Abstract
Motivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) 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.
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1. Introduction
Let be a metric space and
with
. A geodesic path from
to
is an isometry
such that
and
. The image of a geodesic path is called a geodesic segment. A metric space
is a (uniquely) geodesic space if every two points of
are joined by only one geodesic segment. A geodesic triangle
in a geodesic space
consists of three points
of
and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle
is the triangle
in the Euclidean space
such that
for all
.
A geodesic space is a CAT(0) space if for each geodesic triangle
in
and its comparison triangle
in
, the CAT(0) inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ1_HTML.gif)
is satisfied by all and
. The meaning of the CAT(0) inequality is that a geodesic triangle in
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 CAT(0) space (see [3]).
The concept of -convergence introduced by Lim in 1976 was shown by Kirk and Panyanak [4] in CAT(0) 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 CAT(0) spaces. Our results are proved under weaker assumptions as were the case in previous papers and we do not use
-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 for the the unique point
in the geodesic segment joining from
to
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ2_HTML.gif)
We also denote by the geodesic segment joining from
to
, that is,
. A subset
of a CAT(0) space is convex if
for all
. For elementary facts about CAT(0) 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 is a CAT(0) space if and only if the following inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ3_HTML.gif)
is satisfied by all and all
. In particular, if
are points in a CAT(0) space and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ4_HTML.gif)
Recall that a continuous linear functional on
, the Banach space of bounded real sequences, is called a Banach limit if
and
for all
.
Lemma 1.2 (see [9, Proposition ]).
Let be such that
for all Banach limits
and
. Then
.
Lemma 1.3 (see [10, Lemma ]).
Let be a sequence of nonnegative real numbers,
a sequence of real numbers in
with
,
a sequence of nonnegative real numbers with
, and
a sequence of real numbers with
. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ5_HTML.gif)
Then .
2. Halpern's Iteration for a Single Mapping
Lemma 2.1.
Let be a closed convex subset of a complete CAT(0) space
and let
be a nonexpansive mapping. Let
be fixed. For each
, the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ6_HTML.gif)
has a unique fixed point , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ7_HTML.gif)
Proof.
For , we consider the triangle
and its comparison triangle and we have the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ8_HTML.gif)
This implies that is a contraction mapping and hence the conclusion follows.
The following result is proved by Kirk in [11, Theorem ] under the boundedness assumption on
. We present here a new proof which is modified from Kirk's proof.
Lemma 2.2.
Let ,
be as the preceding lemma. Then
if and only if
given by the formula (2.2) remains bounded as
. In this case, the following statements hold:
(1) converges to the unique fixed point
of
which is nearest
;
(2) for all Banach limits
and all bounded sequences
with
.
Proof.
If , then it is clear that
is bounded. Conversely, suppose that
is bounded. Let
be any sequence in
such that
and define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ9_HTML.gif)
for all . By the boundedness of
, we have
. We choose a sequence
in
such that
. It follows from Lemma 1.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ10_HTML.gif)
Then, by the convexity of ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ11_HTML.gif)
This implies that is a Cauchy sequence in
and hence it converges to a point
. Suppose that
is a point in
satisfying
. It follows then that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ12_HTML.gif)
and hence . Moreover,
is a fixed point of
. To see this, we consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ13_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ14_HTML.gif)
This implies that and hence
.
-
(1)
is proved in [12, Theorem
]. In fact, it is shown that
is the nearest point of
to
. Finally, we prove (2). Suppose that
is a sequence given by the formula (2.2), where
is a sequence in
such that
. We also assume that
is the nearest point of
to
. By the first inequality in Lemma 1.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ15_HTML.gif)
Let be a Banach limit. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ16_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ17_HTML.gif)
Letting gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ18_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ19_HTML.gif)
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 CAT(0) space setting. A part of the following theorem is proved in [14].
Theorem 2.3.
Let be a closed convex subset of a complete CAT(0) space
and let
be a nonexpansive mapping with a nonempty fixed point set
. Suppose that
are arbitrarily chosen and
is iteratively generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ20_HTML.gif)
where is a sequence in
satisfying
(C1);
(C2);
(C3) or
.
Then converges to
which is the nearest point of
to
.
Proof.
We first show that the sequence is bounded. Let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ21_HTML.gif)
By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ22_HTML.gif)
for all . This implies that
is bounded and so is the sequence
.
Next, we show that . To see this, we consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ23_HTML.gif)
By the conditions (C2) and (C3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ24_HTML.gif)
Consequently, by the condition (C1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ25_HTML.gif)
From Lemma 2.2, let where
is given by the formula (2.2). Then
is the nearest point of
to
. We next consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ26_HTML.gif)
By Lemma 2.2, we have for all Banach limits
. Moreover, since
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ27_HTML.gif)
It follows from and Lemma 1.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ28_HTML.gif)
Hence the conclusion follows by Lemma 1.3.
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 be a complete CAT(0) space and
a closed convex subset of
. Let
be nonexpansive mappings with
and let
be arbitrarily chosen. Define an iterative sequence
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ29_HTML.gif)
where is a sequence in
satisfying
(C1);
(C2);
(C3) or
.
Suppose, in addition, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ30_HTML.gif)
Then converges to
which is nearest
.
Here the function takes values in
.
Proof.
By [16, Theorem ], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ31_HTML.gif)
The proof line now follows from the proofs of Theorem 2.3 and [15, Theorem ].
3.2. Countable Mappings
The following concept is introduced by Aoyama et al. [10]. Let be a complete CAT(0) space and
a subset of
. Let
be a countable family of mappings from
into itself. We say that a family
satisfies AKTT-condition if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ32_HTML.gif)
for each bounded subset of of
.
If is a closed subset and
satisfies AKTT-condition, then we can define
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ33_HTML.gif)
In this case, we also say that satisfies AKTT-condition.
Theorem 3.2.
Let be a complete CAT(0) space and
a closed convex subset of
. Let
be a countable family of nonexpansive mappings with
. Suppose that
are arbitrarily chosen and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ34_HTML.gif)
where is a sequence in
satisfying
(C1);
(C2);
(C3) or
.
Suppose, in addition, that
(M1) satisfies AKTT-condition;
(M2).
Then converges to
which is nearest
.
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 and
are bounded and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ35_HTML.gif)
By conditions (C2), (C3), AKTT-condition, and Lemma 1.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ36_HTML.gif)
Consequently, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ37_HTML.gif)
Let be the nearest point of
to
. As in the proof of Theorem 2.3, we have
for all Banach limits
and
. We observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ38_HTML.gif)
and this implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ39_HTML.gif)
Therefore, and hence
converges to
.
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 CAT(0) space setting. The idea using here is from [10].
Theorem 3.3.
Let be a complete CAT(0) space and
a closed convex subset of
. Suppose that
is a countable family of nonexpansive mappings with
. Then there exist a family of nonexpansive mappings
and a nonexpansive mapping
such that
(M1) satisfies AKTT-condition;
(M2).
Lemma 3.4.
Let and
be as above. Suppose that
are nonexpansive mappings and
. Then, for any
, the mapping
is nonexpansive and
.
Proof.
To see that 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
. To see the reverse inclusion, let
and
. Then, by the first inequality of Lemma 1.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ40_HTML.gif)
This implies . As
, we have
, as desired.
Proof of Theorem 3.3.
We first define a family of mappings by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ41_HTML.gif)
By Lemma 3.4, each is a nonexpansive mapping satisfying
. Notice that, for fixed
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ42_HTML.gif)
From the estimation above, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ43_HTML.gif)
for each bounded subset of
. In particular,
is a Cauchy sequence for each
. We now define the nonexpansive mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ44_HTML.gif)
Finally, we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ45_HTML.gif)
The latter equality is clearly verified and holds. On the other hand, let
and
. We consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ46_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ47_HTML.gif)
Letting yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ48_HTML.gif)
Because , we have
. Continuing this procedure we obtain that
and hence
. This completes the proof.
4. Nonself Mappings
From Bridson and Haefliger's book (page 176), the following result is proved.
Theorem 4.1.
Let be a complete CAT(0) space and
a closed convex subset of
. Then the followings hold true.
(i)For each , there exists an element
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ49_HTML.gif)
(ii) for all
.
(iii)The mapping is nonexpansive.
The mapping in the preceding theorem is called the metric projection from
onto
. From this, we have the following result.
Theorem 4.2.
Let be a complete CAT(0) space and
a closed convex subset of
. Let
be a nonself nonexpansive mapping with
and
the metric projection from
onto
. Then the mapping
is nonexpansive and
.
Proof.
It follows from Theorem 4.1 that is nonexpansive. To see the latter, it suffices to show that
. Let
and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ50_HTML.gif)
we have and this finishes the proof.
By the preceding theorem and Theorem 2.3, we obtain the following result.
Theorem 4.3.
Let ,
,
, and
be as the same as Theorem 4.2. Suppose that
are arbitrarily chosen and the sequence
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F471781/MediaObjects/13663_2009_Article_1290_Equ51_HTML.gif)
where is a sequence in
satisfying
(C1);
(C2);
(C3) or
.
Then converges to
which is nearest
.
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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.
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Saejung, S. Halpern's Iteration in CAT(0) Spaces. Fixed Point Theory Appl 2010, 471781 (2009). https://doi.org/10.1155/2010/471781
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DOI: https://doi.org/10.1155/2010/471781