Random Periodic Point and Fixed Point Results for Random Monotone Mappings in Ordered Polish Spaces

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

(1.1)

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

(1.2)

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 .

Lemma 1.2 (see [3, 22]).

Let be a Polish space and a measure space. Let be a continuous random mapping. If is measurable, then is measurable.

Lemma 1.3 (see [3, 4]).

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

(1.3)

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

(1.4)

is said to be decreasing if

(1.5)

a random mapping is said to be mixed monotone if

(1.6)

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 ,

(2.1)

is is said to be order contractive if there exists such that

(2.2)

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 ,

(2.3)

Example 2.2.

Let and . Let and be defined by

(2.4)

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

(2.5)

Since is continuous, we have , that is, . This shows that , and so is closed for all . Similarly, is closed for all . We claim that

(2.6)

In fact, if , then is a closed subset of . Let be an open subset of , , and . Then, we have

(2.7)

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

(2.8)

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

(2.9)

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

(2.10)

Set . Then is a Polish subspace of . Since is order continuous, is continuous. By (2.10), we have

(2.11)

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

(3.1)

Then, it follows from (3.1) that

(3.2)

From (3.2) we see that and are two convergent sequences of numbers. For every there exists a positive integer such that

(3.3)

This shows that and are two Cauchy sequences in . The completeness of implies that and are all convergent. Define and by

(3.4)

Since is order continuous, is order continuous. Then, we have

(3.5)

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.,

(3.6)

This shows that a.e. If , then we have a.e., for all . Thus, for a.e.,

(3.7)

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

(3.8)

where . This shows that a.e., namely, there is a unique . Let . Then we have a.e. and

(3.9)

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

(4.1)

Then, from (4.1) it follows that

(4.2)

From (4.2) we see that and are two convergent sequences of numbers. For every there exists a positive integer such that

(4.3)

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

(4.4)

By the continuity of , we have, for ,

(4.5)

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

(4.6)

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

(4.7)

which is a contradiction. Hence, . If and a.e., then we have

(4.8)

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

(4.9)

Then, it follows from (4.9) that

(4.10)

This shows that and are two convergent sequences of numbers by the boundedness of . For every there exists a positive integer such that

(4.11)

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

(4.12)

Since , by Theorem 2.3, and are all measurable; by Lemma 1.3, and are all measurable. Therefore, from (4.12) we have

(4.13)

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

(5.1)

By induction, we have

(5.2)

Thus, from (5.2) it follows that

(5.3)

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

(5.4)

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

(5.5)

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,

(5.6)

Since is a mixed monotone mapping, we have , and . By induction, we have

(5.7)

Thus, from (5.7) it follows that

(5.8)

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

(5.9)

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

(5.10)

Since is order continuous, we have a.e. This completes the proof.