- Research Article
- Open Access

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

- Xing-Hua Zhu
^{1}and - Jian-Zhong Xiao
^{1}Email author

**2010**:723216

https://doi.org/10.1155/2010/723216

© Xing-Hua Zhu and Jian-Zhong Xiao. 2010

**Received: **5 November 2010

**Accepted: **24 December 2010

**Published: **30 December 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.

## Keywords

- Random Mapping
- Periodic Point
- Cauchy Sequence
- Polish Space
- Convergent Sequence

## 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]).

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]).

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]).

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.

Example 2.2.

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.

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.

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.

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.

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.

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.

## 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.

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.

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

## Declarations

### Acknowledgments

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).

## Authors’ Affiliations

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