An existence result for functional-integral equations associated with discontinuous functions

Let I := [0 , 1]. In this paper we consider the implicit functional integral equation y(t, u(t)) = g(t) + f (t, integral(I) k(t, s) u(phi(s)) ds) for a.e. t is an element of I, where psi : I x R -> R, g : I -> R, k : IxI -> [0, +infinity[, phi: I -> I and f : IxR -> R are given functions. We prove an existence result for solutions u is an element of L-p(I) (with p is an element of]1, +infinity]), where the regularity assumptions on f are considerably weaker than the usual Caratheodory condition required in the literature. In fact, a function f satisfying the assumptions of our main result can be discontinuous, with respect to the second variable, even at all points x is an element of R.