VECTOR IMPLICIT FUNCTIONAL-INTEGRAL EQUATIONS ASSOCIATED WITH DISCONTINUOUS FUNCTIONS

Let I := [0, 1]. In this paper we deal with the implicit vector functional-integral equation h(t, u(t)) = g(t) f (t, integral(1)k(t, s) u(c,a(s)) ds) for a.e. t is an element of I, where h : I x (RR)-R-n , phi : I-+ I, g : I R, k:IxI-+ [0,-Foc:4 and f : I x-+ R are given. We prove an existence theorem for solutions u is an element of L-P (I, R-n) (with p is an element of] 1 , +infinity), which extends a very recent result proved for the case n = 1. Such an extension is not immediate and requires a more articulated technical construction. The main peculiarity of our result is the regularity assumption on f, considerably weaker than the usual Caratheodory condition required in the literature. As a matter of fact, a function f satisfying the assumptions of our main result could be discontinuous, with respect to the second variable, even at each point x is an element of R-n.