Implicit vector integral equations associated with discontinuous operators

Let I := [0, 1]. We consider the vector integral equation h(u(l)) = f (l, integral(I) y(l,z),u(z),dz) for a.e. l is an element of I, where f : I Chi J -> R,y : I Chi I -> [0, +infinity] and h : X -> R are given functions and X,J are suitable subsets of R-n. We prove an existence result for solutions u is an element of L-s(I,R-n), where the continuity of f with respect to the second variable is not assumed. More precisely, f is assumed to be a.e. equal (with respect to second variable) to a function f : I Chi J -> R which is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a function f can be discontinuous at each point x is an element of J. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar case n = 1.