Random Periodic Point and Fixed Point Results for Random Monotone Mappings in Ordered Polish Spaces
© Xing-Hua Zhu and Jian-Zhong Xiao. 2010
Received: 5 November 2010
Accepted: 24 December 2010
Published: 30 December 2010
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 , Hanš , and Wang . 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 , Chang , Ding , Fierro et al. , Itoh , Li and Duan , O'Regan et al. , Xiao and Tao , Xu , and Zhu and Xu .
In 1976, Caristi  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 , Guo and Lakshmikantham , and Bhaskar and Lakshmikantham  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."
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 1.4 (cf. ).
Lemma 1.5 (see ).
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. ).
2. Measurability of Order Continuous Random Mappings
Now we prove the following theorem which plays an important role in the sequel.
Theorem 2.3 is a generalization of Lemma 1.2.
3. Random Periodic Points and Fixed Points for Increasing Random Mappings
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 .
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 .
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 .
4. Random Periodic Points and Fixed Points for Decreasing Random Mappings
5. Coupled Random Periodic Point and Fixed Point Theorems
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.
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.
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.
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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