On the existence and uniqueness of a solution of the random integral equation with advancing argument

Abstract

In this paper we shall prove two theorems on the existence of a unique random solution of the following random integral equation with advancing argument: (1) x(t,co) = h(t,w) t+<5(t) * f 0 K(t,r,co)f(r ,x(r ,cj) )dr in the class of all continuous and bounded functions defined on R+ with values in LglQ,A,P), where (Q,A,P) denotes the probability space. Nonrandom differential equations with advancing argument have been investigated by other authors (see [l], [2], [3]). The problem of the existence of a solution for random integral equation of the Volterra type with advancing argument has been considered in the class of all mappings x:R^S2 —— R such that, for every t,x(t,«) is a random variable (see [5]). The fundamental theorems of this paper generalize some results of Christ. P.Tsokos and W.J.Padgett [6j, by making use of certain ideas of that paper (Fragment tekstu).