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Fixed point theorems for a semigroup of generalized asymptotically nonexpansive mappings in CAT(0) spaces

Abstract

In this paper, we prove the existence of common fixed points for a generalized asymptotically nonexpansive semigroup { T s :sS} in CAT(0) spaces, when S is a left reversible semitopological semigroup. We also prove Δ- and strong convergence of such a semigroup when S is a right reversible semitopological semigroup. Our results improve and extend the corresponding results existing in the literature.

MSC:47H09, 47H10.

1 Introduction

Let S be a semitopological semigroup, i.e., S is a semigroup with a Hausdorff topology such that for each sS, the mappings sts and sst from S to S are continuous, and let BC(S) be the Banach space of all bounded continuous real-valued functions with supremum norm. For fBC(S) and cR, we write f(s)c as s R if for each ε>0, there exists wS such that |f(tw)c|<ε for all tS; see [1].

A semitopological semigroup S is said to be left (resp. right) reversible if any two closed right (resp. left) ideals of S have nonvoid intersection. If S is left reversible, (S,) is a directed system when the binary relation ‘’ on S is defined by ts if and only if {t} t S ¯ {s} s S ¯ , for t,sS. Similarly, we can define the binary relation ‘’ on a right reversible semitopological semigroup S. Left reversible semitopological semigroups include all commutative semigroups and all semitopological semigroups which are left amenable as discrete semigroups; see [2]. S is called reversible if it is both left and right reversible.

In 1969, Takahashi [3] proved the first fixed point theorem for a noncommutative semigroup of nonexpansive mappings which generalizes De Marr’s fixed point theorem [4]. He proved that any discrete left amenable semigroup has a common fixed point. In 1970, Mitchell [5] generalized Takahashi’s result by showing that any discrete left reversible semigroup has a common fixed point. In 1981, Takahashi [6] proved a nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space. In 1987, Lau and Takahashi [7] considered the problem of weak convergence of a nonexpansive semigroup of a right reversible semitopological semigroup in a uniformly convex Banach space with Fréchet differentiable norm. After that Lau [812] proved the existence of common fixed points for nonexpansive maps related to reversibility or amenability of a semigroup. Takahashi and Zhang [13, 14] established the weak convergence of an almost-orbit of Lipschitzian semigroups of a noncommutative semitopological semigroup. Kim and Kim [15] proved weak convergence for semigroups of asymptotically nonexpansive type of a right reversible semitopological semigroup and strong convergence for a commutative case. In [16], Kakavandi and Amini proved a nonlinear ergodic theorem for a nonexpansive semigroup in CAT(0) spaces as well as a strong convergence theorem for a commutative semitopological semigroup. In 2011, Anakkanmatee and Dhompongsa [17] extended Rodé’s theorem [18] on common fixed points of semigroups of nonexpansive mappings in Hilbert spaces to the CAT(0) space setting. For works related to semigroups of nonexpansive, asymptotically nonexpansive, and asymptotically nonexpansive type related to reversibility of a semigroup, we refer the reader to [1926].

In this paper, we introduce a new semigroup for a left (or right) reversible semitopological semigroup on metric spaces, called a generalized asymptotically nonexpansive semigroup, and prove the existence and convergence theorems for this semigroup in CAT(0) spaces.

2 Preliminaries

Let S be a semitopological semigroup and C be a nonempty closed subset of a metric space (X,d). A family T={ T s :sS} of mappings of C into itself is said to be a semigroup if it satisfies the following:

(S1) T s t x= T s T t x for all s,tS and xC;

(S2) for every xC, the mapping s T s x from S into C is continuous.

We denote by F(T) the set of common fixed points of T, i.e.,

F(T)= s S F( T s )= s S {xC: T s x=x}.

Remark 2.1 If T={ T s :sS} is a semigroup of continuous mappings of C into itself and d( T s x,y)0 as s R for x,yC, then yF(T).

Proof Let ε>0 be given. Fix tS. By the continuity of T t at y, there exists δ>0 such that d(x,y)<δ implies d( T t x, T t y)< ε 2 for xC. Since d( T s x,y)0 as s R , there exists wS such that d( T a w x,y)<min{ ε 2 ,δ} for each aS. Then d( T t T a w x, T t y)< ε 2 . Therefore, we have

d ( T t y , y ) d ( T t y , T t a w x ) + d ( T t a w x , y ) < ε 2 + ε 2 = ε .

Since ε is arbitrary, we get T t y=y for each tS, so yF(T). □

Let S be a left (or right) reversible semitopological semigroup. A semigroup T={ T s :sS} of mappings of C into itself is said to be

  1. (i)

    nonexpansive if d( T s x, T s y)d(x,y) for all x,yC and sS;

  2. (ii)

    asymptotically nonexpansive if there exists a nonnegative real number k s 0 with lim s k s =0 such that d( T s x, T s y)(1+ k s )d(x,y) for each x,yC and sS.

  3. (iii)

    generalized asymptotically nonexpansive if each T s is continuous and there exist nonnegative real numbers k s , μ s 0 with lim s k s =0 and lim s μ s =0 such that

    d( T s x, T s y)(1+ k s )d(x,y)+ μ s for each x,yC and sS.

Remark 2.2 If μ s =0 for all sS, a generalized asymptotically nonexpansive semigroup reduces to an asymptotically nonexpansive semigroup. If k s =0 and μ s =0 for all sS, a generalized asymptotically nonexpansive semigroup reduces to a nonexpansive semigroup.

We recall a CAT(0) space; see more details in [27]. Let (X,d) be a metric space. A geodesic path joining xX to yX (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]R to X such that c(0)=x, c(l)=y and d(c( t 1 ),c( t 2 ))=| t 1 t 2 | for all t 1 , t 2 [0,l]. In particular, c is an isometry and d(x,y)=l. The image α of c is called a geodesic (or metric) segment joining x and y. When unique, this geodesic is denoted [x,y]. The space (X,d) is said to be a geodesic metric space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,yX. A subset C of X is said to be convex if C includes every geodesic segment joining any two of its points.

A geodesic triangle ( x 1 , x 2 , x 3 ) in a geodesic metric space (X,d) consists of three points x 1 , x 2 , x 3 in X (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A comparison triangle for the geodesic triangle ( x 1 , x 2 , x 3 ) in (X,d) is a triangle ¯ ( x 1 , x 2 , x 3 ):=( x ¯ 1 , x ¯ 2 , x ¯ 3 ) in the Euclidean plane E 2 such that d E 2 ( x ¯ i , x ¯ j )=d( x i , x j ) for i,j{1,2,3}.

A geodesic metric space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom: Let be a geodesic triangle in X and let ¯ be a comparison triangle for . Then is said to satisfy the CAT(0) inequality if for all x,y and all comparison points x ¯ , y ¯ ¯ , d(x,y) d E 2 ( x ¯ , y ¯ ).

If z, x, y are points in a CAT(0) space and if m is the midpoint of the segment [x,y], then the CAT(0) inequality implies

d ( z , m ) 2 1 2 d ( z , x ) 2 + 1 2 d ( z , y ) 2 1 4 d ( x , y ) 2 .
(CN)

This is the (CN) inequality of Bruhat and Tits [28]. By using the (CN) inequality, it is easy to see the CAT(0) spaces are uniformly convex. In fact [27], a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality. Moreover, for each x,yX and λ[0,1], there exists a unique point λx(1λ)y[x,y] such that d(x,λx(1λ)y)=(1λ)d(x,y), d(y,λx(1λ)y)=λd(x,y) and the following inequality holds:

d ( z , λ x ( 1 λ ) y ) λd(z,x)+(1λ)d(z,y)for each zX.

For any nonempty subset C of a CAT(0) space X, let π:= π D be the nearest point projection mapping from C to a subset D of C. In [27], it is known that if D is closed and convex, the mapping π is well defined, nonexpansive, and the following inequality holds:

d ( x , y ) 2 d ( x , π x ) 2 +d ( π x , y ) 2 for all xC and yD.

Let { x α } be a bounded net in a nonempty closed convex subset C of a CAT(0) space X. For xX, we set

r ( x , { x α } ) = lim sup α d(x, x α ).

The asymptotic radius of { x α } on C is given by

r ( C , { x α } ) = inf x C r ( x , { x α } ) ,

and the asymptotic center of { x α } on C is given by

A ( C , { x α } ) = { x C : r ( x , { x α } ) = r ( C , { x α } ) } .

It is known that a CAT(0) space X, A(C,{ x α }) consists of exactly one point; see [29].

In 1976, Lim [30] introduced the concept of Δ-convergence in a general metric space. Later, Kirk and Panyanak [31] extended the concept of Lim to a CAT(0) space.

Definition 2.3 ([31])

A net { x α } in a CAT(0) space X is said to Δ-converge to xX if x is the unique asymptotic center of { u α } for every subnet { u α } of { x α }. In this case, we write Δ- lim α x α =x and call x the Δ-limit of { x α }.

Lemma 2.4 ([31])

Every bounded net in a complete CAT(0) space X has a Δ-convergent subnet.

3 Existence theorems

In this section, we study the existence theorems for a generalized asymptotically nonexpansive semigroup in a complete CAT(0) space.

Theorem 3.1 Let S be a left reversible semitopological semigroup, C be a nonempty closed convex subset of a complete CAT(0) space X, and T={ T s :sS} be a generalized asymptotically nonexpansive semigroup of C into itself. If { T s x:sS} is bounded for some xC and zA(C,{ T s x}), then zF(T).

Proof Let { T s x:sS} be a bounded net and let zA(C,{ T s x}). Then

R:=r ( z , { T s x } ) =r ( C , { T s x } ) = inf y C r ( y , { T s x } ) .

If R=0, then lim sup s d(z, T s x)=0. This implies T s xz. It is obvious by Remark 2.1 that zF(T). Next, we assume R>0. Suppose that zF(T). By Remark 2.1, { T s z} does not converge to z. Then there exists ε>0 and a subnet { s α } in S such that

s α αandd(z, T s α z)>εfor each αS.
(3.1)

We choose a positive number η such that

( R + η ) 2 ε 2 4 < ( R η ) 2 .

Since T is a generalized asymptotically nonexpansive semigroup, there exists s 0 S such that

d ( T s z , T s y ) lim sup a d ( T a z , T a y ) + η 2 lim sup a ( ( 1 + k a ) d ( z , y ) + μ a ) + η 2 = d ( z , y ) + η 2
(3.2)

for each sS with s s 0 , and yC.

It is known by [1] that inf t sup s d(z, T t s x)= lim sup u d(z, T u x). Then inf t sup s d(z, T t s x)=R. So, there exists t 0 S such that for all tS with t t 0 ,

d(z, T t s x)<R+ η 2 for each sS.
(3.3)

Since S is left reversible, there exists γS with γ s 0 and γ t 0 . Then, by (3.1), s γ γ and

d(z, T s γ z)>ε.
(3.4)

Let t s γ γ. Since S is left reversible, we have t{ s γ γ} s γ γ S ¯ . Then we may assume t s γ γ S ¯ . So, there exists { t β } in S such that s γ γ t β t. It follows by (3.2) and (3.3) that

d( T s γ z, T s γ T γ t β x)d(z, T γ t β x)+ η 2 R+ηfor each β.

By (3.3) and s γ γ t β t, we have

d( T s γ z, T t x)R+ η 2 for all t s γ γ.
(3.5)

It follows by (3.3) that

d(z, T s γ T γ t β x)<R+ η 2 for each β.

By s γ γ t β t, we have

d(z, T t x)R+ η 2 <R+ηfor all t s γ γ.
(3.6)

So, by the (CN) inequality, (3.4), (3.5), and (3.6), we have

d 2 ( z T s γ z 2 , T t x ) 1 2 d 2 ( z , T t x ) + 1 2 d 2 ( T s γ z , T t x ) 1 4 d 2 ( z , T s γ z ) < ( R + η ) 2 1 4 ε 2 < ( R η ) 2 .

Thus, d( z T s γ z 2 , T t x)<Rη. This implies that

r ( z T s γ z 2 , { T t x } ) <r ( C , { T t x } ) ,

which is a contradiction. Hence, zF(T). □

Theorem 3.2 Let S be a left reversible semitopological semigroup, C be a nonempty closed convex subset of a complete CAT(0) space X, and T={ T s :sS} be a generalized asymptotically nonexpansive semigroup of C into itself. Then F(T) if and only if { T s x:sS} is bounded for some xC.

Proof Necessity is obvious. Conversely, assume that xC such that { T s x:sS} is bounded. Then there exists a unique element zC such that zA(C,{ T s x}). It follows by Theorem 3.1 that F(T). □

Theorem 3.3 Let S be a left or right reversible semitopological semigroup, C be a nonempty closed convex subset of a complete CAT(0) space X, and T={ T s :sS} be a generalized asymptotically nonexpansive semigroup of C into itself with F(T). Then F(T) is a closed convex subset of C.

Proof First, we show that F(T) is closed. Let { x t } be a net in F(T) such that x t x. By the definition of T t , we have

d ( T t x , x ) d ( T t x , x t ) + d ( x , x t ) ( 2 + k t ) d ( x , x t ) + μ t 0 .

Thus, T t xx. This implies xF(T), and so F(T) is closed.

Next, we show F(T) is convex. Let x,yF(T) and z= x y 2 . For tS, we have

d( T t z,x)(1+ k t )d(z,x)+ μ t = 1 + k t 2 d(x,y)+ μ t

and

d( T t z,y)(1+ k t )d(z,y)+ μ t = 1 + k t 2 d(x,y)+ μ t .

Thus, by the (CN) inequality, we have

d 2 ( T t z , z ) 1 2 d 2 ( T t z , x ) + 1 2 d 2 ( T t z , y ) 1 4 d 2 ( x , y ) ( 1 + k t 2 d ( x , y ) + μ t ) 2 1 4 d 2 ( x , y ) 0 .

Therefore, T t zz. This implies zF(T). Hence, F(T) is convex. □

Taking S=N in Theorems 3.2 and 3.3, we obtain the following existence theorem of a generalized asymptotically nonexpansive mapping in CAT(0) spaces.

Theorem 3.4 Let C be a nonempty closed convex subset of a complete CAT(0) space X and T:CC be a continuous generalized asymptotically nonexpansive mapping. Then F(T) if and only if { T n x:nN} is bounded for some xC. Moreover, F(T) is closed and convex.

4 Δ- and strong convergence theorems

In this section, we study the Δ-convergence and strong convergence theorems for a generalized asymptotically nonexpansive semigroup in a CAT(0) space.

Lemma 4.1 Let S be a right reversible semitopological semigroup, C be a nonempty closed convex subset of a complete CAT(0) space X, and T={ T s :sS} be a generalized asymptotically nonexpansive semigroup of C into itself with F(T). Then lim s d( T s x,z) exists for each zF(T).

Proof Let zF(T) and R= inf s d( T s x,z). For ε>0, there is s 0 S such that

d( T s 0 x,z)<R+ ε 2 .

Since T is a generalized asymptotically nonexpansive semigroup, there exists t 0 S such that

d ( T t T s 0 x , z ) lim sup u d ( T u T s 0 x , z ) + ε 2 d ( T s 0 x , z ) + ε 2

for each t t 0 . Let b t 0 s 0 . Since S is right reversible, we have b{ t 0 s 0 } S t 0 s 0 ¯ . Then we may assume b S t 0 s 0 ¯ . So, there exists { s α } in S such that s α t 0 s 0 b. Therefore,

d( T s α t 0 s 0 x,z)d( T s 0 x,z)+ ε 2 for each α.

Hence, d( T b x,z)d( T s 0 x,z)+ ε 2 . This implies that

R inf s sup t s d( T t x,z) sup b t 0 s 0 d( T b x,z)d( T s 0 x,z)+ ε 2 <R+ε.

Since ε is arbitrary, we get

inf s sup t s d( T t x,z)=R= inf s d( T s x,z).

Thus, lim s d( T s x,z) exists. □

Theorem 4.2 Let S be a right reversible semitopological semigroup, C be a nonempty closed convex subset of a complete CAT(0) space X, and xC. Assume that T={ T s :sS} is a generalized asymptotically nonexpansive semigroup of C into itself with F(T). If lim s d( T s x, T t s x)=0 for all tS, then { T s x:sS} Δ-converges to a common fixed point of the semigroup T.

Proof By Lemma 4.1, we have lim s d( T s x,z) exists for each zF(T), and so { T s x:sS} is bounded. We now let ω Δ ( T s x):=A(C,{ T s α x}), where the union is taken over all subnets { T s α x} of { T s x}. We claim that ω Δ ( T s x)F(T). Let u ω Δ ( T s x). Then there exists a subnet { T s α x} of { T s x} such that A(C,{ T s α x})={u}. By Lemma 2.4, there exists a subnet { T s α β x} of { T s α x} such that Δ- lim β T s α β x=yC. We will show that yF(T). Let ε>0. Since T is a generalized asymptotically nonexpansive semigroup, there exists t 0 S such that

d ( T t T s α β x , T t y ) lim sup a d ( T a T s α β x , T a y ) + ε d ( T s α β x , y ) + ε ,

for each t t 0 and each β. It follows that

d ( T s α β x , T t y ) d ( T s α β x , T t s α β x ) + d ( T t s α β x , T t y ) d ( T s α β x , T t s α β x ) + d ( T s α β x , y ) + ε

for each t t 0 and each β. By lim s d( T s x, T t s x)=0 for all tS, we have

lim sup β d( T s α β x, T t y) lim sup β d( T s α β x,y)+ε

for all t t 0 . Since ε is arbitrary, we get

lim sup β d( T s α β x, T t y) lim sup β d( T s α β x,y)

for all t t 0 . Since { T s α β x} Δ-converges to y, it follows by the uniqueness of asymptotic centers that T t y=y for all t t 0 . So, d( T t y,y)0. This implies yF(T). By Lemma 4.1, lim s d( T s x,y) exists. Suppose that uy. By the uniqueness of asymptotic centers,

lim sup β d ( T s α β x , y ) < lim sup β d ( T s α β x , u ) lim sup α d ( T s α x , u ) < lim sup α d ( T s α x , y ) = lim sup s d ( T s x , y ) = lim sup β d ( T s α β x , y ) .

This is a contradiction, hence u=yF(T). This shows that ω Δ ( T s x)F(T).

Next, we show that ω Δ ( T s x) consists of exactly one point. Let { T s α x} be a subnet of { T s x} with A(C,{ T s α x})={u} and let A(C,{ T s })={z}. Since u ω Δ ( T s x)F(T), it follows by Lemma 4.1 that lim s d( T s x,u) exists. We can complete the proof by showing that z=u. To show this, suppose not. By the uniqueness of asymptotic centers,

lim sup α d ( T s α x , u ) < lim sup α d ( T s α x , z ) lim sup s d ( T s x , z ) < lim sup s d ( T s x , u ) = lim sup s d ( T s x , y ) = lim sup α d ( T s α x , u ) ,

which is a contradiction, and so z=u. Hence, { T s x} Δ-converges to a common fixed point of the semigroup T. □

The following result is a strong convergence theorem for a right reversible semitopological semigroup.

Theorem 4.3 Let S be a right reversible semitopological semigroup, C be a nonempty closed convex subset of a complete CAT(0) space X, and xC. Assume that T={ T s :sS} is a generalized asymptotically nonexpansive semigroup of C into itself with F(T). Then {π T s x} converges strongly to a point of F(T), where π:CF(T) is the nearest point projection.

Moreover, if S is reversible, then Px:= lim s π T s x is the unique asymptotic center of the net { T s x:sS}.

Proof By Lemma 3.3, F(T) is closed and convex. So, the mapping π is well defined. Put R= inf s d( T s x,π T s x). As in the proof of Lemma 4.1, we have

R= inf s d( T s x,π T s x)= lim sup s d( T s x,π T s x).

We will show that {π T s x} is a Cauchy net. To show this, we divide into two cases.

Case 1: R=0. For ε>0, there exists s 0 S such that

d( T s x,π T s x)< ε 4 for each s s 0 .

Since T is a generalized asymptotically nonexpansive semigroup, there exists t 0 S such that

d ( T t s 0 x , π T s 0 x ) lim sup u d ( T u T s 0 x , T u π T s 0 x ) + ε 4 d ( T s 0 x , π T s 0 x ) + ε 4

for each t t 0 . Let a,b t 0 s 0 . Since S is right reversible, a,b{ t 0 s 0 } S t 0 s 0 ¯ . Then we may assume a,b S t 0 s 0 ¯ . So, there exist { t α } and { s β } in S such that t α t 0 s 0 a and s β t 0 s 0 b. Therefore, we have

d ( T t α t 0 s 0 x , T s β t 0 s 0 x ) d ( T t α t 0 s 0 x , π T s 0 x ) + d ( T s β t 0 s 0 x , π T s 0 x ) 2 d ( T s 0 x , π T s 0 x ) + ε 2 .

This implies

d ( π T a x , π T b x ) 2 d ( T s 0 x , π T s 0 x ) + ε 2 < 2 ( ε 4 ) + ε 2 = ε .

Hence, {π T s x} is a Cauchy net.

Case 2: R>0. Suppose that {π T s x} is not a Cauchy net. Then, there exists ε>0 such that for any sS, there are a s , b s S with a s , b s s and d(π T a s x,π T b s x)ε.

We choose a positive number η such that

( R + η ) 2 ε 2 4 < R 2 .

So, there exists u 0 S such that

d( T t x,π T t x)R+ η 2 for each t u 0 .
(4.1)

Then d(π T a u 0 x,π T b u 0 x)ε. Since T is a generalized asymptotically nonexpansive semigroup, there exists v 0 S such that

d ( T t T s x , π T s x ) lim sup u d ( T u T s x , π T s x ) + η 2 d ( T s x , π T s x ) + η 2
(4.2)

for each t v 0 and each sS.

Since S is right reversible, there exists cS such that c v 0 a u 0 and c v 0 b u 0 . Then, there exist { t α } and { s β } in S such that t α v 0 a u 0 c and s β v 0 b u 0 c. So, by (4.1) and (4.2), we have

d( T t α v 0 a u 0 x,π T a u 0 x)d( T a u 0 x,π T a u 0 x)+ η 2 R+η

and

d( T s β v 0 b u 0 x,π T b u 0 x)d( T b u 0 x,π T b u 0 x)+ η 2 R+η.

This implies

d( T c x,π T a u 0 x)R+ηandd( T c x,π T b u 0 x)R+η.

By the (CN) inequality, we get

and so d( T c x, π T a u 0 x π T b u 0 x 2 )<R. Since π is the nearest point projection of C onto F(T), we have

d( T c x,π T c x)d ( T c x , π T a u 0 x π T b u 0 x 2 ) <R.

This contradicts with R= inf s d( T s x,π T s x).

So, {π T s x} is Cauchy in a closed subset F(T) of a complete CAT(0) space X, hence it converges to some point in F(T), say Px.

Finally, by Lemma 4.1, we have { T s x:sS} is bounded. So, let zA(C,{ T s x}). Since S is reversible, it implies by Theorem 3.1 that zF(T). Thus, by the property of π, we obtain

lim sup s d ( T s x , P x ) lim sup s ( d ( T s x , π T s x ) + d ( π T s x , P x ) ) = lim sup s d ( T s x , π T s x ) lim sup s d ( T s x , z ) .

This implies, by the uniqueness of asymptotic centers, that Px=z. □

Taking S=N in Theorem 4.2, we obtain the following Δ-convergence theorem of a generalized asymptotically nonexpansive mapping in CAT(0) spaces.

Theorem 4.4 Let C be a nonempty closed convex subset of a complete CAT(0) space X and xC. Assume that T:CC is a continuous generalized asymptotically nonexpansive mapping with F(T). If lim n d( T n x, T n + 1 x)=0, then { T n x:nN} Δ-converges to a fixed point of T.

Taking S=N in Theorem 4.3, we obtain the following strong convergence theorem of a generalized asymptotically nonexpansive mapping in CAT(0) spaces.

Theorem 4.5 Let C be a nonempty closed convex subset of a complete CAT(0) space X and xC. Assume that T:CC is a continuous generalized asymptotically nonexpansive mapping with F(T). Then {π T n x} converges strongly to a point of F(T), where π:CF(T) is the nearest point projection. Moreover, Px:= lim n π T n x is the unique asymptotic center of the sequence { T n x:nN}.

Remark 4.6

  1. (i)

    It is well known that every commutative semigroup is both left and right reversible and every discrete amenable semigroup is reversible. Then Theorems 3.1, 3.2, 3.3, 4.2, and 4.3 are also obtained for a class of commutative and discrete amenable semigroups.

  2. (ii)

    Theorem 4.2 extends and generalizes the results of [15, 20] to generalized asymptotically nonexpansive semigroups and to CAT(0) spaces.

  3. (iii)

    Theorem 4.3 extends and generalizes the results of [16] from amenable semigroups to right reversible semigroups and from nonexpansive semigroups to generalized asymptotically nonexpansive semigroups.

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Acknowledgements

The first author is supported by the Office of the Higher Education Commission and the Graduate School of Chiang Mai University, Thailand. The second author would like to thank the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand for financial support.

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Correspondence to Suthep Suantai.

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Phuengrattana, W., Suantai, S. Fixed point theorems for a semigroup of generalized asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl 2012, 230 (2012). https://doi.org/10.1186/1687-1812-2012-230

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Keywords

  • fixed point
  • semitopological semigroup
  • reversible semigroup
  • amenable semigroup
  • generalized asymptotically nonexpansive semigroup
  • CAT(0) spaces