Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 471781, 13 pages
doi:10.1155/2010/471781
Research Article
Halpern’s Iteration in CAT(0) Spaces
Satit Saejung1, 2
1
2
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Centre of Excellence in Mathematics, CHE, Sriayudthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Satit Saejung, saejung@kku.ac.th
Received 26 September 2009; Accepted 24 November 2009
Academic Editor: Mohamed A. Khamsi
Copyright q 2010 Satit Saejung. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Motivated by Halpern’s result, we prove strong convergence theorem of an iterative sequence in
CAT0 spaces. We apply our result to find a common fixed point of a family of nonexpansive
mappings. A convergence theorem for nonself mappings is also discussed.
1. Introduction
Let X, d be a metric space and x, y ∈ X with l dx, y. A geodesic path from x to y is
an isometry c : 0, l → X such that c0 x and cl y. The image of a geodesic path is
called a geodesic segment. A metric space X is a (uniquely) geodesic space if every two points of X
are joined by only one geodesic segment. A geodesic triangle x1 , x2 , x3 in a geodesic space
X consists of three points x1 , x2 , x3 of X and three geodesic segments joining each pair of
vertices. A comparison triangle of a geodesic triangle x1 , x2 , x3 is the triangle x1 , x2 , x3 :
x1 , x2 , x3 in the Euclidean space R2 such that dxi , xj dR2 xi , xj for all i, j 1, 2, 3.
A geodesic space X is a CAT(0) space if for each geodesic triangle : x1 , x2 , x3 in
X and its comparison triangle : x1 , x2 , x3 in R2 , the CAT(0) inequality
d x, y ≤ dR2 x, y
1.1
is satisfied by all x, y ∈ and x, y ∈ . The meaning of the CAT0 inequality is that
a geodesic triangle in X is at least thin as its comparison triangle in the Euclidean plane.
A thorough discussion of these spaces and their important role in various branches of
mathematics are given in 1, 2. The complex Hilbert ball with the hyperbolic metric is an
example of a CAT0 space see 3.
2
Fixed Point Theory and Applications
The concept of Δ-convergence introduced by Lim in 1976 was shown by Kirk and
Panyanak 4 in CAT0 spaces to be very similar to the weak convergence in Banach space
setting. Several convergence theorems for finding a fixed point of a nonexpansive mapping
have been established with respect to this type of convergence e.g., see 5–7. The purpose
of this paper is to prove strong convergence of iterative schemes introduced by Halpern 8
in CAT0 spaces. Our results are proved under weaker assumptions as were the case in
previous papers and we do not use Δ-convergence. We apply our result to find a common
fixed point of a countable family of nonexpansive mappings. A convergence theorem for
nonself mappings is also discussed.
In this paper, we write 1 − tx ⊕ ty for the the unique point z in the geodesic segment
joining from x to y such that
dz, x td x, y ,
d z, y 1 − td x, y .
1.2
We also denote by x, y the geodesic segment joining from x to y, that is, x, y {1 − tx ⊕
ty : t ∈ 0, 1}. A subset C of a CAT0 space is convex if x, y ⊂ C for all x, y ∈ C. For
elementary facts about CAT0 spaces, we refer the readers to 1 or, briefly in 5.
The following lemma plays an important role in our paper.
Lemma 1.1. A geodesic space X is a CAT(0) space if and only if the following inequality
d2 1 − tx ⊕ ty, z ≤ 1 − td2 x, z td2 y, z − t1 − td2 x, y
1.3
is satisfied by all x, y, z ∈ X and all t ∈ 0, 1. In particular, if x, y, z are points in a CAT(0) space
and t ∈ 0, 1, then
d 1 − tx ⊕ ty, z ≤ 1 − tdx, z td y, z .
1.4
Recall that a continuous linear functional μ on ∞ , the Banach space of bounded real
sequences, is called a Banach limit if μ μ1, 1, . . . 1 and μn an μn an1 for all {an } ∈
∞ .
Lemma 1.2 see 9, Proposition 2. Let a1 , a2 , . . . ∈ l∞ be such that μn an ≤ 0 for all Banach
limits μ and lim supn an1 − an ≤ 0. Then lim supn an ≤ 0.
Lemma 1.3 see 10, Lemma 2.3. Let {sn } be a sequence of nonnegative real numbers, {αn } a
sequence of real numbers in 0, 1 with ∞
n1 αn ∞, {un } a sequence of nonnegative real numbers
∞
with n1 un < ∞, and {tn } a sequence of real numbers with lim supn → ∞ tn ≤ 0. Suppose that
sn1 ≤ 1 − αn sn αn tn un
Then limn → ∞ sn 0.
∀n ∈ N.
1.5
Fixed Point Theory and Applications
3
2. Halpern’s Iteration for a Single Mapping
Lemma 2.1. Let C be a closed convex subset of a complete CAT(0) space X and let T : C → C be a
nonexpansive mapping. Let u ∈ C be fixed. For each t ∈ 0, 1, the mapping St : C → C defined by
St x tu ⊕ 1 − tT x
for x ∈ C
2.1
has a unique fixed point xt ∈ C, that is,
xt St xt tu ⊕ 1 − tT xt .
2.2
Proof. For x, y ∈ C, we consider the triangle u, T x, T y and its comparison triangle and we
have the following:
d tu ⊕ 1 − tT x, tu ⊕ 1 − tT y ≤ dR2 tu ⊕ 1 − tT x, tu ⊕ 1 − tT y
1 − tdR2 T x, T y
1 − td T x, T y
≤ 1 − td x, y .
2.3
This implies that St is a contraction mapping and hence the conclusion follows.
The following result is proved by Kirk in 11, Theorem 26 under the boundedness
assumption on C. We present here a new proof which is modified from Kirk’s proof.
Lemma 2.2. Let C, T be as the preceding lemma. Then FT /
∅ if and only if {xt } given by the
formula 2.2 remains bounded as t → 0. In this case, the following statements hold:
1 {xt } converges to the unique fixed point z0 of T which is nearest u;
2 d2 u, z0 ≤ μn d2 u, xn for all Banach limits μ and all bounded sequences {xn } with xn −
T xn → 0.
Proof. If FT /
∅, then it is clear that {xt } is bounded. Conversely, suppose that {xt } is
bounded. Let {tn } be any sequence in 0, 1 such that limn → ∞ tn 0 and define g : C → R by
gz lim sup d2 xtn , z
n→∞
2.4
for all z ∈ C. By the boundedness of {xtn }, we have δ : inf{gz : z ∈ C} < ∞. We choose a
sequence {zm } in C such that limm → ∞ gzm δ. It follows from Lemma 1.1 that
d
2
1
1
xtn , zm ⊕ zk
2
2
≤
1 2
1
1
d xtn , zm d2 xtn , zk − d2 zm , zk .
2
2
4
2.5
4
Fixed Point Theory and Applications
Then, by the convexity of C,
1
1
1
1
1
δ ≤ lim sup d xtn , zm ⊕ zk ≤ gzm gzk − d2 zm , zk .
2
2
2
2
4
n→∞
2
2.6
This implies that {zm } is a Cauchy sequence in C and hence it converges to a point z0 ∈ C.
Suppose that z is a point in C satisfying g
z δ. It follows then that
1
1
1
1
1
z − d2 z0 , z,
δ ≤ lim sup d2 xtn , z0 ⊕ z ≤ gz0 g
2
2
2
2
4
n→∞
2.7
and hence z z0 . Moreover, z0 is a fixed point of T . To see this, we consider
dxtn , T xtn tn
du, xtn −→ 0,
1 − tn
2.8
and
lim sup d2 xtn , T z0 ≤ lim supdxtn , T xtn dT xtn , T z0 2
n→∞
n→∞
≤ lim supdxtn , T xtn dxtn , z0 2
n→∞
2.9
lim sup d2 xtn , z0 δ.
n→∞
∅.
This implies that z0 T z0 and hence FT /
1 is proved in 12, Theorem 26. In fact, it is shown that z0 is the nearest point of FT to u. Finally, we prove 2. Suppose that {ztm } is a sequence given by the formula 2.2, where
{tm } is a sequence in 0, 1 such that limm → ∞ tm 0. We also assume that z0 limm → ∞ ztm is
the nearest point of FT to u. By the first inequality in Lemma 1.1, we have
d2 xn , ztm d2 xn , tm u ⊕ 1 − tm T ztm ≤ tm d2 xn , u 1 − tm d2 xn , T ztm − tm 1 − tm d2 u, T ztm ≤ tm d2 xn , u 1 − tm dxn , T xn dT xn , T ztm 2 − tm 1 − tm d2 u, T ztm ≤ tm d2 xn , u 1 − tm dxn , T xn dxn , ztm 2 − tm 1 − tm d2 u, T ztm .
2.10
Let μ be a Banach limit. Then
μn d2 xn , ztm ≤ tm μn d2 xn , u 1 − tm μn d2 xn , ztm − tm 1 − tm d2 u, T ztm .
2.11
This implies that
μn d2 xn , ztm ≤ μn d2 xn , u − 1 − tm d2 u, T ztm .
2.12
Fixed Point Theory and Applications
5
Letting m → ∞ gives
μn d2 xn , z ≤ μn d2 xn , u − d2 u, z.
2.13
In particular,
d2 u, z ≤ μn d2 xn , u
for all Banach limits μ.
2.14
Inspired by the results of Wittmann 13 and of Shioji and Takahashi 9, we use
the iterative scheme introduced by Halpern to obtain a strong convergence theorem for a
nonexpansive mapping in CAT0 space setting. A part of the following theorem is proved in
14.
Theorem 2.3. Let C be a closed convex subset of a complete CAT(0) space X and let T : C → C be a
nonexpansive mapping with a nonempty fixed point set FT . Suppose that u, x1 ∈ C are arbitrarily
chosen and {xn } is iteratively generated by
xn1 αn u ⊕ 1 − αn T xn
∀n ≥ 1,
2.15
where {αn } is a sequence in 0, 1 satisfying
C1 limn → ∞ αn 0;
C2 ∞
n1 αn ∞;
∞
C3 n1 |αn − αn1 | < ∞ or limn → ∞ αn /αn1 1.
Then {xn } converges to z ∈ FT which is the nearest point of FT to u.
Proof. We first show that the sequence {xn } is bounded. Let p ∈ FT . Then
d xn1 , p d αn u ⊕ 1 − αn T xn , p
≤ αn d u, p 1 − αn d T xn , p
≤ αn d u, p 1 − αn d xn , p
≤ max d u, p , d xn , p .
2.16
d xn1 , p ≤ max d u, p , d x1 , p
2.17
By induction, we have
for all n ∈ N. This implies that {xn } is bounded and so is the sequence {T xn }.
6
Fixed Point Theory and Applications
Next, we show that dxn1 , xn → 0. To see this, we consider the following:
dxn1 , xn dαn u ⊕ 1 − αn T xn , αn−1 u ⊕ 1 − αn−1 T xn−1 ≤ dαn u ⊕ 1 − αn T xn , αn u ⊕ 1 − αn T xn−1 dαn u ⊕ 1 − αn T xn−1 , αn−1 u ⊕ 1 − αn−1 T xn−1 2.18
≤ 1 − αn dT xn , T xn−1 |αn − αn−1 |du, T xn−1 ≤ 1 − αn dxn , xn−1 |αn − αn−1 |du, T xn−1 .
By the conditions C2 and C3, we have
dxn1 , xn −→ 0.
2.19
Consequently, by the condition C1,
dxn , T xn ≤ dxn , xn1 dxn1 , T xn dxn , xn1 dαn u ⊕ 1 − αn T xn , T xn 2.20
dxn , xn1 αn du, T xn −→ 0.
From Lemma 2.2, let z limt → 0 xt where xt is given by the formula 2.2. Then z is the nearest
point of FT to u. We next consider the following:
d2 xn1 , z d2 αn u ⊕ 1 − αn T xn , z
≤ αn d2 u, z 1 − αn d2 T xn , z − αn 1 − αn d2 u, T xn ≤ 1 − αn d2 xn , z αn d2 u, z − 1 − αn d2 u, T xn .
2.21
By Lemma 2.2, we have μn d2 u, z − d2 u, xn ≤ 0 for all Banach limits μ. Moreover, since
xn1 − xn → 0,
lim sup d2 u, z − d2 u, xn − d2 u, z − d2 u, xn1 0.
n→∞
2.22
It follows from xn − T xn → 0 and Lemma 1.2 that
lim sup d2 u, z − 1 − αn d2 u, T xn lim sup d2 u, z − d2 u, xn ≤ 0.
n→∞
Hence the conclusion follows by Lemma 1.3.
n→∞
2.23
Fixed Point Theory and Applications
7
3. Halpern’s Iteration for a Family of Mappings
3.1. Finitely Many Mappings
We use the “cyclic method” 15 and Bauschke’s condition 16 to obtain the following strong
convergence theorem for a finite family of nonexpansive mappings.
Theorem 3.1. Let X be a complete CAT(0) space and C a closed convex subset of X. Let
∅ and let u, x1 ∈ C be
T1 , T2 , . . . , TN : C → C be nonexpansive mappings with N
i1 FTi /
arbitrarily chosen. Define an iterative sequence {xn } by
xn1 αn u ⊕ 1 − αn Tn mod N xn
∀n ≥ 1,
3.1
where {αn } is a sequence in 0, 1 satisfying
C1 limn → ∞ αn 0;
C2 ∞
n1 αn ∞;
∞
C3 n1 |αn − αnN | < ∞ or limn → ∞ αn /αnN 1.
Suppose, in addition, that
N
FTi FTN ◦ TN−1 ◦ · · · ◦ T1 .
3.2
i1
Then {xn } converges to z ∈
N
i1
FTi which is nearest u.
Here the modN function takes values in {1, 2, . . . , N}.
Proof. By 16, Theorem 2, we have
N
FTi FT1 ◦ TN ◦ TN−1 ◦ · · · ◦ T2 · · · FTN−1 ◦ TN ◦ T1 ◦ · · · ◦ TN−2 .
3.3
i1
The proof line now follows from the proofs of Theorem 2.3 and 15, Theorem 3.1.
3.2. Countable Mappings
The following concept is introduced by Aoyama et al. 10. Let X be a complete CAT0 space
and C a subset of X. Let {Tn }∞
n1 be a countable family of mappings from C into itself. We say
that a family {Tn } satisfies AKTT-condition if
∞
sup{dTn1 z, Tn z : z ∈ B} < ∞
n1
for each bounded subset of B of C.
3.4
8
Fixed Point Theory and Applications
If C is a closed subset and {Tn } satisfies AKTT-condition, then we can define T : C → C
such that
T x lim Tn x
n→∞
x ∈ C.
3.5
In this case, we also say that {Tn }, T satisfies AKTT-condition.
Theorem 3.2. Let X be a complete CAT(0) space and C a closed convex subset of X. Let {Tn } : C →
∅. Suppose that u, x1 ∈ C are
C be a countable family of nonexpansive mappings with ∞
n1 FTn /
arbitrarily chosen and {xn } is defined by
xn1 αn u ⊕ 1 − αn Tn xn
∀n ≥ 1,
3.6
where {αn } is a sequence in 0, 1 satisfying
C1 limn → ∞ αn 0;
C2 ∞
n1 αn ∞;
∞
C3 n1 |αn − αn1 | < ∞ or limn → ∞ αn /αn1 1.
Suppose, in addition, that
M1 {Tn }, T satisfies AKTT-condition;
M2 FT ∞
n1 FTn .
Then {xn } converges to z ∈ ∞
n1 FTn which is nearest u.
Proof. Since the proof of this theorem is very similar to that of Theorem 2.3, we present here
only the sketch proof. First, we notice that both sequences {xn } and {Tn xn } are bounded and
dxn1 , xn dαn u ⊕ 1 − αn Tn xn , αn−1 u ⊕ 1 − αn−1 Tn−1 xn−1 ≤ dαn u ⊕ 1 − αn Tn xn , αn u ⊕ 1 − αn Tn xn−1 dαn u ⊕ 1 − αn Tn xn−1 , αn u ⊕ 1 − αn Tn−1 xn−1 dαn u ⊕ 1 − αn Tn−1 xn−1 , αn−1 u ⊕ 1 − αn−1 Tn−1 xn−1 ≤ 1 − αn dTn xn , Tn xn−1 1 − αn dTn xn−1 , Tn−1 xn−1 |αn − αn−1 |du, Tn−1 xn−1 3.7
≤ 1 − αn dxn , xn−1 dTn xn−1 , Tn−1 xn−1 |αn − αn−1 |du, Tn−1 xn−1 ≤ 1 − αn dxn , xn−1 |αn − αn−1 |du, Tn−1 xn−1 sup d Tn y, Tn−1 y : y ∈ {xn } .
By conditions C2, C3, AKTT-condition, and Lemma 1.3, we have
dxn1 , xn −→ 0.
3.8
Fixed Point Theory and Applications
9
Consequently, dxn , Tn xn → 0 and hence
dxn , T xn ≤ dxn , Tn xn dTn xn , T xn ≤ dxn , Tn xn sup{dTn z, T z : z ∈ {xn }}
≤ dxn , Tn xn ∞
3.9
sup{dTk z, Tk1 z : z ∈ {xn }} −→ 0.
kn
Let z ∈ FT ∞
n1 FTn be the nearest point of FT to u. As in the proof of Theorem 2.3,
we have d2 u, z ≤ μn d2 u, xn for all Banach limits μ and lim supn → ∞ d2 u, z − d2 u, xn −
d2 u, z − d2 u, xn1 0. We observe that
d2 xn1 , z d2 αn u ⊕ 1 − αn Tn xn , z
≤ αn d2 u, z 1 − αn d2 Tn xn , z − αn 1 − αn d2 u, Tn xn ≤ 1 − αn d2 xn , z αn d2 u, z − 1 − αn d2 u, Tn xn ,
3.10
and this implies that
lim sup d2 u, z − 1 − αn d2 u, Tn xn lim sup d2 u, z − d2 u, xn ≤ 0.
n→∞
n→∞
3.11
Therefore, limn → ∞ d2 xn , z 0 and hence {xn } converges to z.
We next show how to generate a family of mappings from a given family of mappings
to satisfy conditions M1 and M2 of the preceding theorem. The following is an analogue
of Bruck’s result 17 in CAT0 space setting. The idea using here is from 10.
Theorem 3.3. Let X be a complete CAT(0) space and C a closed convex subset of X. Suppose that
∅. Then there
{Tn } : C → X is a countable family of nonexpansive mappings with ∞
n1 FTn /
exist a family of nonexpansive mappings {Sn } : C → X and a nonexpansive mapping S : C → X
such that
M1 {Sn }, S satisfies AKTT-condition;
M2 FS ∞
n1
FTn .
Lemma 3.4. Let X and C be as above. Suppose that S, T : C → X are nonexpansive mappings and
FS ∩ FT /
∅. Then, for any 0 < t < 1, the mapping U : 1 − tS ⊕ tT : C → X is nonexpansive
and FU FS ∩ FT .
Proof. To see that U is nonexpansive, we only apply the triangle inequality and two
applications of the second inequality in Lemma 1.1. We next prove the latter. It is clear that
10
Fixed Point Theory and Applications
FS ∩ FT ⊂ FU. To see the reverse inclusion, let p ∈ FU and q ∈ FS ∩ FT . Then, by
the first inequality of Lemma 1.1,
d2 q, p d2 q, Up
d2 q, 1 − tSp ⊕ tT p
≤ 1 − td2 q, Sp td2 q, T p − t1 − td2 Sp, T p
≤ d2 q, p − t1 − td2 Sp, T p .
3.12
This implies Sp T p. As p Up, we have p ∈ FS ∩ FT , as desired.
Proof of Theorem 3.3. We first define a family of mappings {Sn } : C → X by
S1 x S2 x 1
1
x ⊕ T1 x
2
2
22 − 1
1
S1 x ⊕ 2 T 2 x
2
2
2
..
.
3.13
2n − 1
1
Sn−1 x ⊕ n Tn x
2n
2
Sn x ..
.
By Lemma 3.4, each Sn is a nonexpansive mapping satisfying FSn for fixed p ∈ ∞
n1 FTn ,
d Sn1 x, Sn x d
2
≤
2
2n1 − 1
1
Sn x ⊕ n1 Tn1 x, Sn x
n1
2
2
1 2
d Tn1 x, Sn x
2n1
1 2n1
1
2n−1
n
dTn1 x, p dp, Sn x
k1
FTk . Notice that,
3.14
2
d2 x, p .
From the estimation above, we have
∞
n1
sup{dSn1 x, Sn x : x ∈ B} < ∞
3.15
Fixed Point Theory and Applications
11
for each bounded subset B of C. In particular, {Sn x} is a Cauchy sequence for each x ∈ C. We
now define the nonexpansive mapping S : C → X by
Sx lim Sn x.
3.16
n→∞
Finally, we prove that
FS ∞
∞
FSn n1
FTn .
3.17
n1
The latter equality is clearly verified and ∞
n1 FSn ⊂ FS holds. On the other hand, let
∞
p ∈ FS and q ∈ n1 FTn . We consider the following:
1
2n − 1
S
p
⊕
T
p
d2 q, Sn p d2 q,
n−1
n
2n
2n
≤
1 2n − 1 2 d q, Sn−1 p n d2 q, Tn p
n
2
2
≤
1 2n − 1 2 d q, Sn−1 p n d2 q, p .
2n
2
3.18
Then
d q, Sn p ≤
2
n
2k − 1
2k
k2
≤
1−
n
2k − 1
k2
2k
d2 q, p
1 2
1 2
d q, p d q, T1 p − d p, T1 p
2
2
4
2k
1−
n
2k − 1
k2
n
2k − 1
1
k2
d q, S1 p 2
2
3.19
2k
2
d q, p
n
n
1 2
2k − 1
2k − 1
2
d2 q, p .
≤
d q, p − d p, T1 p 1 −
k
k
4
2
2
k2
k2
Letting n → ∞ yields
d q, p ≤
2
∞ k
2 −1
k2
2k
∞ k
2 −1
1 2
d2 q, p .
d q, p − d p, T1 p 1 −
k
4
2
k2
2
3.20
12
Fixed Point Theory and Applications
k
k
have p T1 p. Continuing this procedure we obtain that
Because ∞
k2 2 − 1/2 > 0, we
∞
p ∈ n1 FTn and hence FS ⊂ ∞
n1 FTn . This completes the proof.
4. Nonself Mappings
From Bridson and Haefliger’s book page 176, the following result is proved.
Theorem 4.1. Let X be a complete CAT(0) space and C a closed convex subset of X. Then the
followings hold true.
i For each x ∈ X, there exists an element πx ∈ C such that
dx, πx distx, C.
4.1
ii πx πx for all x ∈ x, πx.
iii The mapping x → πx is nonexpansive.
The mapping π in the preceding theorem is called the metric projection from X onto C.
From this, we have the following result.
Theorem 4.2. Let X be a complete CAT(0) space and C a closed convex subset of X. Let T : C → X
be a nonself nonexpansive mapping with FT /
∅ and π : X → C the metric projection from X onto
C. Then the mapping π ◦ T is nonexpansive and Fπ ◦ T FT .
Proof. It follows from Theorem 4.1 that π ◦ T is nonexpansive. To see the latter, it suffices to
show that Fπ ◦ T ⊂ FT . Let p ∈ Fπ ◦ T and q ∈ FT . Since
1
1
Tp ⊕ p
d2 q, p d2 π q , π
2
2
1
1
≤ d2 q, T p ⊕ p
2
2
1 1 1 ≤ d2 q, T p d2 q, p − d2 T p, p
2
2
4
4.2
1 ≤ d2 q, p − d2 T p, p ,
4
we have p T p and this finishes the proof.
By the preceding theorem and Theorem 2.3, we obtain the following result.
Theorem 4.3. Let X, C, T : C → X, and π : X → C be as the same as Theorem 4.2. Suppose that
u, x1 ∈ C are arbitrarily chosen and the sequence {xn } is defined by
xn1 αn u ⊕ 1 − αn π ◦ T xn ∀n ≥ 1,
4.3
Fixed Point Theory and Applications
13
where {αn } is a sequence in 0, 1 satisfying
C1 limn → ∞ αn 0;
C2 ∞
n1 αn ∞;
∞
C3 n1 |αn − αn1 | < ∞ or limn → ∞ αn /αn1 1.
Then {xn } converges to z ∈ FT which is nearest u.
Acknowledgments
The author would like to thank the referee for the information that a part of Theorem 2.3 was
proved in 14. This work was supported by the Centre of Excellence in Mathematics, the
Commission on Higher Education, Thailand.
References
1 M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, vol. 319 of Grundlehren der
Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
2 D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in
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