- Research Article
- Open access
- Published:
Strong and
Convergence Theorems for Multivalued Mappings in
Spaces
Journal of Inequalities and Applications volume 2009, Article number: 730132 (2009)
Abstract
We show strong and convergence for Mann iteration of a multivalued nonexpansive mapping whose domain is a nonempty closed convex subset of a CAT(0) space. The results we obtain are analogs of Banach space results by Song and Wang [2009, 2008]. Strong convergence of Ishikawa iteration are also included.
1. Introduction
Let be a nonempty subset of a Banach space
We shall denote by
the family of nonempty closed bounded subsets of
by
the family of nonempty bounded proximinal subsets of
and by
the family of nonempty compact subsets of
. Let
be the Hausdorff distance on
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ1_HTML.gif)
where is the distance from the point
to the set
A multivalued mapping is said to be a nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ2_HTML.gif)
A point is called a fixed point of
if
We denote by
the set of all fixed points of
In 2005, Sastry and Babu [1] introduced the Mann and Ishikawa iterations for multivalued mappings as follows: let be a real Hilbert space and
be a multivalued mapping for which
. Fix
and define
-
(A)
the sequence of Mann iterates by
(1.3)where
is such that
-
(B)
the sequence of Ishikawa iterates by
(1.4)
where is such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ5_HTML.gif)
where is such that
They proved the following results.
Theorem 1.1.
Let be a nonempty compact convex subset of a Hilbert space
Suppose
is nonexpansive and has a fixed point
Assume that (i)
and (ii)
Then the sequence of Mann iterates defined by (A) converges to a fixed point
of
Theorem 1.2.
Let be a nonempty compact convex subset of a Hilbert space
Suppose that a nonexpansive map
has a fixed point
Assume that (i)
(ii)
and (iii)
Then the sequence of Ishikawa iterates defined by (B) converges to a fixed point
of
In 2007, Panyanak [2] extended Sastry-Babu's results to uniformly convex Banach spaces as the following results.
Theorem 1.3.
Let be a nonempty compact convex subset of a uniformly convex Banach spaces
Suppose that a nonexpansive map
has a fixed point
Let
be the sequence of Mann iterates defined by (A). Assume that (i)
and (ii)
Then the sequence
converges to a fixed point of
Theorem 1.4.
Let be a nonempty compact convex subset of a uniformly convex Banach spaces
Suppose that a nonexpansive map
has a fixed point
. Let
be the sequence of Ishikawa iterates defined by (B). Assume that (i)
(ii)
and (iii)
Then the sequence
converges to a fixed point of
Recently, Song and Wang [3, 4] pointed out that the proof of Theorem 1.4 contains a gap. Namely, the iterative sequence defined by (B) depends on the fixed point
Clearly, if
and
then the sequence
defined by
is different from the one defined by
Thus, for
defined by
, we cannot obtain that
is a decreasing sequence from the monotony of
. Hence, the conclusion of Theorem 1.4 (also Theorem 1.3) is very dubious.
Motivated by solving the above gap, they defined the modified Mann and Ishikawa iterations as follows.
Let be a nonempty convex subset of a Banach space
and
be a multivalued mapping. The sequence of Mann iterates is defined as follows: let
and
such that
Choose
and
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ6_HTML.gif)
There exists such that
(see [5, 6]). Take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ7_HTML.gif)
Inductively, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ8_HTML.gif)
where such that
The sequence of Ishikawa iterates is defined as follows: let ,
and
such that
Choose
and
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ9_HTML.gif)
There exists such that
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ10_HTML.gif)
There is such that
Take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ11_HTML.gif)
There exists such that
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ12_HTML.gif)
Inductively, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ13_HTML.gif)
where and
such that
and
They obtained the following results.
Theorem 1.5 (see [3, Theorem 2.3]).
Let be a nonempty compact convex subset of a Banach space
Suppose that
is a multivalued nonexpansive mapping for which
and
for each
Let
be the sequence of Mann iteration defined by (1.8). Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ14_HTML.gif)
Then the sequence strongly converges to a fixed point of
Recall that a multivalued mapping is said to satisfy Condition I ([7]) if there exists a nondecreasing function
with
and
for all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ15_HTML.gif)
Theorem 1.6 (see [3, Theorem 2.4]).
Let be a nonempty closed convex subset of a Banach space
Suppose that
is a multivalued nonexpansive mapping that satisfies Condition I. Let
be the sequence of Mann iteration defined by (1.8). Assume that
and satisfies
for each
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ16_HTML.gif)
Then the sequence strongly converges to a fixed point of
Theorem 1.7 (see [3, Theorem 2.5]).
Let be a Banach space satisfying Opial's condition and
be a nonempty weakly compact convex subset of
Suppose that
is a multivalued nonexpansive mapping. Let
be the sequence of Mann iteration defined by (1.8). Assume that
and satisfies
for each
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ17_HTML.gif)
Then the sequence weakly converges to a fixed point of
Theorem 1.8 (see [4, Theorem  1]).
Let be a nonempty compact convex subset of a uniformly convex Banach space
Suppose that
is a multivalued nonexpansive mapping and
satisfying
for any fixed point
Let
be the sequence of Ishikawa iterates defined by (1.13). Assume that (i)
  (ii)
and (iii)
Then the sequence
strongly converges to a fixed point of
Theorem 1.9 (see [4, Theorem  2]).
Let be a nonempty closed convex subset of a uniformly convex Banach space
Suppose that
is a multivalued nonexpansive mapping that satisfy Condition I. Let
be the sequence of Ishikawa iterates defined by (1.13). Assume that
satisfying
for any fixed point
and
Then the sequence
strongly converges to a fixed point of
In this paper, we study the iteration processes defined by (1.8) and (1.13) in a CAT(0) space and give analogs of Theorems 1.5–1.9 in this setting.
2.
Spaces
A metric space is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in
is at least as "thin" as its comparison triangle in the Euclidean plane. The precise definition is given below. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT
space. Other examples include Pre-Hilbert spaces,
-trees (see [8]), Euclidean buildings (see [9]), the complex Hilbert ball with a hyperbolic metric (see [10]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry (see Bridson and Haefliger [8]). Burago, et al. [11] contains a somewhat more elementary treatment, and Gromov [12] a deeper study.
Fixed point theory in a CAT(0) space was first studied by Kirk (see [13] and [14]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed and many of papers have appeared (see, e.g., [15–24]). It is worth mentioning that the results in CAT(0) spaces can be applied to any CAT() space with
since any CAT(
) space is a CAT(
) space for every
(see [8], page 165).
Let be a metric space. A geodesic path joining
to
(or, more briefly, a geodesic from
to
) is a map
from a closed interval
to
such that
and
for all
In particular,
is an isometry and
The image
of
is called a geodesic (or metric) segment joining
and
. When it is unique this geodesic is denoted by
. The space
is said to be a geodesic space if every two points of
are joined by a geodesic, and
is said to be uniquely geodesic if there is exactly one geodesic joining
and
for each
A subset
is said to be convex if
includes every geodesic segment joining any two of its points.
A geodesic triangle in a geodesic space
consists of three points
in
(the vertices of
) and a geodesic segment between each pair of vertices (the edges of
). A comparison triangle for geodesic triangle
in
is a triangle
in the Euclidean plane
such that
for
A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom.
CAT(0): let be a geodesic triangle in
and let
be a comparison triangle for
. Then
is said to satisfy the CAT(0) inequality if for all
and all comparison points
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ18_HTML.gif)
Let by [24, Lemma 2.1(iv)] for each
there exists a unique point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ19_HTML.gif)
From now on we will use the notation for the unique point
satisfying (2.2). By using this notation Dhompongsa and Panyanak [24] obtained the following lemma which will be used frequently in the proof of our main theorems.
Lemma 2.1.
Let be a CAT (0)space . Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ20_HTML.gif)
for all and
If are points in a CAT(0) space and if
then the CAT(0) inequality implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ21_HTML.gif)
This is the (CN) inequality of Bruhat and Tits [25]. In fact (cf. [8, page 163]), a geodesic metric space is a CAT(0) space if and only if it satisfies (CN).
The following lemma is a generalization of the (CN) inequality which can be found in [24].
Lemma 2.2.
Let be a CAT(0) space. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ22_HTML.gif)
for all and
The preceding facts yield the following result.
Proposition 2.3.
Let be a geodesic space. Then the following are equivalent:
-
(i)
is a CAT (0) space;
-
(ii)
satisfies (CN);
-
(iii)
satisfies (2.5).
The existence of fixed points for multivalued nonexpansive mappings in a CAT(0) space was proved by S. Dhompongsa et al. [17], as follows.
Theorem 2.4.
Let be a closed convex subset of a complete CAT(0) space
, and let
be a nonexpansive nonself-mapping. Suppose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ23_HTML.gif)
for some bounded sequence in
Then
has a fixed point.
3. The Setting
Let be a Banach space, and let
be a bounded sequence in
for
we let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ24_HTML.gif)
The asymptotic radius of
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ25_HTML.gif)
and the asymptotic center of
is the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ26_HTML.gif)
The notion of asymptotic centers in a Banach space can be extended to a CAT(0) space
as well, simply replacing
with
It is known (see, e.g., [18, Proposition 7]) that in a CAT(0) space,
consists of exactly one point.
Next we provide the definition and collect some basic properties of -convergence.
Definition 3.1 (see [23]).
A sequence in a CAT(0) space
is said to
-converge to
if
is the unique asymptotic center of
for every subsequence
of
. In this case one must write
and call
the
-limit of
Remark 3.2.
In a CAT(0) space , strong convergence implies
convergence and they are coincided when
is a Hilbert space. Indeed, we prove a much more general result. Recall that a Banach space is said to satisfy Opial's condition ([26]) if given whenever
converges weakly to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ27_HTML.gif)
Proposition 3.3.
Let be a reflexive Banach space satisfying Opial's condition and let
be a bounded sequence in
and let
Then
converges weakly to
if and only if
for all subsequence
of
Proof.
() Let
be a subsequence of
. Then
converges weakly to
By Opial's condition
(
) Suppose
for all subsequence
of
and assume that
does not converge weakly to
Then there exists a subsequence
of
such that for each
is outside a weak neighborhood of
Since
is bounded, without loss of generality we may assume that
converges weakly to
By Opial's condition
a contradiction.
Lemma 3.4.
(i) Every bounded sequence in has a
convergent subsequence (see [23, page 3690]). (ii) If
is a closed convex subset of
and if
is a bounded sequence in
then the asymptotic center of
is in
(see [17, Proposition 2.1]).
Now, we define the sequences of Mann and Ishikawa iterates in a CAT(0) space which are analogs of the two defined in Banach spaces by Song and Wang [3, 4].
Definition 3.5.
Let be a nonempty convex subset of a CAT(0) space
and
be a multivalued mapping. The sequence of Mann iterates is defined as follows: let
and
such that
Choose
and
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ28_HTML.gif)
There exists such that
Take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ29_HTML.gif)
Inductively, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ30_HTML.gif)
where such that
Definition 3.6.
Let be a nonempty convex subset of a CAT(0) space
and
be a multivalued mapping. The sequence of Ishikawa iterates is defined as follows: let
,
and
such that
Choose
and
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ31_HTML.gif)
There exists such that
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ32_HTML.gif)
There is such that
Take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ33_HTML.gif)
There exists such that
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ34_HTML.gif)
Inductively, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ35_HTML.gif)
where and
such that
and
Lemma 3.7.
Let be a nonempty compact convex subset of a complete CAT (0) space
and let
be a nonexpansive nonself-mapping. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ36_HTML.gif)
for some sequence in
Then
has a fixed point. Moreover, if
converges for each
, then
strongly converges to a fixed point of
Proof.
By the compactness of there exists a subsequence
of
such that
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ37_HTML.gif)
This implies that is a fixed point of
Since the limit of
exists and
we have
This show that the sequence
strongly converges to
Before proving our main results we state a lemma which is an analog of Lemma 2.2 of [27]. The proof is metric in nature and carries over to the present setting without change.
Lemma 3.8.
Let and
be bounded sequences in a CAT (0)space
and let
be a sequence in
with
Suppose that
for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ38_HTML.gif)
Then
4. Strong and
Convergence of Mann Iteration
Theorem 4.1.
Let be a nonempty compact convex subset of a complete CAT (0)space
Suppose that
is a multivalued nonexpansive mapping and
satisfying
for any fixed point
If
is the sequence of Mann iterates defined by (3.7) such that one of the following two conditions is satisfied:
-
(i)
and
-
(ii)
Then the sequence strongly converges to a fixed point of
Proof
Case 1.
Suppose that (i) is satisfied. Let by Lemma 2.2 and the nonexpansiveness of
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ39_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ41_HTML.gif)
It follows from (4.2) that for all
This implies that
is bounded and decreasing. Hence
exists for all
On the other hand, (4.3) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ42_HTML.gif)
Since diverges, we have
and hence
Then there exists a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ43_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ44_HTML.gif)
By Lemma 3.7, converges to a point
Since the limit of
exists, it must be the case that
and hence the conclusion follows.
Case 2.
If (ii) is satisfied. As in the Case 1, exists for each
It follows from the definition of Mann iteration (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ45_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ46_HTML.gif)
By Lemma 3.8, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ47_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ48_HTML.gif)
so the conclusion follows from Lemma 3.7.
Theorem 4.2.
Let be a nonempty closed convex subset of a complete CAT(0) space
Suppose that
is a multivalued nonexpansive mapping that satisfies Condition I. Let
be the sequence of Mann iterates defined by (3.7). Assume that
satisfying
for any fixed point
and
Then the sequence
strongly converges to a fixed point of
Proof.
It follows from the proof of the Case 1 in Theorem 4.1 that exists for each
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ49_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ50_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ51_HTML.gif)
Thus, and hence
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ52_HTML.gif)
Therefore, Furthermore Condition I implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ53_HTML.gif)
The proof of remaining part closely follows the proof of of [2, Theorem 3.8], simply replacing with
.
Next we show a convergence theorem of Mann iteration in a CAT(0) space setting which is an analog of Theorem 1.7. For this we need more lemmas.
Lemma 4.3 (see [24, Lemma  2.8]).
If is a bounded sequence in a complete CAT (0)space
with
and
is a subsequence of
with
and the sequence
converges, then
Lemma 4.4.
Let be a nonempty closed convex subset of a complete CAT (0) space
and let
be a nonexpansive nonself-mapping. Suppose that
is a sequence in
which
converges to
in
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ54_HTML.gif)
Then
Proof.
Notice from Lemma 3.4(ii) that Since
is compact-valued, for each
there exists
and
such that
and
It follows from (4.16) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ55_HTML.gif)
By the compactness of there exists a subsequence
of
such that
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ56_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ57_HTML.gif)
Since and hence
by (4.19). Therefore
is a fixed point of
Lemma 4.5.
Let be a closed convex subset of a complete CAT (0)space
and let
be a nonexpansive mapping. Suppose
is a bounded sequence in
such that
and
converges for all
then
Here
where the union is taken over all subsequences
of
Moreover,
consists of exactly one point.
Proof.
Let , then there exists a subsequence
of
such that
By Lemma 3.4(i) and (ii) there exists a subsequence
of
such that
. By Lemma 4.4,
. By Lemma 4.3,
This shows that
Next, we show that
consists of exactly one point. Let
be a subsequence of
with
and let
. Since
is convergent by the assumption. By Lemma 4.3,
This completes the proof.
Theorem 4.6.
Let be a nonempty closed convex subset of a complete CAT (0) space
Suppose that
is a multivalued nonexpansive mapping. Let
be the sequence of Mann iterates defined by (3.7). Assume that
satisfying
for any fixed point
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ58_HTML.gif)
Then the sequence converges to a fixed point of
Proof.
Let it follows from (4.2) in the proof of Theorem 4.1 that
for all
This implies that
is bounded and decreasing. Hence
exists for all
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ59_HTML.gif)
Thus by (4.9). By Lemma 4.5,
consists of exactly one point and is contained in
. This shows that
converges to an element of
5. Strong Convergence of Ishikawa Iteration
The following lemma can be found in [2].
Lemma 5.1.
Let be two real sequences such that
-
(i)
-
(ii)
as
-
(iii)
Let be a nonnegative real sequence such that
is bounded. Then
has a subsequence which converges to zero.
The following theorem is an analog of Theorem 1.8.
Theorem 5.2.
Let be a nonempty compact convex subset of a complete CAT (0) space
Suppose that
is a multivalued nonexpansive mapping and
satisfying
for any fixed point
Let
be the sequence of Ishikawa iterates defined by (3.12). Assume that
-
(i)
-
(ii)
-
(iii)
Then the sequence strongly converges to a fixed point of
Proof.
Let by Lemma 2.2 and the nonexpansiveness of
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ60_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ61_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ62_HTML.gif)
It follows from (5.2) that the sequence is decreasing and hence
exists for each
On the other hand, (5.3) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ63_HTML.gif)
By Lemma 5.1, there exists a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ64_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ65_HTML.gif)
By Lemma 3.7, converges to a point
Since the limit of
exists, it must be the case that
and hence the conclusion follows.
The following theorem is an analog of Theorem 1.9.
Theorem 5.3.
Let be a nonempty closed convex subset of a complete CAT(0) space
Suppose that
is a multivalued nonexpansive mapping that satisfies Condition I. Let
be the sequence of Ishikawa iterates defined by (3.12). Assume that
satisfying
for any fixed point
and
Then the sequence
strongly converges to a fixed point of
Proof.
Similar to the proof of Theorem 5.2, we obtain exists for each
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ66_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ67_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ68_HTML.gif)
Thus, and hence
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ69_HTML.gif)
Therefore, Furthermore Condition I implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F730132/MediaObjects/13660_2008_Article_1999_Equ70_HTML.gif)
The proof of remaining part closely follows the proof of [2, Theorem 3.8], simply replacing with
.
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Acknowledgments
We are grateful to Professor Sompong Dhompongsa for his suggestion and advice during the preparation of the article. The research was supported by the Commission on Higher Education and Thailand Research Fund under Grant MRG5080188.
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Laowang, W., Panyanak, B. Strong and Convergence Theorems for Multivalued Mappings in
Spaces.
J Inequal Appl 2009, 730132 (2009). https://doi.org/10.1155/2009/730132
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DOI: https://doi.org/10.1155/2009/730132