Integrable solutions of highly discontinuous implicit functional-integral equations

Let I be a real compact interval.We deal with the problem of the existence of solutions u L p (I) of the implicit functional-integral equation f t, u(t), I k(t, s) u((s)) ds = 0 for a.e.t I, where Y is a closed interval, and f : I Y R R, k : I I [0, +[ and : I I are given functions.Such an equation includes, as special cases, many integral equations studied in the literature.We prove an existence result whose main peculiarity is the following: a function f (t, y, x) satisfying our assumptions can be discontinuous, with respect to the third variable, even at all points x R. As regards the function y f (t, y, x), we only require that it is continuous, that it changes its sign over Y , and that it is not identically zero over any interval.No assumption of monotonicity is made on f .Our result extends and improves several results in the literature.Examples and also counter-examples to possible improvements are presented.