SECOND-ORDER IMPLICIT DIFFERENTIAL INCLUSIONS WITH DISCONTINUOUS RIGHT-HAND SIDE

Given a multifunction F: [a, b] x R -> 2(R), we consider the implicit multivalued boundary value problem {h(u ''(t)) is an element of F(t, u(t)) a.e. in [a, b] u(a) = u(b) = 0. We prove an existence theorem for solutions u is an element of W-2,W-p([a, b]), where for each t is an element of [a, b] the multifunction F(t,.) can fail to be lower semicontinuous even at all points x is an element of R. In particular, our assumptions are satisfied, for instance, if there exist a neglegible set E subset of R and a multifunction G : [a, b] x R -> 2(R) such that for a.a. t is an element of [a, b] one has {x is an element of R : G(t, .) is not l.s.c. at x} boolean OR {x is an element of R : G(t, x) not equal F(t, x)} subset of E. No monotonicity assumption is required for h or F. Our result extends Theorem 3 of [5], in which the explicit case is considered.