A Common Random Fixed Point Theorem for Weakly Compatible Mappings in Cone Random Metric Spaces

R. A. Rashwan,H. A. Hammad

doi:10.13189/ujcmj.2016.040403

Abstract

In this paper, we prove a unique common ran- dom fixed p oint t heorem i n t he f ramework o f c one random metric spaces for four weakly random compatible mappings under strict contractive condition. Some corollaries of this theorem for three and two weakly random compatible mappings and for one random mapping are derived. Two examples to justify our theorem are given. Our results extend some previous work related to cone random metric spaces from the current existing literature.

Highlights

Fixed point theory has the diverse applications in different branches of mathematics, statistics, engineering, and economics in dealing with the problems arising in approximation theory, potential theory, game theory, theory of differential equations, theory of integral equations, and others
Common random fixed points and random coincidence points of a pair of compatible random operators and fixed point theorems for contractive random operators in Polish spaces are obtained by Papageorgiou [15, 16] and Beg [3, 4]
[12] Huang and Zhang generalized the concept of metric spaces, replacing the set of real numbers by an ordered Banach space, they have defined the cone metric spaces

Summary

Introduction

Fixed point theory has the diverse applications in different branches of mathematics, statistics, engineering, and economics in dealing with the problems arising in approximation theory, potential theory, game theory, theory of differential equations, theory of integral equations, and others. The results of Spacek and Hansin multi-valued contractive mappings was extended by Itoh [14] This theory has become the full fledged research area and various ideas associated with random fixed point theory are used to give the solution of nonlinear system see [5,6,7, 11, 20]. In [12] Huang and Zhang generalized the concept of metric spaces, replacing the set of real numbers by an ordered Banach space, they have defined the cone metric spaces They described the convergence of sequences and introduced the notion of completeness in cone metric spaces. They have proved some fixed point theorems of contractive mappings on complete cone metric space with the assumption of normality of a cone. The aim of this paper is to extends the contractive condition (2.1) for four, three and two random mappings and establish a unique random fixed point results under this condition in random cone metric spaces using the concept of weakly random compatible mappings

Preliminaries

Theorem

Corollary

Example