 Research
 Open access
 Published:
Fixed point theorems for a semigroup of generalized asymptotically nonexpansive mappings in CAT(0) spaces
Fixed Point Theory and Applications volume 2012, Article number: 230 (2012)
Abstract
In this paper, we prove the existence of common fixed points for a generalized asymptotically nonexpansive semigroup \{{T}_{s}:s\in S\} 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 s\in S, the mappings s\mapsto ts and s\mapsto st from S to S are continuous, and let BC(S) be the Banach space of all bounded continuous realvalued functions with supremum norm. For f\in BC(S) and c\in \mathbb{R}, we write f(s)\to c as s\to {\mathrm{\infty}}_{\mathbb{R}} if for each \epsilon >0, there exists w\in S such that f(tw)c<\epsilon for all t\in S; 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,\u2ab0) is a directed system when the binary relation ‘⪰’ on S is defined by t\u2ab0s if and only if \{t\}\cup \overline{tS}\subseteq \{s\}\cup \overline{sS}, for t,s\in S. 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 [8–12] 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 almostorbit 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 [19–26].
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 \mathfrak{T}=\{{T}_{s}:s\in S\} of mappings of C into itself is said to be a semigroup if it satisfies the following:
(S1) {T}_{st}x={T}_{s}{T}_{t}x for all s,t\in S and x\in C;
(S2) for every x\in C, the mapping s\mapsto {T}_{s}x from S into C is continuous.
We denote by F(\mathfrak{T}) the set of common fixed points of \mathfrak{T}, i.e.,
Remark 2.1 If \mathfrak{T}=\{{T}_{s}:s\in S\} is a semigroup of continuous mappings of C into itself and d({T}_{s}x,y)\to 0 as s\to {\mathrm{\infty}}_{\mathbb{R}} for x,y\in C, then y\in F(\mathfrak{T}).
Proof Let \epsilon >0 be given. Fix t\in S. By the continuity of {T}_{t} at y, there exists \delta >0 such that d(x,y)<\delta implies d({T}_{t}x,{T}_{t}y)<\frac{\epsilon}{2} for x\in C. Since d({T}_{s}x,y)\to 0 as s\to {\mathrm{\infty}}_{\mathbb{R}}, there exists w\in S such that d({T}_{aw}x,y)<min\{\frac{\epsilon}{2},\delta \} for each a\in S. Then d({T}_{t}{T}_{aw}x,{T}_{t}y)<\frac{\epsilon}{2}. Therefore, we have
Since ε is arbitrary, we get {T}_{t}y=y for each t\in S, so y\in F(\mathfrak{T}). □
Let S be a left (or right) reversible semitopological semigroup. A semigroup \mathfrak{T}=\{{T}_{s}:s\in S\} of mappings of C into itself is said to be

(i)
nonexpansive if d({T}_{s}x,{T}_{s}y)\le d(x,y) for all x,y\in C and s\in S;

(ii)
asymptotically nonexpansive if there exists a nonnegative real number {k}_{s}\ge 0 with {lim}_{s}{k}_{s}=0 such that d({T}_{s}x,{T}_{s}y)\le (1+{k}_{s})d(x,y) for each x,y\in C and s\in S.

(iii)
generalized asymptotically nonexpansive if each {T}_{s} is continuous and there exist nonnegative real numbers {k}_{s},{\mu}_{s}\ge 0 with {lim}_{s}{k}_{s}=0 and {lim}_{s}{\mu}_{s}=0 such that
d({T}_{s}x,{T}_{s}y)\le (1+{k}_{s})d(x,y)+{\mu}_{s}\phantom{\rule{1em}{0ex}}\text{for each}x,y\in C\text{and}s\in S.
Remark 2.2 If {\mu}_{s}=0 for all s\in S, a generalized asymptotically nonexpansive semigroup reduces to an asymptotically nonexpansive semigroup. If {k}_{s}=0 and {\mu}_{s}=0 for all s\in S, 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 x\in X to y\in X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]\subset \mathbb{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}\in [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,y\in X. 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 \mathrm{\u25b3}({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 \mathrm{\u25b3}({x}_{1},{x}_{2},{x}_{3}) in (X,d) is a triangle \overline{\mathrm{\u25b3}}({x}_{1},{x}_{2},{x}_{3}):=\mathrm{\u25b3}({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3}) in the Euclidean plane {\mathbb{E}}^{2} such that {d}_{{\mathbb{E}}^{2}}({\overline{x}}_{i},{\overline{x}}_{j})=d({x}_{i},{x}_{j}) for i,j\in \{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 \overline{\mathrm{\u25b3}} be a comparison triangle for △. Then △ is said to satisfy the CAT(0) inequality if for all x,y\in \mathrm{\u25b3} and all comparison points \overline{x},\overline{y}\in \overline{\mathrm{\u25b3}}, d(x,y)\le {d}_{{\mathbb{E}}^{2}}(\overline{x},\overline{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
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,y\in X and \lambda \in [0,1], there exists a unique point \lambda x\oplus (1\lambda )y\in [x,y] such that d(x,\lambda x\oplus (1\lambda )y)=(1\lambda )d(x,y), d(y,\lambda x\oplus (1\lambda )y)=\lambda d(x,y) and the following inequality holds:
For any nonempty subset C of a CAT(0) space X, let \pi :={\pi}_{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:
Let \{{x}_{\alpha}\} be a bounded net in a nonempty closed convex subset C of a CAT(0) space X. For x\in X, we set
The asymptotic radius of \{{x}_{\alpha}\} on C is given by
and the asymptotic center of \{{x}_{\alpha}\} on C is given by
It is known that a CAT(0) space X, A(C,\{{x}_{\alpha}\}) consists of exactly one point; see [29].
In 1976, Lim [30] introduced the concept of Δconvergence 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}_{\alpha}\} in a CAT(0) space X is said to Δconverge to x\in X if x is the unique asymptotic center of \{{u}_{\alpha}\} for every subnet \{{u}_{\alpha}\} of \{{x}_{\alpha}\}. In this case, we write \mathrm{\Delta}\text{}{lim}_{\alpha}{x}_{\alpha}=x and call x the Δlimit of \{{x}_{\alpha}\}.
Lemma 2.4 ([31])
Every bounded net in a complete CAT(0) space X has a Δconvergent 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 \mathfrak{T}=\{{T}_{s}:s\in S\} be a generalized asymptotically nonexpansive semigroup of C into itself. If \{{T}_{s}x:s\in S\} is bounded for some x\in C and z\in A(C,\{{T}_{s}x\}), then z\in F(\mathfrak{T}).
Proof Let \{{T}_{s}x:s\in S\} be a bounded net and let z\in A(C,\{{T}_{s}x\}). Then
If R=0, then {lim\hspace{0.17em}sup}_{s}d(z,{T}_{s}x)=0. This implies {T}_{s}x\to z. It is obvious by Remark 2.1 that z\in F(\mathfrak{T}). Next, we assume R>0. Suppose that z\notin F(\mathfrak{T}). By Remark 2.1, \{{T}_{s}z\} does not converge to z. Then there exists \epsilon >0 and a subnet \{{s}_{\alpha}\} in S such that
We choose a positive number η such that
Since \mathfrak{T} is a generalized asymptotically nonexpansive semigroup, there exists {s}_{0}\in S such that
for each s\in S with s\u2ab0{s}_{0}, and y\in C.
It is known by [1] that {inf}_{t}{sup}_{s}d(z,{T}_{ts}x)={lim\hspace{0.17em}sup}_{u}d(z,{T}_{u}x). Then {inf}_{t}{sup}_{s}d(z,{T}_{ts}x)=R. So, there exists {t}_{0}\in S such that for all t\in S with t\u2ab0{t}_{0},
Since S is left reversible, there exists \gamma \in S with \gamma \u2ab0{s}_{0} and \gamma \u2ab0{t}_{0}. Then, by (3.1), {s}_{\gamma}\u2ab0\gamma and
Let t\u2ab0{s}_{\gamma}\gamma. Since S is left reversible, we have t\in \{{s}_{\gamma}\gamma \}\cup \overline{{s}_{\gamma}\gamma S}. Then we may assume t\in \overline{{s}_{\gamma}\gamma S}. So, there exists \{{t}_{\beta}\} in S such that {s}_{\gamma}\gamma {t}_{\beta}\to t. It follows by (3.2) and (3.3) that
By (3.3) and {s}_{\gamma}\gamma {t}_{\beta}\to t, we have
It follows by (3.3) that
By {s}_{\gamma}\gamma {t}_{\beta}\to t, we have
So, by the (CN) inequality, (3.4), (3.5), and (3.6), we have
Thus, d(\frac{z\oplus {T}_{{s}_{\gamma}}z}{2},{T}_{t}x)<R\eta. This implies that
which is a contradiction. Hence, z\in F(\mathfrak{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 \mathfrak{T}=\{{T}_{s}:s\in S\} be a generalized asymptotically nonexpansive semigroup of C into itself. Then F(\mathfrak{T})\ne \mathrm{\varnothing} if and only if \{{T}_{s}x:s\in S\} is bounded for some x\in C.
Proof Necessity is obvious. Conversely, assume that x\in C such that \{{T}_{s}x:s\in S\} is bounded. Then there exists a unique element z\in C such that z\in A(C,\{{T}_{s}x\}). It follows by Theorem 3.1 that F(\mathfrak{T})\ne \mathrm{\varnothing}. □
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 \mathfrak{T}=\{{T}_{s}:s\in S\} be a generalized asymptotically nonexpansive semigroup of C into itself with F(\mathfrak{T})\ne \mathrm{\varnothing}. Then F(\mathfrak{T}) is a closed convex subset of C.
Proof First, we show that F(\mathfrak{T}) is closed. Let \{{x}_{t}\} be a net in F(\mathfrak{T}) such that {x}_{t}\to x. By the definition of {T}_{t}, we have
Thus, {T}_{t}x\to x. This implies x\in F(\mathfrak{T}), and so F(\mathfrak{T}) is closed.
Next, we show F(\mathfrak{T}) is convex. Let x,y\in F(\mathfrak{T}) and z=\frac{x\oplus y}{2}. For t\in S, we have
and
Thus, by the (CN) inequality, we have
Therefore, {T}_{t}z\to z. This implies z\in F(\mathfrak{T}). Hence, F(\mathfrak{T}) is convex. □
Taking S=\mathbb{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:C\to C be a continuous generalized asymptotically nonexpansive mapping. Then F(T)\ne \mathrm{\varnothing} if and only if \{{T}^{n}x:n\in \mathbb{N}\} is bounded for some x\in C. Moreover, F(T) is closed and convex.
4 Δ and strong convergence theorems
In this section, we study the Δconvergence 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 \mathfrak{T}=\{{T}_{s}:s\in S\} be a generalized asymptotically nonexpansive semigroup of C into itself with F(\mathfrak{T})\ne \mathrm{\varnothing}. Then {lim}_{s}d({T}_{s}x,z) exists for each z\in F(\mathfrak{T}).
Proof Let z\in F(\mathfrak{T}) and R={inf}_{s}d({T}_{s}x,z). For \epsilon >0, there is {s}_{0}\in S such that
Since \mathfrak{T} is a generalized asymptotically nonexpansive semigroup, there exists {t}_{0}\in S such that
for each t\u2ab0{t}_{0}. Let b\u2ab0{t}_{0}{s}_{0}. Since S is right reversible, we have b\in \{{t}_{0}{s}_{0}\}\cup \overline{S{t}_{0}{s}_{0}}. Then we may assume b\in \overline{S{t}_{0}{s}_{0}}. So, there exists \{{s}_{\alpha}\} in S such that {s}_{\alpha}{t}_{0}{s}_{0}\to b. Therefore,
Hence, d({T}_{b}x,z)\le d({T}_{{s}_{0}}x,z)+\frac{\epsilon}{2}. This implies that
Since ε is arbitrary, we get
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 x\in C. Assume that \mathfrak{T}=\{{T}_{s}:s\in S\} is a generalized asymptotically nonexpansive semigroup of C into itself with F(\mathfrak{T})\ne \mathrm{\varnothing}. If {lim}_{s}d({T}_{s}x,{T}_{ts}x)=0 for all t\in S, then \{{T}_{s}x:s\in S\} Δconverges to a common fixed point of the semigroup \mathfrak{T}.
Proof By Lemma 4.1, we have {lim}_{s}d({T}_{s}x,z) exists for each z\in F(\mathfrak{T}), and so \{{T}_{s}x:s\in S\} is bounded. We now let {\omega}_{\mathrm{\Delta}}({T}_{s}x):=\bigcup A(C,\{{T}_{{s}_{\alpha}}x\}), where the union is taken over all subnets \{{T}_{{s}_{\alpha}}x\} of \{{T}_{s}x\}. We claim that {\omega}_{\mathrm{\Delta}}({T}_{s}x)\subset F(\mathfrak{T}). Let u\in {\omega}_{\mathrm{\Delta}}({T}_{s}x). Then there exists a subnet \{{T}_{{s}_{\alpha}}x\} of \{{T}_{s}x\} such that A(C,\{{T}_{{s}_{\alpha}}x\})=\{u\}. By Lemma 2.4, there exists a subnet \{{T}_{{s}_{{\alpha}_{\beta}}}x\} of \{{T}_{{s}_{\alpha}}x\} such that \mathrm{\Delta}\text{}{lim}_{\beta}{T}_{{s}_{{\alpha}_{\beta}}}x=y\in C. We will show that y\in F(\mathfrak{T}). Let \epsilon >0. Since \mathfrak{T} is a generalized asymptotically nonexpansive semigroup, there exists {t}_{0}\in S such that
for each t\u2ab0{t}_{0} and each β. It follows that
for each t\u2ab0{t}_{0} and each β. By {lim}_{s}d({T}_{s}x,{T}_{ts}x)=0 for all t\in S, we have
for all t\u2ab0{t}_{0}. Since ε is arbitrary, we get
for all t\u2ab0{t}_{0}. Since \{{T}_{{s}_{{\alpha}_{\beta}}}x\} Δconverges to y, it follows by the uniqueness of asymptotic centers that {T}_{t}y=y for all t\u2ab0{t}_{0}. So, d({T}_{t}y,y)\to 0. This implies y\in F(\mathfrak{T}). By Lemma 4.1, {lim}_{s}d({T}_{s}x,y) exists. Suppose that u\ne y. By the uniqueness of asymptotic centers,
This is a contradiction, hence u=y\in F(\mathfrak{T}). This shows that {\omega}_{\mathrm{\Delta}}({T}_{s}x)\subset F(\mathfrak{T}).
Next, we show that {\omega}_{\mathrm{\Delta}}({T}_{s}x) consists of exactly one point. Let \{{T}_{{s}_{\alpha}}x\} be a subnet of \{{T}_{s}x\} with A(C,\{{T}_{{s}_{\alpha}}x\})=\{u\} and let A(C,\{{T}_{s}\})=\{z\}. Since u\in {\omega}_{\mathrm{\Delta}}({T}_{s}x)\subset F(\mathfrak{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,
which is a contradiction, and so z=u. Hence, \{{T}_{s}x\} Δconverges to a common fixed point of the semigroup \mathfrak{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 x\in C. Assume that \mathfrak{T}=\{{T}_{s}:s\in S\} is a generalized asymptotically nonexpansive semigroup of C into itself with F(\mathfrak{T})\ne \mathrm{\varnothing}. Then \{\pi {T}_{s}x\} converges strongly to a point of F(\mathfrak{T}), where \pi :C\to F(\mathfrak{T}) is the nearest point projection.
Moreover, if S is reversible, then Px:={lim}_{s}\pi {T}_{s}x is the unique asymptotic center of the net \{{T}_{s}x:s\in S\}.
Proof By Lemma 3.3, F(\mathfrak{T}) is closed and convex. So, the mapping π is well defined. Put R={inf}_{s}d({T}_{s}x,\pi {T}_{s}x). As in the proof of Lemma 4.1, we have
We will show that \{\pi {T}_{s}x\} is a Cauchy net. To show this, we divide into two cases.
Case 1: R=0. For \epsilon >0, there exists {s}_{0}\in S such that
Since \mathfrak{T} is a generalized asymptotically nonexpansive semigroup, there exists {t}_{0}\in S such that
for each t\u2ab0{t}_{0}. Let a,b\u2ab0{t}_{0}{s}_{0}. Since S is right reversible, a,b\in \{{t}_{0}{s}_{0}\}\cup \overline{S{t}_{0}{s}_{0}}. Then we may assume a,b\in \overline{S{t}_{0}{s}_{0}}. So, there exist \{{t}_{\alpha}\} and \{{s}_{\beta}\} in S such that {t}_{\alpha}{t}_{0}{s}_{0}\to a and {s}_{\beta}{t}_{0}{s}_{0}\to b. Therefore, we have
This implies
Hence, \{\pi {T}_{s}x\} is a Cauchy net.
Case 2: R>0. Suppose that \{\pi {T}_{s}x\} is not a Cauchy net. Then, there exists \epsilon >0 such that for any s\in S, there are {a}_{s},{b}_{s}\in S with {a}_{s},{b}_{s}\u2ab0s and d(\pi {T}_{{a}_{s}}x,\pi {T}_{{b}_{s}}x)\ge \epsilon.
We choose a positive number η such that
So, there exists {u}_{0}\in S such that
Then d(\pi {T}_{{a}_{{u}_{0}}}x,\pi {T}_{{b}_{{u}_{0}}}x)\ge \epsilon. Since \mathfrak{T} is a generalized asymptotically nonexpansive semigroup, there exists {v}_{0}\in S such that
for each t\u2ab0{v}_{0} and each s\in S.
Since S is right reversible, there exists c\in S such that c\u2ab0{v}_{0}{a}_{{u}_{0}} and c\u2ab0{v}_{0}{b}_{{u}_{0}}. Then, there exist \{{t}_{\alpha}\} and \{{s}_{\beta}\} in S such that {t}_{\alpha}{v}_{0}{a}_{{u}_{0}}\to c and {s}_{\beta}{v}_{0}{b}_{{u}_{0}}\to c. So, by (4.1) and (4.2), we have
and
This implies
By the (CN) inequality, we get
and so d({T}_{c}x,\frac{\pi {T}_{{a}_{{u}_{0}}}x\oplus \pi {T}_{{b}_{{u}_{0}}}x}{2})<R. Since π is the nearest point projection of C onto F(\mathfrak{T}), we have
This contradicts with R={inf}_{s}d({T}_{s}x,\pi {T}_{s}x).
So, \{\pi {T}_{s}x\} is Cauchy in a closed subset F(\mathfrak{T}) of a complete CAT(0) space X, hence it converges to some point in F(\mathfrak{T}), say Px.
Finally, by Lemma 4.1, we have \{{T}_{s}x:s\in S\} is bounded. So, let z\in A(C,\{{T}_{s}x\}). Since S is reversible, it implies by Theorem 3.1 that z\in F(\mathfrak{T}). Thus, by the property of π, we obtain
This implies, by the uniqueness of asymptotic centers, that Px=z. □
Taking S=\mathbb{N} in Theorem 4.2, we obtain the following Δconvergence 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 x\in C. Assume that T:C\to C is a continuous generalized asymptotically nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. If {lim}_{n\to \mathrm{\infty}}d({T}^{n}x,{T}^{n+1}x)=0, then \{{T}^{n}x:n\in \mathbb{N}\} Δconverges to a fixed point of T.
Taking S=\mathbb{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 x\in C. Assume that T:C\to C is a continuous generalized asymptotically nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. Then \{\pi {T}^{n}x\} converges strongly to a point of F(T), where \pi :C\to F(T) is the nearest point projection. Moreover, Px:={lim}_{n\to \mathrm{\infty}}\pi {T}^{n}x is the unique asymptotic center of the sequence \{{T}^{n}x:n\in \mathbb{N}\}.
Remark 4.6

(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.

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

(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.
References
Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.
Holmes RD, Lau AT: Nonexpansive actions of topological semigroups and fixed points. J. Lond. Math. Soc. 1972, 5: 330–336. 10.1112/jlms/s25.2.330
Takahashi W: Fixed point theorem for amenable semigroups of nonexpansive mappings. Kodai Math. Semin. Rep. 1969, 21: 383–386. 10.2996/kmj/1138845984
De Marr R: Common fixed points for commuting contraction mappings. Pac. J. Math. 1963, 13: 1139–1141. 10.2140/pjm.1963.13.1139
Mitchell T: Fixed points of reversible semigroups of nonexpansive mappings. Kodai Math. Semin. Rep. 1970, 22: 322–323. 10.2996/kmj/1138846168
Takahashi W: A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space. Proc. Am. Math. Soc. 1981, 81: 253–256. 10.1090/S0002993919810593468X
Lau AT, Takahashi W: Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings. Pac. J. Math. 1987, 126: 177–194.
Lau AT, Takahashi W: Invariant means and semigroups of nonexpansive mappings on uniformly convex Banach spaces. J. Math. Anal. Appl. 1990, 153: 497–505. 10.1016/0022247X(90)902288
Lau AT, Takahashi W: Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure. J. Funct. Anal. 1996, 142: 79–88. 10.1006/jfan.1996.0144
Lau AT, Takahashi W: Nonlinear submeans on semigroups. Topol. Methods Nonlinear Anal. 2003, 22: 345–353.
Lau AT, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. 2007, 67: 1211–1225. 10.1016/j.na.2006.07.008
Lau AT, Zhang Y: Fixed point properties of semigroups of nonexpansive mappings. J. Funct. Anal. 2008, 254: 2534–2554. 10.1016/j.jfa.2008.02.006
Takahashi W, Zhang PJ: Asymptotic behavior of almostorbits of semigroups of Lipschitzian mappings in Banach spaces. Kodai Math. J. 1988, 11: 129–140. 10.2996/kmj/1138038824
Takahashi W, Zhang PJ: Asymptotic behavior of almostorbits of semigroups of Lipschitzian mappings. J. Math. Anal. Appl. 1989, 142: 242–249. 10.1016/0022247X(89)901777
Kim HS, Kim TH: Asymptotic behavior of semigroups of asymptotically nonexpansive type on Banach spaces. J. Korean Math. Soc. 1987, 24: 169–178.
Kakavandi, BA, Amini, M: Nonlinear ergodic theorem in complete nonpositive curvature metric spaces. Bull. Iran. Math. Soc. (in press)
Anakkanmatee W, Dhompongsa S: Rodé’s theorem on common fixed points of semigroup of nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 34
Rodé G: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space. J. Math. Anal. Appl. 1982, 85: 172–178. 10.1016/0022247X(82)900324
Ishihara H, Takahashi W: A nonlinear ergodic theorem for a reversible semigroup of Lipschitzian mappings in a Hilbert space. Proc. Am. Math. Soc. 1988, 104: 431–436. 10.1090/S0002993919880962809X
Kim HS, Kim TH: Weak convergence of semigroups of asymptotically nonexpansive type on a Banach space. Commun. Korean Math. Soc. 1987, 2: 63–69.
Lim TC: Characterization of normal structure. Proc. Am. Math. Soc. 1974, 43: 313–319. 10.1090/S0002993919740361728X
Lau AT: Semigroup of nonexpansive mappings on a Hilbert space. J. Math. Anal. Appl. 1985, 105: 514–522. 10.1016/0022247X(85)900666
Lau AT, Shioji N, Takahashi W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. J. Funct. Anal. 1999, 191: 62–75.
Lau AT: Invariant means and fixed point properties of semigroup of nonexpansive mappings. Taiwan. J. Math. 2008, 12: 1525–1542.
Lau AT, Takahashi W: Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces. Nonlinear Anal. 2009, 70: 3837–3841. 10.1016/j.na.2008.07.041
Takahashi W: A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space. Proc. Am. Math. Soc. 1986, 97: 55–58. 10.1090/S00029939198608313864
Bridson M, Haefliger A: Metric Spaces of Nonpositive Curvature. Springer, Berlin; 1999.
Bruhat F, Tits J: Groupes réductifs sur un corps local. Publ. Math. Inst. Hautes Études Sci. 1972, 41: 5–251. 10.1007/BF02715544
Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 2006, 65: 762–772. 10.1016/j.na.2005.09.044
Lim TC: Remark on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002993919760423139X
Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contribute equally and significantly in this research work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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/10.1186/168718122012230
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122012230