Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.8, 2014 78 Common Random Fixed Point Theorems of Contractions in Partial Cone Metric Spaces over Non normal Cones PIYUSH M. PATEL (1) Research Scholar of Sainath University,Ranchi(Jarkhand). pmpatel551986@gmail.com RAMAKANT BHARDWAJ (2) Associate Professor in Truba Institute of Engineering & Information Technology, Bhopal (M.P). drrkbhardwaj100@gmail.com SABHAKANT DWIVEDI (3) Department of Mathematics, Institute for Excellence in higher education Bhopal 1. ABSTRACT: The purpose of this paper is to prove existence of common random fixed point in the setting of partial cone metric space over the non-normal cones. Key wards: common fixed point,cone metric space, random variable 2. INTRODUCTION AND PRELIMINARIES Random nonlinear analysis is an important mathematical discipline which is mainly concerned with the study of random nonlinear operators and their properties and is needed for the study of various classes of random equations. The study of random fixed point theory was initiated by the Prague school of Probabilities in the 1950s [4, 13, and 14]. Common random fixed point theorems are stochastic generalization of classical common fixed point theorems. The machinery of random fixed point theory provides a convenient way of modeling many problems arising from economic theory and references mentioned therein. Random methods have revolutionized the financial markets. The survey article by Bharucha-Reid [1] attracted the attention of several mathematicians and gave wings to the theory. Itoh [18] extended Spacek's and Hans's theorem to multivalued contraction mappings. Now this theory has become the full edged research area and various ideas associated with random fixed point theory are used to obtain the solution of nonlinear random system (see [2,3,7,8,9 ]). Papageorgiou [11, 12], Beg [5,6] studied common random fixed points and random coincidence points of a pair of compatible random and proved fixed point theorems for contractive random operators in Polish spaces. In 2007, Huang and Zhang [9] introduced the concept of cone metric space and establish some fixed point theorems for contractive mappings in normal cone metric spaces. Subsequently, several other authors [10, 17, ] studied the existence of fixed points and common fixed points of pings satisfying contractive type condition on a normal cone metric space. In 2008, Rezapour and Hamlbarani [17] omitted the assumption of normality in cone metric space, which is a milestone in developing fixed point theory in cone metric space. In this paper we prove existence of common random fixed point in the setting of cone random metric spaces under weak contractive condition. Recently, Dhagat et al. [19] introduced the concept of cone random metric space and proved an existence of random fixed point under weak contraction condition in the setting of cone random metric spaces. The purpose of this paper to find common random fixed point theorems of contractions in partial cone metric spaces over non normal cones. Definition 2.1. Let X be a nonempty set and let  be a cone of a topological vector space E . A partial cone metric on X is a mapping P X X : p     such that, for each X t h t g t f  ) ( ), ( ), ( ,   t ,