Nonlinear Stochastic Integral-Equation of Hammerstein Type

A nonlinear stochastic integral equation of the Hammerstein type in the form x(t; c) = h(t; co) + f k(t, s; co)f (s, x(s; co)) dy (s) is studied where t E S, a v-finite measure space with certain properties, co E Q, the supporting set of a probability measure space (Q, A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process x(t; co) on S which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on S, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.