic theorems for nonlinear semigroups and their applications. Nonlinear Anal. 1997, 30: 1283–1293. 10.1016/S0362-546X(96)00262-3MathSciNetView ArticleGoogle Scholar
  23. Zhu LP, Li G: Fixed points theorems for reversible semigroups of asymptotically nonexpansive type mappings in general Banach spaces. Chin. J. Contemp. Math. 2009, 30: 329–336.Google Scholar
  24. Li G: Weak convergence and nonlinear ergodic theorem for reversible semigroups of non-Lipschitzian mappings. J. Math. Anal. Appl. 1997, 206: 451–464. 10.1006/jmaa.1997.5237MathSciNetView ArticleGoogle Scholar
  25. Kirk W, Torrejon R: Asymptotically nonexpansive semigroup in Banach space. Nonlinear Anal. 1979, 3: 111–121. 10.1016/0362-546X(79)90041-5MathSciNetView ArticleGoogle Scholar
  26. Li G: Nonlinear ergodic theorem for semitopological semigroups of non-Lipschitzian mappings in Banach space. Chin. Sci. Bull. 1997, 42: 8–11. 10.1007/BF02882509View ArticleGoogle Scholar
  27. Miyadera I: Nonlinear ergodic theorems for semigroups of non-Lipschitzian mappings in Banach spaces. Nonlinear Anal. 2002, 50: 27–39. 10.1016/S0362-546X(01)00723-4MathSciNetView ArticleGoogle Scholar
  28. Day M: Amenable semigroups. Ill. J. Math. 1957, 1: 509–544.Google Scholar
  29. Yosida K: Functional Analysis. 6th edition. Springer, Berlin; 1998.Google Scholar
  30. Suzuki T: Strong convergence theorem to common fixed points of two nonexpansive mappings in general Banach spaces. J. Nonlinear Convex Anal. 2002, 3: 381–391.MathSciNetGoogle Scholar
  31. Royden HL: Real Analysis. 3rd edition. Pearson Education, Upper Saddle River; 2004.Google Scholar

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