MEASURABLE SOLUTIONS OF IMPLICIT INTEGRAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDE

We prove an existence theorem for measurable solutions u : [a, b] -> Y of the integral equation psi(t, u(t)) = phi(t, integral(t)(a) h(s, u(s)) ds), where Y is a compact, connected and locally connected metric space, and h : [a, b] x Y -> R-n, psi : [a, b] x Y -> R and phi : [a, b] x R-n -> R are given functions. Our result extends and improves a previous result, valid for the case where n = 1 and psi does not depend on t explicitly. A function phi : [a, b] x R-n -> R satisfying our assumptions can be discontinuous (with respect to the second variable) even at all points x is an element of R-n.