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Fixed Point Theorems for Random Lowersemi-continuous Mappings
Fixed Point Theory and Applications volume 2009, Article number: 584178 (2009)
Abstract
We prove a general principle in Random Fixed Point Theory by introducing a condition named () which was inspired by some of Petryshyn's work, and then we apply our result to prove some random fixed points theorems, including generalizations of some Bharucha-Reid theorems.
1. Introduction
Let be a metric space and
a closed and nonempty subset of
. Denote by
(resp.,
) the family of all nonempty (resp., nonempty and closed) subsets of
. A mapping
is said to satisfy
if, for every closed ball
of
with radius
and any sequence
in
for which
and
as
, there exists
such that
where
. If
is any nonempty set, we say that the operator
satisfies
if for each
, the mapping
satisfies
. We should observe that this latter condition is related to a condition that was originally introduced by Petryshyn [1] for single-valued operators, in order to prove existence of fixed points. However, in our case, the condition is used to prove the measurability of a certain operator. On the other hand, in the year 2001, Shahzad (cf. [2]) using an idea of Itoh (cf. [3]), see also ([4]), proved that under a somewhat more restrictive condition, named condition (A), the following result.
Theorem 1 S.
Let be a nonempty separable complete subset of a metric space
and
a continuous random operator satisfying condition (A). Then
has a deterministic fixed point if and only if
has a random fixed point.
We shall show that the above result is still valid if the operator is only lower semi-continuous. In addition, the assumption that each value
is closed has been relaxed without an extra assumption. Furthermore we state a new condition which generalizes condition (A) and allow us to generalize several known results, such as, Bharucha-Reid [5, Theorem 7], Domínguez Benavides et al. [6, Theorem 3.1] and Shahzad [2, Theorem 2.1].
2. Preliminaries
Let be a measurable space and let
be a metric space. A mapping
, is said to be measurable if
is measurable for each open subset
of
. This type of measurability is usually called weakly (cf. [7]), but since this is the only type of measurability we use in this paper, we omit the term "weakly". Notice that if
is separable and if, for each closed subset
of
, the set
is measurable, then
is measurable.
Let be a nonempty subset of
and
, then we say that
is lower (upper) semi-continuous if
is open (closed) for all open (closed) subsets
of
. We say that
is continuous if
is lower and upper semi-continuous.
A mapping is called a random operator if, for each
, the mapping
is measurable. Similarly a multivalued mapping
is also called a random operator if, for each
,
is measurable. A measurable mapping
is called a measurable selection of the operator
if
for each
. A measurable mapping
is called a random fixed point of the random operator
(or
) if for every
(or
). For the sake of clarity, we mention that
Let be a closed subset of the Banach space
, and suppose that
is a mapping from
into the topological vector space
. We say the
is demiclosed at
if, for any sequences
in
and
in
with
,
converges weakly to
and
converges strongly to
, then it is the case that
and
. On the other hand, we say that
is hemicompact if each sequence
in
has a convergent subsequence, whenever
as
.
3. Main Results
Theorem 3.1.
Let be a closed separable subset of a complete metric space
, and let
be measurable in
and enjoy
. Suppose, for each
, that
is upper semi-continuous and the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ1_HTML.gif)
Then has a random fixed point.
Proof.
Let be a countable dense subset of
. Define
by
. Firstly, we show that
is measurable. To this end, let
be an arbitrary closed ball of
, and set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ2_HTML.gif)
where and
. We claim that
. To see this, let
. Then there exists
such that
. Since
is upper semi-continuous, for each
, there exists
such that
. Therefore
. On the other hand, if
, then there exists a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ3_HTML.gif)
for all . This means that
and
as
. Consequently, by
, there exists
such that
. Hence
. Then we conclude that
, and thus
is measurable. To complete the proof, let
be an arbitrary open subset of
. Then by the separability of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ4_HTML.gif)
Since , we conclude that
is measurable. Additionally, we show that
is closed for each
. To see this, let
such that
. Then, let
be a degenerated ball centered at
and radius
, and since
,
implies that
. Hence
and thus by the Kuratowski and Ryll-Nardzewski Theorem [8],
has a measurable selection
such that
for each
.
As a consequence of Theorem 3.1, we derive a new result for a lower semi-continuous random operator.
Theorem 3.2.
Let be a closed separable subset of a complete metric space
, and let
be a lower semi-continuous random operator, which enjoys
. Suppose, for each
, that the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ5_HTML.gif)
Then has a random fixed point.
Proof.
Due to Theorem 3.1, it is enough to show that is upper semi-continuous. To see this, we will prove that
is open in
for
. Let
and select
. Then there exists
so that
. Since
is lower semi-continuous, there exists a positive number
such that
for all
. Hence, we may choose
for which,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ6_HTML.gif)
and consequently, . Therefore,
is open, and proof is complete.
We observe that if the mapping is upper semi-continuous, then not necessarily the mapping
is lower semi-continuous. Consider the following example.
Let be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ7_HTML.gif)
Then for
while
, which is upper semi-continuous. On the other hand,
is not lower semi-continuous.
Now, we derive several consequences of Theorem 3.2. We first obtain an extension of one of the main results of [6].
Theorem 3.3.
Let be a weakly compact separable subset of a Banach space
, and let
be a lower semi-continuous random operator. Suppose, for each
, that
is demiclosed at
and the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ8_HTML.gif)
Then has a random fixed point.
Proof.
In order to apply Theorem 3.2, we just need to prove that enjoys
. To this end, let
be fixed in
. Suppose that
is a closed ball of
with radius
where
is a sequence in
such that
and
as
. Since
is separable, the weak topology on
is metrizable, and thus there exists a weakly convergent subsequence
of
, so that
weakly, while
as
. Consequently, for each
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ9_HTML.gif)
Hence, the demiclosedness of implies that
, and thus
enjoys
.
Before we give an extension of the main result of [4], we observe that is basically applied to those closed balls directly used to prove the measurability of the mapping
, as will be seen in the proof of the next result.
Theorem 3.4.
Let be a closed separable subset of a complete metric space
, and let
be a continuous hemicompact random operator. If, for each
, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ10_HTML.gif)
then has a random fixed point.
Proof.
Due to Theorem 3.2, it would be enough to show that enjoys
for every
. To see this, let
be a closed ball of
, and let
be a sequence in
such that
and
as
. Then by the hemicompactness of
, there exists a convergent subsequence
of
, so that
. Hence
as
. This means that, for each
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ11_HTML.gif)
Consequently, . On the other hand, since
is upper semi-continuous at
, for every
there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ12_HTML.gif)
Hence, . Since
is arbitrary and
is closed, we derive that
, and thus
satisfies
.
Corollary 3.5.
Let be a locally compact separable subset of a complete metric space
, and let
be a continuous random operator. Suppose, for each
, that the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ13_HTML.gif)
Then has a random fixed point.
Proof.
Let be an arbitrary open subset of
, and let
. Since
is locally compact, there exists a compact ball
centered at
such that
. Now, we prove that
holds with respect to
. To see this, let
, and let
be a sequence in
such that
and
as
. Then there exists a sequence
in
so that
as
. Since
is compact, there exists a convergent subsequence
of
such that
, and consequently
with
as well as
as
. Since
is upper semi-continuous, we derive, as in the proof of Theorem 3.4, that
. In addition, since
is lower semi-continuous, we may follow the proof of Theorem 3.1, to conclude that
is measurable. Hence, the separability of
implies that we can select countably many compact balls
centered at corresponding points
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ14_HTML.gif)
Therefore, is measurable.
Next, we get a stochastic version of Schauder's Theorem, which is also an extension of a Theorem of Bharucha-Reid (see [5, Theorem 10]). We also observe that our proof is much easier and quite short.
Corollary 3.6.
Let be a compact and convex subset of a Fréchet space
, and let
be a continuous random operator. Then
has a random fixed point.
Proof.
As we know, every Fréchet space is a complete metric space, and since is compact,
itself is a complete separable metric space. In addition, for each
, there exists
such that
. This means that the set
, defined in Theorem 3.1, is nonempty. Since
is compact, any sequence in
contains a convergent subsequence, which means that
is trivially a hemicompact operator. Consequently, by Theorem 3.4,
has a random fixed point.
Before obtaining an extension of Bharucha-Reid [5, Theorem 3.7], we define a contraction mapping for metric spaces. Let be a metric space, and let
be a measurable space. A random operator
is said to be a random contraction if there exists a mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ15_HTML.gif)
Theorem 3.7.
Let be a complete separable metric space, and let
be a continuous random operator such that
is a contraction with constant
for each
. Then
has a unique random fixed point.
Proof.
For each , the mapping
has a unique fixed point,
, which is also the unique fixed point of
. It remains to show that the mapping
defined by
is measurable. To see this, let
be an arbitrary measurable function. Then, we claim that
is measurable. To this end, let
be a countable dense set of
. Let
and let
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ16_HTML.gif)
where is the smallest natural number for which
. Since
is measurable, so are the sets
, which, as a matter of fact, conform a disjoint covering of
. Consequently,
is a sequence of measurable functions that converges pointwise to
. On the other hand, the range of each
is a subset of
, and hence constant on each set
. Since the mapping
is measurable in
, then, for each
,
is also measurable. Therefore the continuity of
on the second variable implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ17_HTML.gif)
for each . Hence
is measurable. Define the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F584178/MediaObjects/13663_2009_Article_1158_Equ18_HTML.gif)
Then is a sequence of measurable functions. Since
, the fact that
is a contraction implies that
. Therefore, the mapping
is measurable, which completes the proof.
As a direct consequence of Theorem 3.7, we derive the extension mentioned earlier where the space is more general, and the randomness on the mapping
has been removed.
Corollary 3.8.
Let be a complete separable metric space, and let
be a random contraction operator with constant
for each
. Then
has a unique random fixed point.
Next, one can derive a corollary of the proof of Theorem 3.7, which is a theorem of Hans [9].
Corollary 3.9.
Let be a complete separable metric space, and let
be a continuous random operator. Suppose, for each
, that there exists
such that
is a contraction with constant
. Then
has a unique random fixed point.
Proof.
As in the proof of the theorem, the mapping has a unique fixed point for each
. The rest of the proof follows the proof of the theorem with the appropriate changes of the second power of
by the
power of
.
Notice that Theorem 3.7 holds for single-valued operators. The following question is formulated for multivalued operators taking closed and bounded values in .
Open Question
Suppose that is a complete separable metric space, and let
be a continuous random operator such that
is a contraction with constant
for each
. Then does
have a unique random fixed point?
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Acknowledgments
This work was partially supported by Dirección de Investigación e Innovación de la Pontificia Universidad Católica de Valparaíso under grant 124.719/2009. In addition, the first author was supported by Laboratory of Stochastic Analysis PBCT-ACT 13.
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Fierro, R., Martínez, C. & Morales, C.H. Fixed Point Theorems for Random Lowersemi-continuous Mappings. Fixed Point Theory Appl 2009, 584178 (2009). https://doi.org/10.1155/2009/584178
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DOI: https://doi.org/10.1155/2009/584178