This article appeared in a journal published by Elsevier. The attached
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling or
licensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of the
article (e.g. in Word or Tex form) to their personal website or
institutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies are
encouraged to visit:
http://www.elsevier.com/copyright
Author's personal copy
Nonlinear Analysis 75 (2012) 445–452
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
Mann’s algorithm for nonexpansive mappings in CAT(κ ) spaces
J.S. He a,∗ , D.H. Fang b , G. López c , C. Li d,e
a
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, PR China
b
College of Mathematics and Computer Science, Jishou University, Jishou 416000, PR China
c
Department of Mathematical Analysis, University of Sevilla, Apdo. 1160, 41080-Sevilla, Spain
d
Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China
e
Department of Mathematics, College of Sciences, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia
article
info
Article history:
Received 1 July 2010
Accepted 24 July 2011
Communicated by W.A. Kirk
abstract
In this paper, it is proved that the sequence defined by Mann’s algorithm ∆-converges to a
fixed point in complete CAT(κ ) spaces.
© 2011 Elsevier Ltd. All rights reserved.
MSC:
47H09
65K05
Keywords:
CAT(κ ) spaces
Nonexpansive mapping
Mann’s iteration
∆-convergence
1. Introduction
Let X be a complete metric space and let T be a self-mapping defined on X . It is well known that, if the mapping T is a
contraction, then, by Banach’s contraction principle, for each x ∈ X , the iterative sequence {T n x} converges strongly to the
unique fixed point of T . But, if T is only assumed to be a nonexpansive mapping, we must add some additional conditions
to ensure the existence of a fixed point of T and, even when a fixed point exists, the sequences of iterates in general do
not converge to a fixed point. This fact explains why the study of the asymptotic behavior of nonexpansive mappings has
become a very attractive research topic in nonlinear analysis.
One of the most successful methods to approach fixed points of a nonexpansive mapping is Mann’s algorithm, which, in
the case when X is a linear space, generates a sequence {xn } via the following iteration:
xn+1 = tn xn + (1 − tn )T (xn ),
n ≥ 0.
(1.1)
The convergence properties of this algorithm have been studied extensively; see, for example, [1–4] and the references
therein.
Because of the convex structure of the algorithms, few results on approximation of fixed points have been obtained out
of the setting of linear spaces except for ones in some special metric spaces. For example, Mann’s iteration has been studied
in hyperbolic spaces (see [5,6]), the Hilbert ball (see [7]), and Hadamard manifolds (see [8]).
∗
Corresponding author.
E-mail addresses: hejinsu@zjnu.cn (J.S. He), dh_fang@jsu.edu.cn (D.H. Fang), glopez@us.es (G. López), cli@zju.edu.cn (C. Li).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2011.07.070
Author's personal copy
446
J.S. He et al. / Nonlinear Analysis 75 (2012) 445–452
Our interest here is to study Mann’s iteration scheme in complete CAT(κ ) spaces. Roughly speaking, CAT(κ ) spaces are
geodesic spaces of bounded curvature and generalizations of Riemannian manifolds of sectional curvature bounded above.
The study of nonexpansive mappings in the realm of CAT(κ ) spaces was initiated by Kirk [9,10]. In his seminal works,
several results are proved, mainly concerning fixed point theorems in CAT(0) spaces. Recently, by using the ∆-convergence
introduced by Lim [11], Dhompongsa and Panyanak established in [12] the ∆-convergence result of Mann’s iteration in
CAT(0) spaces.
The aim of this paper is to establish in complete CAT(κ ) spaces the ∆-convergence results of the sequence defined by
Mann’s algorithm for any nonnegative number κ . Our results obtained in the present paper extend the corresponding ones
in [8,12].
2. Preliminaries
In this section, we introduce some fundamental definitions, properties, and notations of model spaces Mk2 and CAT(κ )
spaces. We basically follow [13].
Let (X , ρ) be a metric space and let x, y ∈ X . As usual, we use B(x, δ) (respectively, B(x, δ)) to denote the ball (respectively,
closed ball) centered at x with radius δ . A geodesic path joining x to y (or, more briefly, a geodesic from x to y) is a map
c : [0, l] ⊆ R → X such that c (0) = x, c (l) = y, and ρ(c (t ), c (t ′ )) = |t − t ′ | for all t , t ′ ∈ [0, l]. In particular, c is an
isometry between [0, l] and c ([0, l]), and ρ(x, y) = l. Usually, the image c ([0, l]) of c is called a geodesic segment (or metric
segment) joining x and y. A metric segment joining x and y is not necessarily unique in general. In particular, in the case
when the geodesic segment joining x and y is unique, we use [x, y] to denote the unique geodesic segment joining x and y;
this means that z ∈ [x, y] if and only if there exists t ∈ [0, 1] such that ρ(z , x) = (1 − t )ρ(x, y) and ρ(z , y) = t ρ(x, y). In
this case, we will write z = tx ⊕ (1 − t )y for simplicity. For fixed D ∈ (0, +∞], the space (X , ρ) is called a D-geodesic space
if any two points of X with their distance smaller than D are joined by a geodesic segment. An ∞-geodesic space is simply
called a geodesic space. Recall that a geodesic triangle ∆ := ∆(x, y, z ) in the metric space (X , ρ) consists of three points
in X (the vertices of ∆) and three geodesic segments between each pair of vertices (the edges of ∆). For the sake of saving
printing space, we write p ∈ ∆ when a point p ∈ X lies in the union of [x, y], [x, z ] and
∑m[y, z ].
Let m ∈ N and let ⟨·|·⟩ denote the Euclidean scalar product in Rm , that is, ⟨x|y⟩ =
i=1 xi yi , where x = (x1 , . . . , xm ) and
y = (y1 , . . . , ym ). Let Sm denote the m-dimensional sphere defined by
Sm := {x = (x1 , . . . , xm+1 ) ∈ Rm+1 : ⟨x|x⟩ = 1}.
Define d : Sm × Sm → [0, +∞) by
d(x, y) = arccos⟨x|y⟩,
∀(x, y) ∈ Sm × Sm .
(2.1)
Then (S , d) is a metric space (see [13, Proposition 2.1, page 16]). By definition (see [13, page 17]), the spherical angle
between two geodesics from a point of Sm , with initial vectors u and v , is the unique number α ∈ [0, π] such that
cos α = ⟨u|v⟩. Given ∆(x, y, z ) a triangle in Sm , the vertex angle at z is defined to be the spherical angle between the
sides of ∆ joining z to x and z to y. Hence, by [13, page 17], the spherical law of cosines holds:
m
cos d(x, y) = cos d(y, z ) cos d(x, z ) + sin d(y, z ) sin d(x, z ) cos γ ,
(2.2)
where γ is the vertex angle at z.
From now on, we always assume that κ ≥ 0, and write
 π
√
Dκ :=
κ
+∞
κ > 0,
κ = 0.
Let Mκ denote the metric space obtained from (Sm , d) by multiplying the distance function by the constant √1κ if κ > 0
m
and the m-dimension Euclidean space Rm if κ = 0. We use the same symbol d to denote the distance in Mκm if no confusion
arises. Clearly, Mκm is a geodesic metric space.
Consider triangles ∆ := ∆(x, y, z ) ⊂ X and ∆ := ∆(x̄, ȳ, z̄ ) ⊂ Mk2 . The triangle ∆ is called a comparison triangle for ∆ if
ρ(x, y) = d(x̄, ȳ),
ρ(x, z ) = d(x̄, z̄ ) and ρ(z , y) = d(z̄ , ȳ).
By [13, Lemma 2.14, page 24], a comparison triangle for ∆ always exists provided that the perimeter ρ(x, y) + ρ(y, z ) +
ρ(z , x) < 2Dκ . A point p̄ ∈ [x̄, ȳ] ⊂ ∆ is called a comparison point for p ∈ [x, y] ⊂ ∆ if d(p̄, x̄) = d(p, x).
¯ a
Recall that a geodesic triangle ∆ in X with perimeter less than 2Dκ is said to satisfy the CAT(κ ) inequality if, given ∆
comparison triangle in Mκ2 for ∆, one has that
ρ(p, q) ≤ d(p̄, q̄),
∀p, q ∈ ∆,
where p̄ and q̄ are respectively the comparison points of p and q. Thus we are ready to introduce the notion of a CAT(κ ) space
in the following definition taken from [13].
Definition 2.1. The metric space (X , ρ) is called a CAT(κ ) space if it is Dκ -geodesic and any geodesic triangle in X of
perimeter less than 2Dκ satisfies the CAT(κ ) inequality.
Author's personal copy
J.S. He et al. / Nonlinear Analysis 75 (2012) 445–452
447
By definition, it is clear that (X , ρ) is a CAT(0) space if and only if it is geodesic and any geodesic triangle in X satisfies
the CAT(0) inequality.
Recall that a set Y ⊆ X is said to be convex if any two points x, y ∈ Y can be joined by a geodesic in X and all geodesics
joining them are contained in Y . Then the following proposition is known in [13, Proposition 1.4, page 160] and will be used
in the next section.
Proposition 2.1. Let X be a CAT(κ ) space. Then any ball in X of radius smaller than
CAT(0) space is convex.
Dκ
2
is convex. In particular, any ball in a
Remark 2.1. For κ < 0, a CAT(κ ) space is defined in terms of comparison triangles in the hyperbolic plane; see [13] for
details. Here, for the sake of simplicity, we omit its definition, since it is known (see [13, page 165]) that any CAT(κ1 ) space
is also a CAT(κ2 ) space for any pair (κ1 , κ2 ) with κ2 ≥ κ1 . This means that any CAT(κ ) space with κ < 0 is a CAT(0) space.
The remainder of this section is concerned with the notion of ∆-convergence and some relative properties that are useful
for our study in the next section. Let {xn } be a bounded sequence in X . We define
r(x, {xn }) := lim sup ρ(x, xn ),
n→∞
∀x ∈ X .
The asymptotic radius r({xn }) and the asymptotic center A({xn }) of {xn } are defined respectively by
r({xn }) := inf{r(x, {xn }) : x ∈ X }
and
A({xn }) := {x ∈ X : r(x, {xn }) = r({xn })}.
Therefore, the following equivalence holds for any point u ∈ X :
u ∈ A({xn }) ⇐⇒ lim sup ρ(u, xn ) ≤ lim sup ρ(x, xn ),
n→∞
n→∞
∀x ∈ X .
(2.3)
The notion of ∆-convergence in the following definition is taken from [11].
Definition 2.2. Let {xn } ⊆ X be a bounded sequence and let x ∈ X . The sequence {xn } is said to ∆-converge to x if, for any
subsequence {xnk } of {xn }, the point x is the unique asymptotic center of {xnk }. In this case, we write ∆ − limn xn = x and
call x the ∆-limit of {xn }.
The notion of ∆-convergence was introduced by Lim [11] in general metric spaces with some restrictions. In [14], it was
shown that CAT(0) spaces are a natural realm for this notion, and this notion was applied to study the fixed point theory
in CAT(0). Recently, Espínola and Fernández-León [15] extended this study to CAT(κ ) spaces and proved that this notion of
convergence coincides, in this setting, with one of two notions of weak convergence introduced by Sosov [16] in geodesic
metric spaces, and with weak convergence in Hilbert spaces.
Let C be a subset of X . We use conv(C ) and diam(C ) to denote the closed convex hull and the diameter of C , which are
respectively defined by
conv(C ) :=

{E : E ⊇ C and E is closed convex}
and
diam(C ) := sup{ρ(x, y) : x, y ∈ C }.
The following proposition was proved in [15]; see [15, Proposition 4.1] for assertion (i) and [15, Corollary 4.4] for assertion
(ii).
Proposition 2.2. Let X be a complete CAT(κ ) space and let {xn } be a sequence in X with r({xn }) <
assertions hold.
Dκ
2
. Then the following
(i) A({xn }) consists of exactly one point.
(ii) {xn } has a ∆-convergent subsequence.
Assertion (2.4) in the following proposition was proved in [15, Proposition 4.5] under the additional assumption that
diam(X ) < Dκ .
Proposition 2.3. Let X be a complete CAT(κ ) space and let p ∈ X . Suppose that {xn } ⊂ X satisfies r(p, {xn }) <
{xn } ∆-converges to x. Then
x∈
∞

conv({xn , xn+1 , . . .})
Dκ
2
and that
(2.4)
n =1
and
ρ(x, p) ≤ lim inf ρ(xn , p).
n→∞
(2.5)
Author's personal copy
448
J.S. He et al. / Nonlinear Analysis 75 (2012) 445–452
Proof. Without loss of generality, we assume that κ = 1. Then Dκ = π . Let r(p, {xn }) < r̄ < π2 and let p0 := 12 p ⊕ 12 x.
Consider the subset X0 := B(p0 , r̄ ). Then diam(X0 ) < π , and X0 is a CAT(1) space by [13, Example 1.15(1), page 167]. By the
choice of r̄, there exists N ∈ N such that
max{ρ(xn , p), ρ(xn , x)} < r̄ ,
∀n > N ;
that is, x, p ∈ B(xn , r̄ ) for each n > N. This, together with Proposition 2.1, implies that p0 ∈ B(xn , r̄ ), and so {xn : n > N } ⊂
X0 . Moreover, since
ρ(x, p) ≤ r(x, {xn }) + r(p, {xn }) ≤ 2r(p, {xn }) < 2r̄ ,
it follows that ρ(x, p0 ) < r̄ and x ∈ X0 . Hence {xn } ∆-converges to x in X0 as {xn } ⊂ X0 does in X . Thus [15, Proposition 4.5]
is applicable (to X0 and {xn : n > N } in place of X and {xn }), allowing us to conclude that
x∈
∞

conv({xn , xn+1 , . . .}) =
n>N
∞

conv({xn , xn+1 , . . .}),
n=1
and assertion (2.4) is proved.
To show assertion (2.5), set r := lim infn ρ(xn , p), and let ϵ > 0 be such that r + ϵ < π2 . Then there exists a subsequence
{xnk } of {xn } such that
ρ(xnk , p) < r + ϵ <
π
2
,
∀k ∈ N .
This means that {xnk } ⊆ B(p, r + ϵ). By Proposition 2.1, B(p, r + ϵ) is convex. It follows that conv({xnk }) ⊆ B(p, r + ϵ).
Applying (2.4) (to {xnk } in place of {xn }), we have that
x∈
∞

conv({xnk , xnk+1 , . . .}) ⊆ conv({xnk }) ⊆ B(p, r + ϵ).
k=1
Hence ρ(x, p) ≤ r + ϵ . Since ϵ > 0 is arbitrary, it follows that ρ(x, p) ≤ lim infn ρ(xn , p). The proof is complete.
3. Mann’s algorithm in CAT(κ) spaces
In the case when X is a normed linear space and T : X → X is a nonexpansive mapping, for {tn } ⊂ [0, 1], Mann introduced
in [1] the iteration process {xn } defined by the algorithm (1.1), which is known as Mann’s
∑∞iteration for finding fixed points of
nonexpansive mappings. Ishikawa proved in [2] that, if 0 < b ≤ tn < 1 for each n and n=1 tn = ∞, then ‖xn − T (xn )‖ → 0
as n → ∞, which implies the convergence of {xn } to a fixed point of T if the range of T lies in a compact subset of X , and
Reich
in [3] the weak convergence of the sequence in a uniformly convex Banach space, assuming that {tn } ⊂ (0, 1)
∑obtained
∞
and n=1 tn (1 − tn ) = ∞. For further results, see [4] and the references therein.
In the framework of metric spaces (X , ρ), Goebel and Kirk in [5,9] and Reich and Shafrir in [17] provided a iterative
method for finding fixed points of nonexpansive mappings in spaces of hyperbolic type, whose algorithm is defined by
xn+1 ∈ [xn , T (xn )],
ρ(xn , T (xn )) = tn ρ(xn , xn+1 ), ∀n ≥ 0,
where each tn ∈ (0, 1); that is, using the notation fixed in the preliminary section,
xn+1 = tn xn ⊕ (1 − tn )T (xn ),
∀n ≥ 0.
(3.1)
The ∆-convergence of the sequence defined by algorithm (3.1) has been studied in [12] for CAT(0) spaces. Motivated by
these results, we studied here the ∆-convergence of this sequence in CAT(κ ) spaces. In order to get the main result, we need
the following definition and lemmas.
Definition 3.1. Let X be a complete metric space and let C ⊆ X be a nonempty set. A sequence {xn } ⊂ X is called Fejér
monotone with respect to C if
ρ(xn+1 , y) ≤ ρ(xn , y),
∀y ∈ C and ∀n ≥ 0.
For convenience, we say that a point x ∈ X is a ∆-cluster point of the sequence {xn } if there exists a subsequence of {xn }
that ∆-converges to x.
Lemma 3.1. Let X be a complete metric space and let C ⊆ X be a nonempty set. Suppose that the sequence {xn } ⊂ X is Fejér
monotone with respect to C . Let {un } be a subsequence of {xn }. Then we have that A({un }) ∩ C ⊆ A({xn }). In particular, if x ∈ C
is a ∆-cluster point of {xn }, then x ∈ A({xn }).
Author's personal copy
J.S. He et al. / Nonlinear Analysis 75 (2012) 445–452
449
Proof. Let ū ∈ A({un }) ∩ C . By the assumed Fejér monotonicity, one has that limn→∞ ρ(un , ū) = limn→∞ ρ(xn , ū). It follows
from definition that, for any u ∈ X ,
lim ρ(xn , ū) = lim ρ(un , ū) ≤ lim sup ρ(un , u) ≤ lim sup ρ(xn , u).
n→∞
n→∞
n→∞
n→∞
Hence ū ∈ A({xn }) by definition (see (2.3)), and the proof is complete.
Lemma 3.2. Let X be a complete CAT(κ ) space and let C ⊆ X be a nonempty set. Suppose that the sequence {xn } ⊂ X is Fejér
monotone with respect to C and satisfies that r({xn }) < D2κ . Suppose also that any ∆-cluster point x of {xn } belongs to C . Then
{xn } ∆-converges to a point in C .
Proof. By Proposition 2.2(ii), the ∆-cluster point set of {xn } is nonempty. Let x be a ∆-cluster point of {xn }. Then x ∈ C
by assumption, and so A({xn }) = {x} thanks to Lemma 3.1 and Proposition 2.2(i). Below, we show that {xn } ∆-converges
to x. For this purpose, let {un } be an arbitrary subsequence of {xn }. By Proposition 2.2(i), we have that A({un }) = {u} for
some u ∈ X . Applying Proposition 2.2(ii) to the sequence {un }, we conclude that {un } has a ∆-cluster point ū. Clearly, ū is
also a ∆-cluster point of {xn }, and so ū ∈ C by assumptions. It follows from Lemma 3.1 that ū ∈ A({xn }) ∩ A({un }), and so
ū = u = x. Thus, by definition, {xn } ∆-converges to x ∈ C , and the proof is complete. Now we give our main result in this section.
Theorem 3.1. Let X be a complete CAT(κ ) space and let T : X → X be a nonexpansive mapping such that F := Fix(T ) ̸= ∅.
Suppose that {tn } ⊂ (0, 1) satisfies that
∞
−
tn (1 − tn ) = ∞.
(3.2)
n =0
Then, for each x0 ∈ X with ρ(x0 , F ) <
Dκ
4
, the sequence {xn } defined by (3.1) ∆-converges to a point of F .
Proof. Without loss of generality, we assume that κ = 1. Let
 {xn } be the sequence generated by the algorithm (3.1) and let
x0 ∈ X be such that ρ(x0 , F ) < π4 . Write F0 := F ∩ B x0 , π2 . Let n ≥ 0 and let q ∈ F0 be fixed. Since ρ(T (x0 ), q) ≤ ρ(x0 , q)
and since the open ball in X with center q and radius less than π2 is convex (see Proposition 2.1), it follows that
ρ(x1 , q) = ρ(t0 x0 ⊕ (1 − t0 )T (x0 ), q) ≤ ρ(x0 , q).
By mathematical induction, we can easily show that
ρ(xn+1 , q) ≤ ρ(xn , q) ≤ ρ(x0 , q),
∀n ≥ 0.
(3.3)
Hence {xn } is Fejér monotone with respect to F0 . In particular, choose p ∈ F such that ρ(x0 , p) < π4 . Then p ∈ F0 and
ρ(xn+1 , p) ≤ ρ(xn , p) ≤ ρ(x0 , p) <
π
4
,
∀n ∈ N .
(3.4)
This implies that r({xn }) < π4 . Thus, by Lemma 3.2, it suffices to show that any ∆-cluster point of {xn } belongs to F0 . For this
end, let x̄ ∈ X be a ∆-cluster point of {xn }. Then there exists a subsequence {xnk } of {xn } which ∆-converges to x̄. Below, we
will show that x̄ ∈ F0 . We first note by (3.4) that r(p, {xnk }) ≤ ρ(x0 , p) < π4 . It follows from (2.5) in Proposition 2.3 that
ρ(x̄, x0 ) ≤ ρ(x̄, p) + ρ(x0 , p) ≤ lim inf ρ(xnk , p) + ρ(x0 , p) <
k
π
2
;


hence x̄ ∈ B x0 , π2 . To show that x̄ ∈ F , it suffices to prove that
lim ρ(xn , T (xn )) = 0.
(3.5)
k→∞
Indeed, if (3.5) holds, then
lim sup ρ(T (x̄), xnk ) ≤ lim sup ρ(T (x̄), T (xnk )) + lim sup ρ(T (xnk ), xnk )
k
k
k
≤ lim sup ρ(x̄, xnk );
k
hence T (x̄) ∈ A({xnk }) by definition, and so T (x̄) = x̄; that is, x̄ ∈ F .
To show (3.5), we only need to show that {ρ(xn , T (xn ))} is monotone and that
lim inf ρ(xn , T (xn )) = 0.
n→∞
(3.6)
Author's personal copy
450
J.S. He et al. / Nonlinear Analysis 75 (2012) 445–452
Using the nonexpansivity of T and the definition of xn+1 , we conclude that
ρ(xn+1 , T (xn+1 )) ≤
≤
=
=
ρ(xn+1 , T (xn )) + ρ(T (xn ), T (xn+1 ))
ρ(xn+1 , T (xn )) + ρ(xn , xn+1 )
tn ρ(xn , T (xn )) + (1 − tn )ρ(xn , T (xn ))
ρ(xn , T (xn )).
This means that {ρ(xn , T (xn ))} is monotone. Moreover, we have by (3.4) that
ρ(x0 , T (x0 )) ≤ ρ(x0 , p) + ρ(p, T (x0 )) ≤ 2ρ(x0 , p) <
π
2
,
and so
ρ(xn , T (xn )) ≤ ρ(x0 , T (x0 )) <
π
2
,
∀n ∈ N .
(3.7)
It remains to show assertion (3.6). For this end, we fix n ∈ N and write qn := T (xn ). Then ρ(p, qn ) ≤ ρ(p, xn ) < π4 by (3.4)
and the nonexpansivity of T . Therefore, we get that
ρ(xn , p) + ρ(xn , qn ) + ρ(qn , p) ≤ 4ρ(xn , p) ≤ 4ρ(x0 , p) < π.
Thus we can take ∆(x̄n , p̄, q̄n ) to be a comparison triangle of ∆(xn , p, qn ) in S2 . Hence,
ρ(xn , p) = d(x̄n , p̄),
ρ(qn , p) = d(q̄n , p̄) and ρ(xn , qn ) = d(x̄n , q̄n ).
(3.8)
Let ȳn+1 := tn x̄n ⊕ (1 − tn )q̄n ∈ [x̄n , q̄n ] be the comparison point of xn+1 = tn xn ⊕ (1 − tn )qn . Then
d(ȳn+1 , x̄n ) = (1 − tn )d(x̄n , q̄n ) and
d(ȳn+1 , q̄n ) = tn d(x̄n , q̄n ).
(3.9)
Moreover, by the CAT(1) inequality, we have that
ρ(xn+1 , p) ≤ d(ȳn+1 , p̄).
(3.10)
Note that points x̄n , q̄n and ȳn+1 are located in the two-dimensional subspace of the Euclidean space R3 spanned by {x̄n , q̄n },
and that the point ȳn+1 is in the arc connecting x̄n and q̄n . Then, by elementary geometry, ȳn+1 can be expressed as
ȳn+1 = αn x̄n + βn q̄n ,
(3.11)
where αn and βn are positive numbers satisfying αn + βn ≥ 1. By the definition of the metric in (2.1),
cos d(ȳn+1 , x̄n ) = ⟨ȳn+1 |x̄n ⟩ = ⟨αn x̄n + βn q̄n |x̄n ⟩ = αn + βn cos d(x̄n , q̄n )
(3.12)
cos d(ȳn+1 , q̄n ) = ⟨ȳn+1 |q̄n ⟩ = ⟨αn x̄n + βn q̄n |q̄n ⟩ = βn + αn cos d(x̄n , q̄n ).
(3.13)
and
Combining the two equalities above and (3.9) gives that

αn + βn cos d(x̄n , q̄n ) = cos((1 − tn )d(x̄n , q̄n ))
βn + αn cos d(x̄n , q̄n ) = cos(tn d(x̄n , q̄n )).
Hence,
αn + βn =
cos(tn d(x̄n , q̄n )) + cos((1 − tn )d(x̄n , q̄n ))
1 + cos d(x̄n , q̄n )
.
(3.14)
By (2.1) and (3.11), we conclude that
cos d(ȳn+1 , p̄) = ⟨ȳn+1 |p̄⟩ = αn ⟨x̄n |p̄⟩ + βn ⟨q̄n |p̄⟩ = αn cos d(x̄n , p̄) + βn cos d(q̄n , p̄).
(3.15)
By the nonexpansivity of T and (3.8), one sees that
0 ≤ d(q̄n , p̄) = ρ(qn , p) ≤ ρ(xn , p) = d(x̄n , p̄) <
π
4
.
It follows that
√
cos d(q̄n , p̄) ≥ cos d(x̄n , p̄) ≥ cos d(x̄0 , p̄) >
2
2
.
The first inequality in (3.16) together with (3.15) entails that
cos d(ȳn+1 , p̄) ≥ (αn + βn ) cos d(x̄n , p̄).
(3.16)
Author's personal copy
J.S. He et al. / Nonlinear Analysis 75 (2012) 445–452
451
Substituting expression (3.14) of αn + βn into the above inequality, we get that
cos(tn d(x̄n , q̄n )) + cos((1 − tn )d(x̄n , q̄n ))
cos d(ȳn+1 , p̄) ≥
1 + cos d(x̄n , q̄n )
cos d(x̄n , p̄).
(3.17)
For simplicity, we write
an := d(x̄n , q̄n )
and
cos(tn an ) + cos((1 − tn )an ) − 1 − cos an
sn :=
1 + cos an
.


Then an ∈ 0, π2 by (3.7) and, by elementary trigonometry,
sn =
2 sin
· sin (1−2tn )an
tn an
2
cos
≥ 0.
an
2
(3.18)
Furthermore, by (3.8) and the definition of qn , condition (3.6) is equivalent to
lim inf an = 0.
(3.19)
n→∞
Thus, to complete the proof, we only need to show that condition (3.19) holds. For this purpose, we note that (3.8), (3.10),
(3.16) and (3.17) imply that
√
cos ρ(xn+1 , p) − cos ρ(xn , p) ≥ cos d(ȳn+1 , p̄) − cos d(x̄n , p̄) ≥ sn cos d(x̄n , p̄) >
2
2
sn .
(3.20)
It follows that
∞
−
sn ≤
∞
√ −
2
(cos ρ(xn+1 , p) − cos ρ(xn , p)) < ∞.
(3.21)
n =0
n =0
Now, we suppose on the contrary that (3.19) does not hold. Then, there exist a > 0 and integer N ∈ N such that an > a
holds for each n > N. Since each an < π2 as noted earlier, it follows from (3.18) that
sn
tn (1 − tn )
≥
· sin (1−2tn )a
,
tn (1 − tn ) cos 2a
2 sin
tn a
2
∀n > N .
(3.22)
Consider the function h on (0, 1) defined by
h(t ) :=
2 sin
· sin (1−2t )a
,
t (1 − t )
ta
2
∀t ∈ (0, 1).
Then h is positive, continuous on (0, 1), and
lim h(t ) = lim h(t ) = a sin
t →0+
t →1−
a
2
.
Hence, hmin := inft ∈(0,1) h(t ) > 0. This, together with (3.22), implies that
lim inf
n→∞
sn
tn (1 − tn )
≥
hmin
cos
a
2
> 0.
Since n=0 tn (1 − tn ) = ∞ by assumption (3.2), it follows that
holds, and the proof is complete. ∑∞
∑∞
n=0 sn
= ∞, which contradicts (3.21). Therefore, (3.19)
Recall that X is boundedly compact if each closed bounded subset of X is compact.
Corollary 3.1. Let X be a boundedly compact, complete CAT(κ ) space and let T : X → X be a nonexpansive mapping such that
F := Fix(T ) ̸= ∅. Suppose that {tn } ⊂ (0, 1) satisfies (3.2). Then, for each x0 ∈ X with ρ(x0 , F ) < D4κ , the sequence {xn } defined
by (3.1) converges strongly to a point of F .
Proof. Let {xn } be the sequence generated by (3.1) and let x0 ∈ X be such that ρ(x0 , F ) < D4κ . Then, by Theorem 3.1,
{xn }is bounded and ∆-converges to x ∈ F . Suppose on the contrary that {xn } does not converge strongly to x. Then, by the
Author's personal copy
452
J.S. He et al. / Nonlinear Analysis 75 (2012) 445–452
compactness assumption, one can find a converging subsequence {xnk } of {xn } and a point x̄ ∈ X with x̄ ̸= x such that
xnk → x̄ strongly. Therefore,
lim ρ(xnk , x̄) = 0 ≤ lim ρ(xnk , x).
k→∞
k→∞
Since x is the unique asymptotic center of {xnk }, it follows that x̄ ∈ A({xnk }) and x̄ = x, which is a contradiction. Thus we
complete the proof. The following corollary is a direct consequence of Theorem 3.1 and Corollary 3.1, which extends the corresponding ones
in [8,12].
Corollary 3.2. Let X be a complete CAT(0) space and let T : X → X be a nonexpansive mapping such that Fix(T ) ̸= ∅. Suppose
that {tn } ⊂ (0, 1) is such that (3.2) holds. Then, for each x0 ∈ X , the sequence {xn } defined by (3.1) ∆-converges to a fixed point
of T . Furthermore, if X is additionally boundedly compact, then the sequence {xn } converges strongly to a fixed point of T .
Acknowledgments
The first and the last authors’ researches were supported in part by the National Natural Science Foundation of China
(grant 11171300) and by Zhejiang Provincial Natural Science Foundation of China (grant Y6110006). The second author’s
research was supported in part by the National Natural Science Foundation of China (grant 11101186) and by the Scientific
Research Fund of Hunan Provincial Education Department (Grant 08A057). The third author’s research was supported by
DGES, Grant MTM2009-13997-C02-01 and Junta de Andaluc, Grant FQM-127.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (3) (1953) 506–510.
S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976) 65–71.
S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274–276.
H.K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66 (2002) 240–256.
K. Goebel, W.A. Kirk, Iteration processes for nonexpansive mappings, Contemp. Math. 21 (1983) 115–123.
W.A. Kirk, Krasnoselskii’s Iteration process in hyperbolic space, Numer. Funct. Anal. Optim. 4 (4) (1981–82) 371–381.
K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometric and Nonexpansive Mappings, in: Pure Appl. Math., Marcel Dekker, Inc., New York, Basel,
1984.
C. Li, G. López, V. Martín-Márquez, Iterative algorithms for nonexpansive mappings on Hadamard manifolds, Taiwanese J. Math. 14 (2) (2010) 541–559.
W.A. Kirk, Geodesic Geometry and Fixed Point Theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Univ. Sevilla Secr. Publ., Seville,
2003, pp. 195–225.
W.A. Kirk, Geodesic Geometry and Fixed Point Theory II, in: Proc. Intern. Conf. on Fixed Point Theory and Applications, Yokohama Publ., Yokohama,
2004, pp. 113–142.
T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182.
S. Dhompongsa, B. Panyanak, On ∆-convergence in CAT(0) spaces, Comput. Math. Appl. 56 (2008) 2572–2579.
M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, 1999.
W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008) 3689–3696.
R. Espínola, A. Fernández-León, CAT(κ ) spaces, weak convergence and fixed points, J. Math. Anal. Appl. 353 (2009) 410–427.
E.N. Sosov, On analogues of weak convergence in a special metric space, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (2004) 84–89. Translation in Russian
Math.(Iz. VUZ) 48 (5) (2004) 79–83.
S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990) 537–558.