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Random Periodic Point and Fixed Point Results for Random Monotone Mappings in Ordered Polish Spaces
Fixed Point Theory and Applications volume 2010, Article number: 723216 (2010)
Abstract
The measurability of order continuous random mappings in ordered Polish spaces is studied. Using order continuity, some random fixed point theorems and random periodic point theorems for increasing, decreasing, and mixed monotone random mappings are presented.
1. Introduction and Preliminaries
The study of random fixed points forms a central topic in probabilistic functional analysis. It was initiated by Špaček [1], Hanš [2], and Wang [3]. Some random fixed point theorems play an important role in the theory of random differential and random integral equations (see Bharucha-Reid [4, 5]). Since the recent 30 years, many interesting random fixed point theorems and applications have been developed, for example, see Beg and Shahzad [6, 7], Beg and Abbas [8], Chang [9], Ding [10], Fierro et al. [11], Itoh [12], Li and Duan [13], O'Regan et al. [14], Xiao and Tao [15], Xu [16], and Zhu and Xu [17].
In 1976, Caristi [18] introduced a partial ordering in metric spaces by a function and proved the famous Caristi fixed point theorem, which is one of the most important results in nonlinear analysis. From then on, there appeared many papers concerning fixed point theory and abstract monotone iterative technique in ordered metric spaces or ordered Banach spaces. In particular, some useful fixed point theorems for monotone mappings were proved by Zhang [19], Guo and Lakshmikantham [20], and Bhaskar and Lakshmikantham [21] under some weak assumptions.
In this paper, motivated by ideas in [18–21], we study random version of fixed point theorems for increasing, decreasing, and mixed monotone random mappings in ordered Polish spaces. In Section 2, we introduce order continuous random mapping and discuss its measurability. A well-known result is generalized (see Remark 2.4). In Sections 3–5, we present some existence results of random periodic point and fixed point for increasing, decreasing and, mixed monotone random mappings, respectively.
We begin with some definitions that are essential for this work. Let be a metric space and
be a Borel algebra of
, where
is a metric function on
. If
is separable and complete, then
is called a Polish space. We denote by
a complete probability measure space (briefly, a measure space), where
is a measurable space,
is a sigma algebra of subsets of
, and
is a probability measure. The notation "a.e." stands for "almost every."
Definition 1.1 (see [3, 5, 9, 12]).
A mapping is said to be measurable if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ1_HTML.gif)
for each open subset of
. A measurable mapping is also called a random variable. A mapping
is called a random mapping, if for each fixed
, the mapping
is measurable. A random mapping is said to be continuous, if for
a.e., the mapping
is continuous. A measurable mapping
is said to be a random fixed point of the random mapping
, if
, for
a.e. Let
be the family of all nonempty subsets of
and
a set-valued mapping.
is said to be measurable, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ2_HTML.gif)
for each open subset of
. A mapping
is said to be a measurable selection of a measurable mapping
, if
is measurable and
a.e.
We denote by the set of all random fixed points of a random mapping
. If
is a positive integer and
, then
is a random
-periodic points of a random mapping
. By
we denote the
th iterate
of
, where
,
is defined by
.
Let be a Polish space and
a measure space. Let
be a continuous random mapping. If
is measurable, then
is measurable.
Let be a Polish space and
a measure space. If
is a sequence of measurable mappings in
and
a.e., then
is measurable.
Lemma 1.4 (cf. [23]).
Let be a Polish space and
a measure space. Let
be a set-valued mapping. Then,
(1) is measurable if and only if Graph
is
measurable;
(2)if is measurable and
is closed a.e., then there exists a measurable selection of
.
Lemma 1.5 (see [18]).
Let be a metric space and
a functional. Then the relation
on
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ3_HTML.gif)
is a partial ordering.
By Lemma 1.5, if is the partial ordering induced by
, then
implies
. If
is a Polish space and
is the partial ordering induced by
, then
is called an ordered Polish space. If
and
, then
is called an order interval in
.
Definition 1.6 (cf. [19]).
Let be an ordered Polish space and
a measure space. Let
is a random mapping.
is is said to be increasing if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ4_HTML.gif)
is said to be decreasing if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ5_HTML.gif)
a random mapping is said to be mixed monotone if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ6_HTML.gif)
It is evident that, if is mixed monotone, then
is increasing and
is decreasing, for every fixed
.
2. Measurability of Order Continuous Random Mappings
Definition 2.1.
Let be an ordered Polish space and
a measure space. Let
be a random mapping.
is said to be order continuous if for every monotone sequence
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ7_HTML.gif)
is is said to be order contractive if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ8_HTML.gif)
It is evident that continuity implies order continuity. If is order contractive, then
is order continuous. A mixed monotone random mapping
is said to be order continuous if and only if for monotone sequences
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ9_HTML.gif)
Example 2.2.
Let and
. Let
and
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ10_HTML.gif)
It is easy to check that is order continuous, but
is not continuous at
.
Now we prove the following theorem which plays an important role in the sequel.
Theorem 2.3.
Let be an ordered Polish space and
a measure space, where
is continuous. Let
be an order continuous random mapping. If
is measurable, then
is measurable.
Proof.
Let ,
, and
, where
. Clearly,
,
, and
are all nonempty subsets of
for all
. Since
is continuous,
is closed for all
. Let
and
. Then, from
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ11_HTML.gif)
Since is continuous, we have
, that is,
. This shows that
, and so
is closed for all
. Similarly,
is closed for all
. We claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ12_HTML.gif)
In fact, if , then
is a closed subset of
. Let
be an open subset of
,
, and
. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ13_HTML.gif)
Since is measurable and
is closed,
is measurable. From
, we see that
is measurable. Hence,
is measurable. Similarly,
is measurable. Now we prove that
is measurable. Since
is continuous and
is measurable,
is measurable. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ14_HTML.gif)
is measurable. Using Lemma 1.4(1), we obtain that
is measurable. Therefore, (2.6) holds. Let
. Then,
is nonempty and closed for all
. By (2.6),
is measurable. By Lemma 1.4(2), we can take
, where
is measurable. For
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ15_HTML.gif)
Then, is nonempty and closed for all
. When
is measurable, from (2.6), we obtain that
is measurable. Using Lemma 1.4(2), we can take
, where
is measurable. By induction, there exists a measurable sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ16_HTML.gif)
Set . Then
is a Polish subspace of
. Since
is order continuous,
is continuous. By (2.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ17_HTML.gif)
By Lemma 1.2, is measurable for all
. Thus, from (2.11) and Lemma 1.3 it follows that
is measurable. This completes the Proof.
Remark 2.4.
Theorem 2.3 is a generalization of Lemma 1.2.
3. Random Periodic Points and Fixed Points for Increasing Random Mappings
Theorem 3.1.
Let be an ordered Polish space, where
is continuous. Let
be an order continuous and increasing random mapping with
and
for
a.e., where
is a positive integer. Then there exist a minimum random
-periodic point
and a maximum random
-periodic point
in
such that
a.e., for all
.
Proof.
Without loss of generality, we may assume that ,
,
is order continuous for all
, and
,
for all
. Let
,
,
, and
. Since
,
, and
is increasing, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ18_HTML.gif)
Then, it follows from (3.1) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ19_HTML.gif)
From (3.2) we see that and
are two convergent sequences of numbers. For every
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ20_HTML.gif)
This shows that and
are two Cauchy sequences in
. The completeness of
implies that
and
are all convergent. Define
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ21_HTML.gif)
Since is order continuous,
is order continuous. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ22_HTML.gif)
Note that . By Theorem 2.3,
and
are all measurable. By Lemma 1.3,
and
are all measurable. Therefore, from (3.5) we see that
and
are all random fixed points of
, that is,
. Since
is continuous, we have, for
a.e.,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ23_HTML.gif)
This shows that a.e. If
, then we have
a.e., for all
. Thus, for
a.e.,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ24_HTML.gif)
This shows that a.e., which is the desired conclusion.
Corollary 3.2.
Let be an ordered Polish space, where
is continuous. Let
be an order continuous and increasing random mapping with
and
for
a.e.. Then there exist a minimum random fixed point
and a maximum random fixed point
in
such that
a.e., for all
.
Proof.
It is obtained by taking in Theorem 3.1.
Corollary 3.3.
Let be an ordered Polish space, where
is continuous. Let
be a increasing random mapping with
and
for
a.e., where
is a positive integer. If
is an order contraction mapping, then there exists a unique random fixed point
in
.
Proof.
From order contraction of it follows that
is order continuous. By Theorem 3.1, there exist a minimum random
-periodic point
and a maximum random
-periodic point
in
. Since
is an order contraction mapping, for
a.e., we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ25_HTML.gif)
where . This shows that
a.e., namely, there is a unique
. Let
. Then we have
a.e. and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ26_HTML.gif)
that is, . Hence, we have
. This shows that
. If
and
a.e., then
, and so
, that is, there is a unique
. This completes the proof.
4. Random Periodic Points and Fixed Points for Decreasing Random Mappings
Theorem 4.1.
Let be an ordered Polish space, where
is continuous. Let
be an order continuous and decreasing random mapping with
and
for
a.e. Then there exists a random 2-periodic point
in
such that
a.e.
Proof.
Without loss of generality, we may assume that ,
,
is order continuous for all
and
,
for all
. Let
,
, and
, (
). Since
is decreasing, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ27_HTML.gif)
Then, from (4.1) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ28_HTML.gif)
From (4.2) we see that and
are two convergent sequences of numbers. For every
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ29_HTML.gif)
This shows that and
are two Cauchy sequences in
. By the completeness of
we see that
and
are all convergent. Define
and
by (3.4). Since
is order continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ30_HTML.gif)
By the continuity of , we have, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ31_HTML.gif)
Since , we have
a.e.. By Theorem 2.3,
and
are all measurable. By Lemma 1.3,
and
are all measurable. Therefore, from (4.4) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ32_HTML.gif)
This shows that , which is the desired conclusion.
Corollary 4.2.
Let be an ordered Polish space, where
is continuous. Let
be a decreasing random mapping with
and
for
a.e. If
is an order contraction mapping, then there exists a unique random fixed point
in
.
Proof.
Since is an order contraction mapping,
is order continuous. By Theorem 4.1, there exists a random 2-periodic point
in
such that
a.e. We claim that
a.e. In fact that, from (4.1) we have
a.e. If
a.e., then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ33_HTML.gif)
which is a contradiction. Hence, . If
and
a.e., then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ34_HTML.gif)
where and
are the iterations in the proof of Theorem 4.1. It is easy to check that
, for all
a.e. But
, and so we have
. This completes the proof.
Theorem 4.3.
Let be an ordered Polish space, where
is continuous and
is bounded. Let
be an order continuous and decreasing random mapping with
and
for
a.e. Then there exists a random 2-periodic point
in
.
Proof.
Without loss of generality, we may assume that ,
,
is order continuous for all
, and
,
for all
. Let
,
, and
, (
). Since
is decreasing, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ35_HTML.gif)
Then, it follows from (4.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ36_HTML.gif)
This shows that and
are two convergent sequences of numbers by the boundedness of
. For every
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ37_HTML.gif)
This shows that is a Cauchy sequence in
. The completeness of
implies that
is convergent. Similarly,
is convergent. Define
and
by (3.4). Since
is order continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ38_HTML.gif)
Since , by Theorem 2.3,
and
are all measurable; by Lemma 1.3,
and
are all measurable. Therefore, from (4.12) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ39_HTML.gif)
This shows that , which is the desired conclusion.
5. Coupled Random Periodic Point and Fixed Point Theorems
Theorem 5.1.
Let be an ordered Polish space, where
is continuous. Let
be an order continuous and mixed monotone random mapping with
and
for
a.e., where
is a positive integer. Then there exists a coupled random
-periodic point
such that
,
, and
a.e. If
is a coupled random
-periodic point such that
a.e., then
a.e.
Proof.
Without loss of generality, we may assume that ,
,
is order continuous for all
and
,
for all
. Let
,
,
, and
, (
). Since
is a mixed monotone mapping, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ40_HTML.gif)
By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ41_HTML.gif)
Thus, from (5.2) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ42_HTML.gif)
This shows that and
are two convergent sequences of numbers. In a similar way to the proof of Theorem 3.1, we can check that
and
are two Cauchy sequences in
. The completeness of
implies that
and
are all convergent. Define
and
by (3.4). Since
is continuous, it is easy to prove that
for all
. Since
is order continuous,
is order continuous. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ43_HTML.gif)
Note that . By Theorem 2.3,
and
are all measurable. By Lemma 1.3,
and
are all measurable. Therefore, from (5.4) we see that
is a coupled random fixed point of
, that is, it is a coupled random
-periodic point of
. If
is a coupled random
-periodic point such that
a.e., then, by mixed monotonicity of
, we have
a.e. and
a.e. Then, by induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ44_HTML.gif)
Since is order continuous, we have
a.e.. This completes the proof.
Corollary 5.2.
Let be an ordered Polish space, where
is continuous. Let
be an order continuous and mixed monotone random mapping with
and
for
a.e. Then there exists a coupled random fixed point
such that
,
and
a.e. If
is also a coupled random fixed point such that
a.e., then
a.e.
Proof.
It is obtained by taking in Theorem 5.1.
Theorem 5.3.
Let be an ordered Polish space, where
is continuous and
is bounded. Let
be an order continuous and mixed monotone random mapping with
and
for
a.e., where
. Then there exists a coupled random fixed point
such that
,
, and
a.e. If
is also a coupled random fixed point such that
a.e., then
a.e.
Proof.
Without loss of generality, we may assume that ,
,
is order continuous for all
and
,
for all
. Let
,
, and
, (
). Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ45_HTML.gif)
Since is a mixed monotone mapping, we have
, and
. By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ46_HTML.gif)
Thus, from (5.7) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ47_HTML.gif)
This shows that and
are two convergent sequences of numbers by the boundedness of
. In a similar way to the proof of Theorem 4.3, we can check that
and
are two Cauchy sequences in
. The completeness of
implies that
and
are all convergent. Define
and
by (3.4). Since
is continuous, it is easy to prove that
and
for all
. Since
is order continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ48_HTML.gif)
Note that . By Theorem 2.3,
and
are all measurable. By Lemma 1.3,
and
are all measurable. Therefore, from (5.9) we see that
is a coupled random fixed point of
. If
is a coupled random point of
with
a.e., then, by mixed monotonicity of
, we have
a.e., and
a.e., namely,
a.e. By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F723216/MediaObjects/13663_2010_Article_1333_Equ49_HTML.gif)
Since is order continuous, we have
a.e. This completes the proof.
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The authors are grateful to the referees for their suggestions to improve the legibility of the paper. This work is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 10KJB110006) and by the Natural Science Foundation of Nanjing University of Information Science and Technology of China (20080286).
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Zhu, XH., Xiao, JZ. Random Periodic Point and Fixed Point Results for Random Monotone Mappings in Ordered Polish Spaces. Fixed Point Theory Appl 2010, 723216 (2010). https://doi.org/10.1155/2010/723216
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DOI: https://doi.org/10.1155/2010/723216