Opuscula Mathematica • Vol. 30 • No. 4 • 2010 EXISTENCE AND ATTRACTIVITY RESULTS FOR NONLINEAR FIRST ORDER RANDOM DIFFERENTIAL EQUATIONS Bapurao C. Dhage, Sotiris K. Ntouyas Abstract. In this paper, the existence and attractivity results are proved for nonlinear first order ordinary random differential equations. Two examples are provided to demonstrate the realization of the abstract developed theory. Keywords: random differential equations, locally attractive, globally attractive, local asymptotic attractive, fixed point theorem. Mathematics Subject Classification: 47H40, 60H25. 1. INTRODUCTION Let R denote the real line and R + , the set of nonnegative real numbers, that is, R + = [0, ∞) ⊂ R. Let C(R + , R) denote the class of real-valued functions defined and continuous on R + . Given a measurable space (Ω, A) and a measurable function x :Ω → C(R + , R), we consider the initial value problem of nonlinear first order ordinary random differential equations (in short RDE)  x ′ (t, ω)+ k(t, ω)x(t, ω)= f (t, x(t, ω),ω) a.e. t ∈ R + , x(0,ω)= q(ω) (1.1) for all ω ∈ Ω, where k : R + × Ω → R + , q :Ω → R and f : R + × R × Ω → R. By a random solution of the RDE (1.1) we mean a measurable function x : Ω → AC(R + , R) that satisfies the equations in (1.1), where AC(R + , R) is the space of absolutely continuous real-valued functions defined on R + . The initial value problems of ordinary differential equations have been studied in the literature on bounded as well as unbounded internals of the real line for different aspects of the solution. See for example, Burton and Furumochi [2], Dhage [5,6], Hu and Yan [7] and the references therein. Similarly, the initial value problem of random 411 http://dx.doi.org/10.7494/OpMath.2010.30.4.411