Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 17 (2010) 303-316 Copyright c 2010 Watam Press http://www.watam.org NONLINEAR IMPLICIT IMPULSIVE VOLTERRA TYPE RANDOM INTEGRAL EQUATIONS IN BANACH SPACES Heng-you Lan 1 , Jong Kyu Kim 2 and Yeol Je Cho 3 1 Department of Mathematics Sichuan University of Science & Engineering Zigong, Sichuan 643000, P. R. China 2 Department of Mathematics Education Kyungnam University, Masan 631-701, Korea 3 Department of Mathematics Gyeongsang National University, Chinju 660-701, Korea Abstract. In this paper, by using Banach fixed point theorem, we obtain existence, uniqueness and iterative approximation of solution for first-order nonlinear implicit impul- sive Volterra type random integral equations in Banach spaces. The results presented in this paper improve and generalize some known corresponding results in the literature. Keywords. Nonlinear implicit impulsive Volterra type random integral equation, Banach fixed point theorem, random solution, existence and uniqueness. AMS(MOS) subject Classification: 34G20, 45N05, 45R05. 1 Introduction In this paper, we shall consider the following first-order nonlinear implicit impulsive Volterra type random integral equations: Find x :Ω × J → E such that x(ω,t)= x 0 (ω)+ Z t t0 (t - s)f (ω, s, x(ω,s),x 0 (ω,s),T (ω,x(ω,s)))ds + X t0<t k <t (t - t k )I k (ω,x(ω,t k )), (1.1) where Ω is a measure space, E is Banach space, J =[x 0 ,x 0 + a](a> 0), x 0 :Ω → E, f :Ω × J × E × E × E → E, I k :Ω × E → E (k =1, 2, ··· ,m), T (ω,x(ω,t)) = R t t0 κ(ω, t, s)x(ω,s)ds, κ :Ω × D → R + = [0, ∞) and D = {(t, s)| s, t ∈ J, t ≥ s}. By a solution of the equation (1.1) we mean a function x ∈ PC 1 (J, E) that satisfies (1.1). 2 Corresponding author: jongkyuk@kyngnam.ac.kr (Jong Kyu Kim).