Existence theory for a class of nonlinear random functional integral equations

Abstract

The aim of the present paper is to study a random equation of the general form x(t, ω)=(Ux)(t, ω), t∈R+ and its special case a nonlinear random functional integral equation given by $$x\left( {t,w} \right) = F\left( {t,\int\limits_0^{g_1 \left( t \right)} {f_1 } \left( {t,s,x\left( {s,w} \right),w} \right)ds, \ldots ,\int\limits_0^{g_m \left( t \right)} {f_m } \left( {t,s,x\left( {s,w} \right),w} \right)ds,x\left( {h_1 \left( t \right),w} \right), \ldots ,x\left( {h_p \left( t \right),\omega } \right),\omega } \right).$$ The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered.