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Rodé’s theorem on common fixed points of
semigroup of nonexpansive mappings in CAT(0)
spaces
Watcharapong Anakkamatee
1
and Sompong Dhompongsa
1,2*
* Correspondence:

1
Department of Mathematics,
Faculty of Science, Chiang Mai
University, Chiang Mai, 50200,
Thailand
Full list of author information is
available at the end of the article
Abstract
We extend Rodé’s theorem on common fixed points of semigroups of nonexpansive
mappings in Hilbert spaces to the CAT(0) space setting.
2000 Mathematics Subject Classification: 47H09; 47H10.
Keywords: CAT(0) space, semigroup of nonexpansive mappings, Δ?Δ?-convergence
1 Introduction
In 1976, Lim [1] introduced a co ncept of convergence in a general metric space, called
strong Δ-convergence. In [2], Kirk and Panyanak introduced a concept of co nvergence
in a CAT(0) space, called Δ-convergence (see Section 2 for the definition). Moreover,
they showed that many Banach space concepts and results which involve weak conver-
gence can be extended to the CAT(0) space setting by using the Δ-convergence.
For each semigroup S,letB(S) be the Banach space of all bounded real-valued map-
pings on S with s upremum norm. A continuous linear functional μ Î B(S)* (the dual
space o f B(S)) is called a mean on B(S)if||μ || = μ(1). For any f Î B(S), we use the

following notation:
μ
(
f
)
= μ
s
(
f
(
s
)).
A mean μ on B(S) is said to be left invariant [respectively, right invariant]ifμ
s
(f(ts))
= μ
s
(f(s )) [respectively, μ
s
(f(st)) = μ
s
(f(s ))] for all f Î B(S)andforallt Î S. We will
say that μ is an invariant mean if it is both left and right invariants. I f B(S)hasan
invariant mean, we call S an amenable semigroup. It is well known that every commu-
tative semigroup is amenable [3]. For each s Î S and f Î B(S), we de fine elements l
s
f
and r
s
f in B(S)by(l

s
f)(t)=f (st)and(r
s
f)(t)=f (ts)foranyt Î S, respectively. A net
{μ
a
} of means on B(S) is said to be asymptotically invariant if
lim
α
(μ
α
(l
s
f ) − μ
α
(f ))=0=lim
α
(μ
α
(r
s
f ) − μ
α
(f ))
.
In [4], Rodé proved the following:
Theorem 1.1.[4]If S is an amenable semigro up, C is a closed convex subset of a Hil-
bert space
H, S =
{

T
s
: s ∈ S
}
is a nonexpansive semigroup on C such that the set
F
(S)
of
common fixed points of
S
is nonempty and {μ
a
} is an asymptotically invariant net of
means, then for each x Î C,
{T
μ
α
x
}
converges weakly to an element of
F
(
S
)
.
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
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Commons Attribution License (http ://creativecommons. org/licenses/by/2.0), which permits unrestricted use, distribut ion, and
reproduction in any medium, provided the original work is properly cited.
Further, for each x Î C, the limit point of

{T
μ
α
x
}
is the same for all asymptotically
invariant nets of means {μ
a
}.
It is remarked that if S is amenable, then there is always an asymptotically str ong
invariant net of finite means, i.e., means that are convex combination of point evalua-
tions. This follows from Proposition 3.3 in [5].
Development of this subject had been made by several authors [1,6-8]. The main
purpose of this a rticle is to extend this result of Rodé for a nonexpansive semigroup
on a CAT(0) space in which the Δ-convergence plays the role of weak convergence.
2 Preliminaries
Let (X, d)beametricspace.Ageodesic joining x Î X to y Î X is a mapping c from a
closed interval [0, l] ⊂ ℝ to X such that c(0) = x, c(l)=y and d(c(t), c(t’)) = |t-t’| for all
t, t’Î [0, l]. In particular, c is an isometry and d(x, y)=l.Theimageg of c is called a
geodesic (or metric) segment joining x and y. When it is unique, this geodesic is
denote d [x, y]. Write c(a 0 + (1 - a)l)=ax ⊕(1 - a)y, and for
α
=
1
2
,wewrite
1
2
x ⊕
1

2
y
as
x⊕y
2
, the midpoint of x and y. The space X is said to be a geodesic space if every two
points of X are joined by a geodesic.
Following [2], a metric space X is said to be a CAT(0) space if it is geodesically con-
nected and if every geodesic triangle in X is at least as thin as its comparison triangle
in the Euclidean plane. This latter property, which is what we referred to as the (CN)
inequality, enables one to define the concept of nonpositive curvature in this situation,
generalizing the same concept in Riemannian geometry. In fact (cf. [[9], p. 163]), the
following are equivalent for a geodesic space X:
(i) X is a CAT(0) space.
(ii) X satisfies the (CN) inequality:Ifx
0
, x
1
Î X and
x
0
⊕
x
1
2
is the midpoint of x
0
and
x
1

, then
d
2
(y,
x
0
⊕ x
1
2
) ≤
1
2
d
2
(y, x
0
)+
1
2
d
2
(y, x
1
) −
1
4
d
2
(x
0

, x
1
), for all y ∈ X
.
(iii) X satisfies the Law of cosine:Ifa = d(p, q), b = d(p, r), c = d(q, r) and ξ is the
Alexandrov angle at p between [p, q] and [p, r], then c
2
≥ a
2
+b
2
-2ab cos ξ.
For any subset C of X, let π = π
D
be a nearest point projection mappi ng from C to a
subset D. It is known by [[9], pp. 176-177] (see also [[10], Proposition 2.6]) that if D is
closed and convex, the mapping π is well-defined, nonexpansive, and satisfies
d
2
(
x, y
)
≥ d
2
(
x, π x
)
+ d
2
(

πx, y
)
for all x ∈ Candy∈ D
.
(1)
Definition 2.1. [[11], Definition 5.13] A complete CAT(0) space X has the property of
the nice projection onto geodesics (property (N) for short) if, given any geodesic segment
I ⊂ X, it is the case that π
I
(m) Î [π
I
(x), π
I
(y)] for any x, y in X and m Î [x, y].
As noted in [11], we do not know of any example of a CAT() space which does not
enjoy the property (N).
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 2 of 14
Let S be a semigroup, C be a closed convex subset of a Hilbert space H, and for each
s in S, T
s
is a mapping from C into itsel f. Suppose {T
s
x : s Î S} is bounded for al l x Î
C. Let x Î C and μ be a mean on B(S). By Riesz’s representation theorem, there exists
a unique x
0
Î C such that
μ
s

T
s
x,
y
 = x
0
,
y
(2)
for all y Î H. Here 〈 , 〉 denotes the inner product on H.
The following result is a mild generalization of a result of Kakavandi and Amini
[[12], Lemma 2.1].
Lemma 2.2. LetCbeaclosedconvexsubsetofaCAT(0)spaceXandμ be a mean
on B(S). For a bounded function h : S ® C, define
ϕ
μ
(y):=μ
s
(d
2
(h(s), y)
)
for all y Î X. Then, 
μ
attains its unique minimum at a point of
co{h
(
s
)
: s ∈ S

}
.
For each x Î C,denote
S
(
x
)
:= {T
s
x : s ∈ S
}
.If
S
(
x
)
is bounded, then by Lemma 2.2
we put
T
μ
(h):=argmin{y → μ
s
(d
2
(h(s), y))}
,
and for h(s) of the form T
s
x, we write T
μ

(h) shortly as T
μ
x.
Remark 2.3. If X is a Hilbert space, then
(i) T
μ
x = x
0
where x
0
verifies (2), and
(ii) ||x
0
||
2
= sup
yÎX
(2〈x
0
, y〉 -||y||
2
).
Proof. (i): Let x
0
be such that μ
s
〈T
s
x, y〉 = 〈x
0

, y〉 for all y Î X. We have 
μ
(x
0
)=
μ
(0)
+||x
0
||
2
-2〈x
0
, x
0
〉 = 
μ
(0) - ||x
0
||
2
≤ 
μ
(0) + ||T
μ
x||
2
-2〈x
0
, T

μ
x〉 = 
μ
(T
μ
x). There-
fore, x
0
= T
μ
x.
(ii) : For any x, y Î X, we know that ||T
s
x - y||
2
=||T
s
x||
2
-2〈T
s
x, y〉 +||y||
2
.Bythe
linearity of μ and (2), we have μ
s
(||T
s
x - y||
2

)=μ
s
(||T
s
x||
2
)-2〈x
0
, y〉 +||y||
2
.Thus,
inf
yÎX
μ
s
(||T
s
x - y||
2
)=μ
s
(||T
s
x||
2
)-sup
yÎX
(2〈x
0
, y〉 -||y||

2
). On the other hand, by
(i), inf
yÎX
μ
s
(||T
s
x - y||
2
)=μ
s
(||T
s
x - x
0
||
2
)=μ
s
(||T
s
x||
2
)-2μ
s
〈T
s
x, x
0

〉 +||x
0
||
2
= μ
s
(||
T
s
x||
2
)-||x
0
||
2
. Hence, ||x
0
||
2
= sup
yÎX
(2〈x
0
, y〉 -||y||
2
). ■
Let C be a closed convex subset of a CAT(0) space X and S asemigroup.Wesay
that the set
S(
S

)
:= {T
s
: s ∈ S
}
is a nonexpansive semigroup on C if
(i) T
s
: C ® C is a nonexpansive mapping, i.e., d(T
s
x, T
s
y) ≤ d(x, y) for all x, y Î X,
for all s Î S,
(ii) the mapping s ® T
s
x is bounded for all x Î C, and
(iii) T
ts
= T
t
T
s
, for all s, t Î S.
We denote by
F
(
S
)
the set of a ll common fixed points of mappings in

S
(
S
)
, i.e.,
F(S ):=

s
∈
s
F( T
s
)
, where F (T
s
):={x Î C : T
s
x = x} is the set of fixed points of T
s
.
For any bounded net {x
a
} in a closed convex subset C of a CAT(0) space X, put
r(x, {x
α
}) = lim sup
α
d(x, x
α
)

Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 3 of 14
for each x Î C. The asymptotic radius of {x
a
}onC is given by
r(C, {x
α
})=inf
x
∈
C
r(x, {x
α
})
,
and the asymptotic center of {x
a
}inC is the set
A
(
C, {x
α
}
)
= {x ∈ C : r
(
x, {x
α
}
)

= r
(
C, {x
α
}
)
}
.
It is known that in a complete CAT(0) space, A(C,{x
a
}) consists of exactly one point
and A(X,{x
a
}) = A(C,{x
a
}) (cf. [2]).
Remark 2.4. (i) Let D, E be directions and ν : E ® D. If {x
ν(b)
: b Î E} is a subnet of
a bounded net {x
a
: a Î D}, then r (C,{x
ν(b)
}) ≤ r(C,{x
a
}).
(ii) Let C be a closed convex subset of a CAT(0 ) space X, T : C ® C a nonexpansive
mapping and x Î C. If {T
n
x} is bounded and if z Î A(C,{T

n
x}), then z Î F(T).
Proof.(i)Leta
0
Î D. By the definition of subnets, there exists b
0
Î E such that ν(b)
≽ a
0
for all b ≽ b
0
. For each x Î C,wehave
sup
αα
0
d(x, x
α
) ≥ sup
β

β
0
d(x, x
ν(β)
)
.
Thus,
sup
αα
0

d(x, x
α
) ≥ inf
β
1
sup
β

β
1
d(x, x
ν(β)
)
,andthisholdsforalla
0
. Hence,
r(x, {x
α
})=inf
α
0
sup
α

α
0
d(x, x
α
) ≥ r(x, x
ν(β)

)
,andthisholdsforallx Î C.Conse-
quently, r(C,{x
a
}) = inf
xÎC
r(x,{x
a
}) ≥ inf
xÎC
r(x, x
ν(b)
)=r(C,{x
ν(b)
}).
(ii) Since T is nonexpansive, lim sup
n
d
2
(T
n
x, Tz) = lim sup
n
d
2
(TT
n
x, Tz) ≤ lim sup
n
d

2
(T
n
x, z).
As every asymptotic center is unique, we have z = Tz. □
Definition 2.5. [[2], Definition 3.3] A net {x
a
} in X is said to Δ-converge to x Î Xifx
istheuniqueasymptoticcenterof{u
b
} for every subnet {u
b
} of {x
a
}. In this case, we
write Δ - lim
a
x
a
= x and call x the Δ-limit of {x
a
}.
Proposition 2.6. [[2], Proposition 3.4] Every b ound ed net i n X has a Δ-convergent
subnet.
Remark 2.7. (i) Let D be a direction,{x
a
: a Î D} a net in X and x Î X. If lim sup
a
d(x, x
a

) > r for some r >0, then there exists a subnet
{x
β
α
}
of {x
a
} such that
d(x, x
β
α
) ≥ ρ
for all a.
(ii) Let {x
a
} be a net in X. Then,{x
a
} Δ-converges to x Î X if and only if every subnet
{x
a’
} of {x
a
} has a subnet {x
a“
} which Δ-converges to x.
Proof. (i): For each a Î D,wehavesup
a’ ≽a
d(x, x
a’
) > r. Thus there ex ists b

a
≻ a
such that
d(x, x
β
α
) ≥ ρ
, and this holds for all a. Set a set E ={b
a
: a Î D}. Clearly, E is
a direction, and define ν : E ® D by ν (b
a
)=b
a
.Leta
0
Î D, thus ν(b
a
) ≽ a
0
for all
β
α
 β
α
0
and this shows that
{x
β
α

}
is a subnet of {x
a
} satisfying
d(x, x
β
α
) ≥ ρ
for all a.
ii): It is easy to see that if {x
a
} Δ-converges to x , then every subnet of {x
a
}also
Δ-converges to x. On the other hand, suppose {x
a
} does not Δ-converge to x.Thus,
there exists a subnet {x
b
}of{x
a
} such that x ∉ A (C,{x
b
}), and so lim sup
b
d(x, x
b
)>r
>r(C,{x
b

}) for some r >0.By(i),thereexistsasubnet
{x
γ
β
}
of {x
b
} satisfying
d(x, x
γ
β
) ≥
ρ
for all b. By assumption, there exists a subnet
{x
(γ
β
)
η
}
of
{x
γ
β
}
Δ-conver-
ging to x. Using Remark 2.4,
ρ ≤ lim sup
γ
d(x, x

(γ
β
)
η
)=r(C, {x
(γ
β
)
η
}) ≤ r(C, {x
γ
β
}) ≤ r(C, {x
β
}
)
, a contradiction. □
In [13], Berg and Nikolaev introduced a concept of quasilinearization. Let us formally
denote a pair (a, b) Î X × X by
−→
ab
and call it a vector. Then, quasilinearization is
defined as a map 〈, 〉 :(X × X)×(X × X)by
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 4 of 14

−→
ab,
−→
cd  =

1
2
d
2
(a, d)+
1
2
d
2
(b, c) −
1
2
d
2
(a, c) −
1
2
d
2
(b, d
)
for all a, b, c, d Î X. Recently, Kakavandi and Amini [14] introduced a concept of w-
conv ergence: a sequence {x
n
}issaidtow-converge to x Î X if
lim
n→∞

−→
x

n
x,
−→
ab

=
0
for
all a, b Î X.
Proposition 2.8. [[14], Proposition 2.5] For sequences in a complete CAT(0) space X,
w-convergence implies Δ-convergence (to the same limit).
A simple example shows that the converse of this proposition does not hold:
Example 2.9. Consider an ℝ-tree in ℝ
∞
defined as follow: Let {e
n
} be the standard
basis, x
0
= e
1
= (1, 0, 0, 0, ), and for each n, let x
n
= x
0
+ e
n+1
. An ℝ-tree is formed by
the segments [x
1

, x
n
] for n ≥ 0. It is easy to see that {x
n
} Δ-converges to x
1
. But {x
n
} does
not w-converge to x
1
since

−−→
x
n
x
1
,
−−→
x
0
x
1

= −
1
for all n ≥ 2.
Thus, a bounded sequence does not necessary contain an w-convergent subsequence.
3 Main results

3.1 Δ-convergence
Lemma 3.1. [[12], Lemma 3.1] If C is a closed convex subset of a CAT(0) space X and
T : C ® C is a nonexpansive mapping, then F(T) is closed and convex.
Lemma 3.2. [[12], Proposition 3.2] Let C be a closed convex subset of a CAT(0) space
X and S an amenable semigroup. If
S(
S
)
is a nonexpansive semigroup on C, then the fol-
lowing conditions are equivalent.
(i)
S(
x
)
is bounded for some x Î C;
(ii)
S
(
x
)
is bounded for all x Î C;
(iii)
F
(
S
)
=
∅
.
Proposition 3.3. [[12], Theorem 3.3] LetCbeaclosedconvexsubsetofacomplete

CAT(0) space X, S an amenable semi group, and
S
(
S
)
a nonexpansive semigroup on C
with
F
(S)
=
∅
. Then,
T
μ
x ∈ F(S
)
for any invariant mean μ on B(S).
We now let S be a commutative semigroup and define a partial order ≽ on S by s ≽ t
if s = t or there exists u Î S such that s = ut. When s ≽ t but s ≠ t, we simply write s ≻
t. We can see that (S, ≽) is a directed set. Examples of such S are the usual ordered
sets (N ∪ {0}, +, ≥) and (ℝ
+
∪ {0}, +, ≥). The following fact is well known:
Proposition 3.4. Let μ be a right invariant mean on B(S). Then,
sup
s
inf
t
f (ts) ≤ μ( f (s)) ≤ inf
s

sup
t
f (ts
)
for each f Î B(S). Similarly, let μ be a left invariant mean on B(S). Then,
sup
s
inf
t
f (st) ≤ μ(f (s)) ≤ inf
s
sup
t
f (st
)
for each f Î B(S).
Remark 3.5. If lim
s
f (s)=a for some a Î ℝ and {s’} is a subnet of {s} satisfying s’ ≻ s
for each s, then
μ
s

(
f
(
s

))
= a

.
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 5 of 14
Proof. This is an easy consequence of Proposition 3.4 since μ
s’
(f (s’ )) = μ
s
(f (s’ )) =
lim
s
f (s’)=a. ■
Proposition 3.6. [[12], Proposition 4.1] Let C be a closed convex subset of a complete
CAT(0) space X, S a commutative semigroup, and
S
(
S
)
a nonexpansive semigrou p on C
with
F
(S)
=
∅
. Then, for each x Î C,, the net {πT
s
x}
sÎS
converges to a point Px in
F
(

S
)
,
where
π = π
F
(
S
)
: C → F(
S)
is the nearest point projection.
Propos ition 3.7. Let C be a closed convex subset o f a complete CAT(0) space X, S a
commutat ive semigroup, and
S
(
S
)
a nonexpansive semigroup on C with
F
(
S
)
=
∅
. Then,
for any invariant mean μ on B(S), T
μ
x = lim
s

πT
s
x = Px for all x Î C.
Proof.Fixx Î C and let ε >0. From Proposition 3.6, we see that there exists s
0
Î S
such that d(πT
s
x, Px) < ε for all s ≽ s
0
.WeknowbyProposition3.3that
T
μ
x ∈ F(S
)
.
So, d(Px, T
s
x) ≤ d (Px, πT
s
x)+d(π T
s
x, T
s
x) <d(πT
s
x, T
s
x)+ε ≤ d(T
μ

x, T
s
x)+ε for all
s ≽ s
0
.Since{T
s
x : s Î S} is b ounded by Lemma 3. 2, there exis ts M>0 such that d
(T
μ
x, T
s
x) <Mfor all s Î S. Therefore, d
2
(Px, T
s
x) ≤ d
2
(T
μ
x, T
s
x)+2Mε + ε
2
for each s
≽ s
0
.Sinceμ is an invariant mean, we have
μ
s

(d
2
(Px, T
s
x)) = μ
s
(d
2
(Px, T
ss
0
x)) ≤ μ
s
(d
2
(T
μ
x, T
ss
0
x))+2Mε+ε
2
= μ
s
(d
2
(T
μ
x, T
s

x))+2Mε+ε
2
for any ε
>0. By the argminimality of T
μ
x (see Lemma 2.2), T
μ
x = Px. □
In order to obtain the Rodé’s theorem (Theorem 1.1) in the framework of CAT(0)
spaces, we need to restrict the asymptotically invariant nets of means {μ
a
}tothose
that satisfy an additional condition: for each t Î S,
μ
α
s
(d
2
(T
s
x, y)) − μ
α
s
(d
2
(T
st
x, y)) → 0 uniformly for y ∈ C
.
(3)

In the Hilbert space setting, condition (3) is not required because the weak conver-
gence can obtain from (2) directly.
Lemma 3.8. Let X be a comp lete CAT(0) space th at has property (N), C be a closed
convex s ubset of X, S a commutative semigroup, and
S
(
S
)
a nonexpansive semigroup on
Cwith
F
(
S
)
=
∅
. Suppose {μ
a
} is an asymptotically invariant nets of means on B(S)
satisfying condition (3). If
{
T
μ
α
x
}
Δ-converges to x
0
, then
x

0
∈ F
(S)
.
Proof. First, we show that, for each r Î S,
lim
α
d(T
μ
α
x, T
r
T
μ
α
x)=0.
(4)
If this is not the case, there must be some δ >0sothatforeacha, there exist s a’ ≻
a satisfying
d(T
μ
α

x, T
r
T
μ
α

x) ≥

δ
.Put
ε =
δ
2
2
. Since the asymptoticall y invariant net {μ
a
}
satisfies (3), there exists a
0
for which for each a ≽ a
0
,
ϕ
μ
α
(T
r
T
μ
α
x)=μ
α
s
(d
2
(T
s
x, T

r
T
μ
α
x)) <μ
α
s
(d
2
(T
r
T
s
x, T
r
T
μ
α
x)) + ε ≤ μ
α
s
(d
2
(T
s
x, T
μ
α
x))+ε = ϕ
μ

α
(T
μ
α
x)+
ε
.Set
w =
T
μ
α

0
x ⊕ T
r
T
μ
α

0
x
2
. B y the (CN ) inequality, the foll owing in equalities hold for each
s Î S:
d
2
(T
s
x, w) ≤
1

2
d
2
(T
s
x, T
μ
α

0
x)+
1
2
d
2
(T
s
x, T
r
T
μ
α

0
x) −
1
4
d
2
(T

μ
α

0
x, T
r
T
μ
α

0
x
)
≤
1
2
d
2
(T
s
x, T
μ
α

0
x)+
1
2
d
2

(T
s
x, T
r
T
μ
α

0
x) −
δ
2
4
.
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 6 of 14
Therefore,
ϕ
μ
α

0
(w) ≤
1
2
ϕ
μ
α

0

(T
μ
α

0
x)+
1
2
ϕ
μ
α

0
(T
r
T
μ
α

0
x) −
δ
2
4
<ϕ
μ
α

0
(T

μ
α

0
x)+
ε
2
−
δ
2
4
= ϕ
μ
α

0
(T
μ
α

0
x),
which is a contradiction and thus (4) holds.
Next, we show that
x
0
∈ F
(
S
)

. We suppose on the contrary that
x
0
/∈ F
(
S
)
. Thus, for
some r Î S, T
r
x
0
≠ x
0
,i.e.,d(x
0
, T
r
x
0
):=g >0. Since
{T
μ
α
x}⊂co{T
s
x
}
,itisbounded.
We can get an M>0sothat

d(T
μ
α
x, x
0
) ≤
M
for all a.Welet
0 <ε<min{
γ
2
1
6
M
,2M
}
.From(4),thereexistsa
0
with the pro perty that
d(T
r
T
μ
α
x, T
μ
α
x) <
ε
for all a ≽ a

0
.Now,foreacha ≽ a
0
,
d(T
μ
α
x, T
r
x
0
) ≤ d(T
μ
α
x, T
r
T
μ
α
x)+d(T
r
T
μ
α
x, T
r
x
0
) < d(T
μ

α
x, x
0
)+
ε
.Thus,
d
2
(T
μ
α
x, T
r
x
0
) < d
2
(T
μ
α
x, x
0
)+2εd(T
μ
α
x, x
0
)+ε
2
.Let

w =
x
0
⊕T
r
x
0
2
.Usingthe(CN)
inequality, we see that
d
2
(T
μ
α
x, w) ≤
1
2
d
2
(T
μ
α
x, x
0
)+
1
2
d
2

(T
μ
α
x, T
r
x
0
) −
1
4
d
2
(x
0
, T
r
x
0
)
≤
1
2
d
2
(T
μ
α
x, x
0
)+

1
2
(d
2
(T
μ
α
x, x
0
)+2εM + ε
2
) −
γ
2
4
= d
2
(T
μ
α
x, x
0
)+εM +
ε
2
2
−
γ
2
4

for all a ≽ a
0
. Consequently,
lim sup
α
d
2
(T
μ
α
x, w) ≤ lim sup
α
d
2
(T
μ
α
x, x
0
)+εM +
ε
2
2
−
γ
2
4
< lim sup
α
d

2
(T
μ
α
x, x
0
),
contradicting to the fact that {x
0
} is the center of
{
T
μ
α
x
}
. Therefore, T
r
x
0
= x
0
for all r
Î S, and this shows that
x
0
∈ F
(S)
as desired. □
Theorem 3.9. Let X be a complete CAT(0) space that has Property (N), C be a closed

convex s ubset of X, S a commutative semigroup, and
S
(
S
)
a nonexpansive semigroup on
C with
F
(S)
= ∅
. Suppose {μ
a
}is an asymptotically invariant net of means on B(S) satis-
fying condition (3). Then,
{T
μ
α
x
}
Δ-converges to
Px ∈ F
(
S
)
for all x Î C. Here, Px is
defined in Proposition 3.6.
Proof.Letx Î C and {μ
a’
} be any subnet of {μ
a

}. There exists a subn et {μ
a”
}of{μ
a’
}
such that {μ
a”
}w*-convergestoμ for some invariant mean μ on B(S). By Proposition
3.7, T
μ
x = Px. Since the net
{T
μ
α

x}⊂co{T
s
x : s ∈ S
}
,itisbounded.ThenbyProposi-
tion 2.6, there exists a subnet
{μ
α
β
}
of {μ
a”
} such that
{T
μ

α
β
x
}
Δ-converges to some x
0
Î C. By Lemma 3.8,
x
0
∈ F
(
S
)
.
We show x
0
= T
μ
x by splitting the proof into three steps.
Step 1.If
T
μ
α
β
x := argmin{y → μ
β
s
(d
2
(T

ss
0
x, y))
}
, then
T
μ
α
β
x ∈ co{T
s
x}
ss
0
.
Suppose
T
μ
α
β
x /∈ co{T
s
x}
ss
0
,by(1),
d
2
(T
ss

0
x, T
μ
α
β
x) ≥ d
2
(T
ss
0
x, π T
μ
α
β
x)+d
2
(T
μ
α
β
x, π T
μ
α
β
x
)
for each s Î S where
π : C → co{T
s
x}

s

s
0
is the nearest point projection. Thus,
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 7 of 14
μ
α
β
(d
2
(T
ss
0
x, T
μ
α
β
x)) ≥ μ
α
β
(d
2
(T
ss
0
x, πT
μ
α

β
x))+d
2
(T
μ
α
β
x, πT
μ
α
β
x) >μ
α
β
(d
2
(T
ss
0
x, πT
μ
α
β
x)
)
.This
impossibility shows that
T
μ
α

β
x ∈ co{T
s
x}
ss
0
.
Step 2.
lim
β
d(T
μ
α
β
x, T
μ
α
β
x)=
0
.
If this does not hold, there must be some h >0 so that for each b, there exists b’ ≻ b
satisfying
d(T
μ
α
β

x, T
μ

α
β

x) ≥
η
.Put
ε =
η
2
2
. Since the asymptotically invariant net {μ
b
}
satisfies ( 3), there exists b
0
such that
|μ
α
β
s
(d
2
(T
s
x, T
μ
α
β
x)) − μ
α

β
s
(d
2
(T
ss
0
x, T
μ
α
β
x))| <
ε
for each b ≽ b
0
. We suppose first that
μ
α
β

0
s
(d
2
(T
ss
0
x, T
μ
α

β

0
x)) ≤ μ
α
β

0
s
(d
2
(T
s
x, T
μ
α
β

0
x)
)
.Set
w =
T
μ
α
β

0
x ⊕ T

μ
α
β

0
x
2
.By(CN)
inequality, the following inequalities hold for each s Î S:
d
2
(T
s
x, w) ≤
1
2
d
2
(T
s
x, T
μ
α
β

0
x)+
1
2
d

2
(T
s
x, T
μ
α
β

0
x) −
1
4
d
2
(T
μ
α
β

0
x, T
μ
α
β

0
x
)
≤
1

2
d
2
(T
s
x, T
μ
α
β

0
x)+
1
2
d
2
(T
s
x, T
μ
α
β

0
x) −
η
2
4
.
Therefore,

ϕ
μ
α
β

0
(w) ≤
1
2
ϕ
μ
α
β

0
(T
μ
α
β

0
x)+
1
2
ϕ
μ
α
β

0

(T
μ
α
β

0
x) −
η
2
4
<
1
2
ϕ
μ
α
β

0
(T
μ
α
β

0
x)+
1
2
μ
α

β

0
s
(d
2
(T
ss
0
x, T
μ
α
β

0
x)) +
ε
2
−
η
2
4
≤ ϕ
μ
α
β

0
(T
μ

α
β

0
x)+
ε
2
−
η
2
4
= ϕ
μ
α
β

0
(T
μ
α
β

0
x),
contradicting to the argminimality of
T
μ
α
β


0
x
. In case
μ
α
β

0
s
(d
2
(T
s
x, T
μ
α
β

0
x)) <μ
α
β

0
s
(d
2
(T
ss
0

x, T
μ
α
β

0
x)
)
,wecanshowinthesamewaythat
μ
α
β

0
s
(d
2
(T
ss
0
x, w)) <μ
α
β

0
s
(d
2
(T
ss

0
x, T
μ
α
β

0
x)
)
for some w which also leads to a
contradiction.
Step 3. x
0
= T
μ
x.
We suppose on the contrary and l et h := d(x
0
, T
μ
x) >0. Let I =[T
μ
x, x
0
]andπ
I
: C
® I be the nearest point projection onto I. S ince {T
s
x} is bounded, there exists M>0

such that d(T
s
x, π
I
(T
s
x)) ≤ M for all s Î S.Set
N
0
>
4(M+η)
5
η
and
ρ =
η
5N
0
.Suppose
there exists s
0
Î S such that d(π
I
(T
s
x), x
0
) ≥ 2r for all s ≽ s
0
. We know, by Step 1, that

T
μ
α
β
x ∈ co{T
s
x}
ss
0
.LetA := {y Î C: d(π
I
(y), x
0
)>2r}. Using property (N), A i s convex
and
co{T
s
x}
s

s
0
⊂
¯
A ⊂{y ∈ C : d(π
1
(y), x
0
) ≥ 2ρ
}

and thus
d(π
I
(T
μ
α
β
x), x
0
) ≥ 2
ρ
.By
Step 2,
lim
β
d(T
μ
α
β
x, T
μ
α
β
x)=
0
. Choose b
0
, using the nonexpansiveness of π
I
,sothat

d(π
I
(T
μ
α
β
x), π
I
(T
μ
α
β
x)) <
ρ
for all b ≽ b
0
. Thus,
d(π
I
(T
μ
α
β
x), x
0
) >
ρ
for all b ≽ b
0
.

But then
x
0
/∈ co{T
μ
α
β
x}
β β
0
which contradicts to the fact that x
0
is the Δ - limit of
{T
μ
α
β
x
}
. Therefore, there must be a subnet {s’}ofS such that s’ ≻ s for all s and
d(π
I
(T
s

x), x
0
) < 2ρ =
2
η

5N
0
(5)
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 8 of 14
for all s’. Hence,
d(π
I
(T
s

x), T
μ
x)=η − d(π
I
(T
s

x), x
0
) >η−
2
η
5N
0
.
(6)
By the property of N
0
,5h

2
N
0
>4hM +4h
2
and so
η
2
−
4η
2
5N
0
>
4ηM
5N
0
.
(7)
From (5), (6), and (7),
d
2
(π
I
(T
s

x), T
μ
x) >η

2
−
4η
2
5N
0
+(
2η
5N
0
)
2
>
4ηM
5N
0
+(
2η
5N
0
)
2
> 2d
(
x
0
, π
I
(
T

s

x
))
d
(
T
s

x, π
I
(
T
s

x
))
+ d
2
(
x
0
, π
I
(
T
s

x
)).

Using (1),
d
2
(T
s

x, T
μ
x) ≥ d
2
(π
I
(T
s

x), T
μ
x)+d
2
(π
I
(T
s

x), T
s

x)
> d
2

(x
0
, π
I
(T
s

x)) + 2d(x
0
, π
I
(T
s

x))d(T
s

x, π
I
(T
s

x)) + d
2
(π
I
(T
s

x), T

s

x
)
=(d(x
0
, π
I
(T
s

x)) + d(T
s

x, π
I
(T
s

x)))
2
≥ d
2
(
T
s

x, x
0
)

for all s’.Sincethepointsx
0
and T
μ
x belong to the set
F
(
S
)
,thenets{d
2
(T
s
x, x
0
)}
and {d
2
(T
s
x, T
μ
x)} are decreasing. So, lim
s
d
2
(T
s
x, x
0

)andlim
s
d
2
(T
s
x, T
μ
x)exist.
Hence, 
μ
(T
μ
x )=lim
s
d
2
(T
s
x, T
μ
x ) = lim
s’
d
2
(T
s’
x , T
μ
x)=μ

s’
(d
2
( T
s’
x, T
μ
x )) ≥ μ
s’
(d
2
(T
s’
x, x
0
)) = lim
s’
d
2
(T
s’
x, x
0
)=lim
s
d
2
(T
s
x, x

0
)=
μ
(x
0
), a contradiction. Thus, x
0
=
T
μ
x.
The above argument shows that, for every subnet {μ
a’
}of{μ
a
}, there exists a subnet
{μ
α
β
}
of {μ
a’
}suchthat
{T
μ
α
β
x
}
Δ-converges to T

μ
x(= Px). By Remark 2.7 (ii),
{T
μ
α
x
}
Δ-converges to Px. □
It is an interesting open problem to determine whether Theorem 3.9 remains valid
when the semigroup is amenable but not commutative.
3.2 Applications
Proposition 3.10. Let C be a closed convex subset of a complete CAT(0) space X and T
: C ® CbeanonexpansivemappingwithF(T) ≠ ∅. Let S =(N ∪ {0}, +),
S
(
S
)
= {T
n
: n ∈ S
}
, Λ = N or ℝ
+
and b
lk
≥ 0 be such that

k
∈
S

β
λk
=
1
for all l Î Λ.
Suppose for all k Î S,
lim
λ
→
∞
β
λk
=0
(8)
and for each m Î S,
lim
λ
→∞
∞

k
=
m
|β
λk
− β
λ(k−m)
| =0
.
(9)

For any f =(a
0
, a
1
, ) Î B(S) let
μ
λ
(f )=

∞
k
=
0
β
λk
a
k
. Then for each x Î C,
{T
μ
λ
x
}
Δ-converges to z for some z in F(T).
In particular, if X is a H ilbert space, we have

∞
k
=
0

β
λk
T
k
x
converges weakly to z for
some z in F(T) as l ® ∞.
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 9 of 14
Proof. For each m Î S,
|μ
λ
(f )−μ
λ
(r
m
f )| =




∞
k=0
β
λk
a
k
−

∞

k=0
β
λk
a
k+m



≤

m−
1
k=0
|
β
λk
||
a
k
|
+

∞
k=m


β
λk
− β
λ(k−m)



|
a
k
|
.
By (8) and (9), we have
lim
λ→∞
|μ
λ
(
f
)
− μ
λ
(
r
m
f
)
| =
0
, and this shows that the net {μ
l
}is
asymptotically invariant. Let x Î C and consider a
k
of the form a

k
= d
2
(T
k
x, y) where y Î
C. We see that {μ
l
} sati sfies (3). By Theorem 3.9, we have
{T
μ
λ
x
}
Δ -convergestoz for
some z in F(T).
In Hilbert spaces, by a well-known result in probability theory, we know that
∞

k
=
0
β
λk





T

k
x −
∞

k
=
0
β
λk
T
k
x





2
≤
∞

k
=
0
β
λk



T

k
x − y



2
for all y Î C. So we have
T
μ
λ
x =

∞
k
=
0
β
λk
T
k
x
. □
Corollary 3.11 (Bailo n Ergodic Theorem). LetCbeaclosedconvexsubsetofaHil-
bert space H and T : C ® C be a nonexpansive mapping with F(T) ≠ ∅. Then, for any
x Î C,
S
n
x =
1
n

n−
1

k
=
0
T
k
x
converges weakly to z for some z in F(T) as n ® ∞.
Proof. Let Λ = N and put, for l Î Λ and k Î S =(N ∪ {0}, +),
β
λk
=

1
λ
, k ≤ λ − 1,
0, k >λ− 1
.
The result now follows from Proposition 3.10. □
Corollary 3.12. [[15], Theorem 3.5.1] LetCbeaclosedconvexsubsetofaHilbert
space H and T : C ® C be a nonexpansive mapping with F(T) ≠ ∅. Then, for any x Î
C,
S
r
x =(1− r)

∞
k

=
0
r
k
T
k
x
converges weakly to z for some z in F(T) as r ↑ 1.
Proof. Let Λ = ℝ
+
and put, for l Î Λ and k Î S =(N ∪{0}, +),
β
λk
=
(λ − 1)
k
λ
k+1
.
Taking
r =
λ−1
λ
, Proposition 3.10 implies the desired result.
Let S =(ℝ
+
∪ {0}, +) and C be a closed convex subset of a Hilbert space H.Then,a
family
S
(

S
)
= {T
(
s
)
: s ∈ S
}
is said to be a continuous nonexpansive semigroup on C if
S
(
S
)
satisfies the following:
(i) T(s):C ® C is a nonexpansive mapping for all s Î S,
(ii) T(t + s)x = T(t)T(s)x for all x Î C and t, s Î S,
(iii) for each x Î C, the mapping s ® T(s)x is continuous, and
(iv) T(0)x = x for all x Î C.
Proposition 3.13. Let C be a closed convex subset of a Hilbert space H. Let S =(ℝ
+
∪
{0}, + ),
S
(
S
)
be a continuous nonexpa nsive semigroup on C with
F
(
S

)
=
∅
, Λ = ℝ
+
and
g
l
be a density function on S,i.e., g
l
≥ 0 and

∞
0
g
λ
(s)ds =
1
for all l Î Λ. Suppose g
l
satisfies the following properties. for each h Î S ,
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 10 of 14
lim
λ
→
∞
g
λ
(s)=

0
(10)
uniformly on [0, h] and
lim
λ
→∞

∞
h
|g
λ
(s) − g
λ
(s − h)|ds =0
.
(11)
Then, for any x Î C,

∞
0
g
λ
(s)T(s) xd
s
converges weakly to some
z
∈ F
(
S
)

as l ® ∞.
Proof. For f Î B(S) we define
μ
λ
(f )=

∞
0
g
λ
(s)f (s)d
s
for all l > 0. Thus, μ
l
is a mean
on B(S). For any h Î S we consider
|
μ
λ
(f ) − μ
λ
(r
h
f )| =





∞

0
g
λ
(s)f (s)ds −

∞
0
g
λ
(s)f (s + h)ds




≤

h
0
|g
λ
(s)||f (s)|ds +

∞
h
|g
λ
(s) − g
λ
(s − h)||f (s)|ds
.

By (10) and (11),
lim
λ
|μ
λ
(
f
)
− μ
λ
(
r
h
f
)
| =
0
.So,{μ
l
} is asymptotically invariant. For
each z Î C,letf(s)=||z - T(s) x||
2
.Weseethat{μ
l
} satisfies (3). For each x Î C,we
know that

∞
0
g

λ
(s)





∞
0
g
λ
(s)T(s) xds − T(s)x




2
ds ≤

∞
0
g
λ
(s)


y − T(s)x


2

d
s
for all y Î C. Thus,
T
μ
λ
x =

∞
0
g
λ
(s)T(s) xd
s
. By Theorem 3.9, we have

∞
0
g
λ
(s)T(s) xd
s
converges weakly to some
z
∈ F
(
S
)
as l ® ∞. □
Corollary 3.14. [[15], Theorem 3.5.2] LetCbeaclosedconvexsubsetofaHilbert

space H. Suppose S =(ℝ
+
∪ {0}, +) and
S(
S
)
be a continuous nonexpansive semigroup
on C with
F
(
S
)
=
∅
. Then, for any x Î C,
S
λ
x =
1
λ

λ
0
T(s)xd
s
converges weakly to some
z
∈ F
(S)
as l ® ∞.

Proof.LetΛ = ℝ
+
and put, for l Î Λ and s Î S,
g
λ
(s)=
1
λ
χ
[0,λ
]
.Theresultnowfol-
lows from Proposition 3.13. □
Corollary 3.15. [[15], Theorem 3.5.3] LetCbeaclosedconvexsubsetofaHilbert
space H. Suppose S =(ℝ
+
∪ {0}, +) and
S
(
S
)
be a continuous nonexpansive semi-group
on C with
F
(
S
)
=
∅
. Then, for any x Î C,

r

∞
0
e
−rs
T(s)xd
s
converges weakly to some
z
∈ F
(
S
)
as r ↓ 0.
Proof.LetΛ = ℝ
+
and put, for l Î Λ and s Î S,
g
λ
(s)=
1
λ
e
−
1
λ
s
. Again, we can then
apply Proposition 3.13 by taking

r =
1
λ
. □
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 11 of 14
By using Lemma 2.2, we can obtain a strong conver gence theorem in Hilbert spaces
stated as Theorem 3.17 below.
Proposition 3.16. Let C be a closed convex subset of a Hilbert space H and T : C ®
C be a nonexpan sive mapping with F(T) ≠ ∅. Given x Î Candletr= r(C,{T
n
x}). Let
z be the unique asymptotic center of {T
n
x}. For each n Î N, define

n
:=
⎧
⎨
⎩
p = {β
nk
}
k≥n
⊂ [0, 1] :

k≥n
β
nk

=1
⎫
⎬
⎭
and
V
n
:= sup
p∈
n

k
≥
n
β
nk



T
k
x −
¯
x
n



2
where

¯
x
n
=

k
≥
n
β
nk
T
k
x
. If V := lim
n®∞
V
n
, then V = r
2
.
Proof. Given ε >0. Since z Î A(C,{T
n
x}), by Remark 2.4 (ii), z Î F(T). Choose n
ε
Î N
such that ||T
n
x - z|| <r+ ε for all n ≥ n
ε
. Fix n ≥ n

ε
and let p ={b
nk
}
k≥n
Î Π
n
. Thus,

k
≥
n
β
nk



T
k
x −
¯
x
n



2
≤

k

≥
n
β
nk



T
k
x − z



2
< (r + ε)
2
.
So
V
n
=sup
p∈
n

k
≥
n
β
nk



T
k
x −
¯
x
n


2
< (r + ε)
2
. Letting n ® ∞, V = lim
n ® ∞
V
n
≤
(r+ε)
2
for any ε > 0. Hence, V ≤ r
2
.
Next, we show that r
2
≤ V. Indeed, since
z
∈ co{T
k
x}
k

≥n
for all n Î N,thereexistsa
sequence
{
¯
x
n
}
with
¯
x
n
∈ co{T
k
x}
k
≥
n
for each n and
¯
x
n
→ z
as n ® ∞.Put
¯
x
n
=

k

≥
n
β
nk
T
k
x
.Since{T
n
x} is bounded, there exists M>0suchthat


T
k
x −
¯
x
n


+


T
k
x − z


≤
M

. For each ε >0, choose n
ε
Î N such that

¯
x
n
− z

<
ε
, V
n
<V+ ε for all n ≥ n
ε
, and ||T
k
x - z|| >r- ε for all k ≥ n
ε
. Thus for any n ≥ n
ε
,

k≥n
β
nk
|||T
k
x −
¯

x
n
||
2
−||T
k
x − z||
2
|
=

k≥n
β
nk
|||T
k
x −
¯
x
n
|| − ||T
k
x − z|||(||T
k
x −
¯
x
n
|| + ||T
k

x − z||
)
=

k
≥
n
β
nk
||
¯
x
n
− z||(||T
k
x −
¯
x
n
|| + ||T
k
x − z||) ≤ εM.
Hence,
(r − ε)
2
<

k≥n
β
nk

||T
k
x − z||
2
=

k≥n
β
nk
||T
k
x − z||
2
+

k≥n
β
nk
||T
k
x −
¯
x
n
||
2
−

k≥n
β

nk
||T
k
x −
¯
x
n
||
2
≤ V
n
+

k
≥
n
β
nk
|||T
k
x −
¯
x
n
||
2
−||T
k
x − z||
2

| < V + ε + εM.
So (r - ε)
2
<V + ε + εM for any ε >0. This implies r
2
≤ V.
Theorem 3.17. Let C be a closed convex subset of a Hilb ert space H and T : C ® C
be a n onexpansive mapping with F(T) ≠ ∅,. Suppose z, Π
n
, V, and
¯
x
n
be defined as in
Proposition 3.16. If the sequence
{
¯
x
n
}
satisfies
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 12 of 14
lim
n→∞
⎛
⎝

k≥n
β

nk
||T
k
x −
¯
x
n
||
2
⎞
⎠
= V
,
(12)
then
{
¯
x
n
}
converges (strongly) to z Î F (T ) as n ® ∞.
Proof.Supposeforsomeε >0, there exists a subsequence
{
¯
x
n
l
}
of
{

¯
x
n
}
such that


¯
x
n
l
− z


≥
ε
for all l Î N. For each y Î C and n Î N,define
ϕ
n
(y):=

k
≥
n
β
nk
||T
k
x − y||
2

.Let
0 <δ<
ε
2
8
.By(12)andz Î A(C,{T
k
x}), we choose
n
δ
such that
r
2
− δ = V − δ<ϕ
n
l
(
¯
x
n
l
) < V + δ = r
2
+
δ
for all n
l
≥ n
δ
and ||T

k
x - z||
2
<r
2
+ δ for all k ≥ n
δ
. Fix l ≥ n
δ
and let
ω =
¯
x
n
l
+z
2
. By the Parallelogram law, we have for
each k ≥ n
l
,
|
|T
k
x − ω||
2
=
1
2
||T

k
x −
¯
x
n
l
||
2
+
1
2
||T
k
x − z||
2
−
1
4
||
¯
x
n
l
− z||
2
.
Hence,
ϕ
n
l

(ω) <
1
2
(r
2
+ δ)+
1
2
(r
2
+ δ) −
1
4
ε
2
< r
2
− δ<ϕ
n
l
(
¯
x
n
l
)
.
Using Lemma 2.2, we see that this contradicts to the minimality of
ϕ
n

l
(
¯
x
n
l
)
. □
Acknowledgements
The authors would like to thank Anthony To-Ming Lau for drawing the problem into our attention and also for giving
valuable advice during the preparation of the manuscript. We thank the referee for valuable and useful comments.
We also wish to thank the National Research University Project under Thailand’s Office of the Higher Education
Commission for financial support.
Author details
1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
2
Materials
Science Research Center, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 1 February 2011 Accepted: 15 August 2011 Published: 15 August 2011
References
1. Lim, TC: Remarks on some fixed point theorems. Proc Am Math Soc. 60, 179–182 (1976). doi:10.1090/S0002-9939-1976-
0423139-X
2. Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008). doi:10.1016/
j.na.2007.04.011
3. Day, MM: Amenable semigroups. Illinois J Math. 1, 509–544 (1957)

4. Rodé, G: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space. J Math Anal Appl. 85,
172–178 (1982). doi:10.1016/0022-247X(82)90032-4
5. Kaniuth, E, Lau, AT, Pym, J: On character amenability of Banach algebras. J Math Anal Appl. 34, 942–955 (2008)
6. Kada, O, Lau, AT, Takahashi, W: Asymptotically invariant nets and fixed point sets for semigroups of nonexpansive
mappings. Nonlinear Anal. 29, 539–550 (1997). doi:10.1016/S0362-546X(96)00063-6
7. Lau, AT: Semigroup of nonexpansive mappings on a Hilbert space. J Math Anal Appl. 105, 514–522 (1985). doi:10.1016/
0022-247X(85)90066-6
8. 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. 191,62–75 (1999)
9. Bridson, M, Haeiger, A: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999)
10. Sturm, KT: Probability measures on metric spaces of non-positive curvature. Heat Kernels and Analysis on Manifolds,
Graphs and Metric Spaces (Paris, 2002), Contemp Math 338, Am Math Soc Providence, RI. 357–390 (2003)
11. Espínola, R, Fernández-León, A: CAT(κ) spaces, weak convergence and fixed points. J Math Anal Appl. 353, 410–427
(2009). doi:10.1016/j.jmaa.2008.12.015
12. Kakavandi, BA, Amini, M: Non-linear ergodic theorem in complete non-positive curvature metric spaces. Bull Iran Math
Soc. (in press)
Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34
/>Page 13 of 14
13. Berg, ID, Nikolaev, IG: Quasilinerization and curvature of Alexandrov spaces. Geom Dedicata. 133, 195–218 (2008).
doi:10.1007/s10711-008-9243-3
14. Kakavandi, BA, Amini, M: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear
Anal. 73, 3450–3455 (2010). doi:10.1016/j.na.2010.07.033
15. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publisher, Yokohama (2000)
doi:10.1186/1687-1812-2011-34
Cite this article as: Anakkamatee and Dhompongsa: Rodé’s theorem on common fixed points of semigroup of
nonexpansive mappings in CAT(0) spaces. Fixed Point Theory and Applications 2011 2011:34.
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