An existence and qualitative result for discontinuous implicit differential equations

Abstract

Let T > 0 and Y ⊆ Rⁿ. Given a function f:[0,T] × Rⁿ ×  Y → R, we consider the Cauchy problem f(t, u, u′) = 0 in [0, T], u(0) = ξ. We prove an existence and qualitative result for the generalized solutions of the above problem. In particular, our result does not require the continuity of f with respect to the first two variables. As a matter of fact, a function f(t, x, y) satisfying our assumptions could be discontinuous (with respect to x) even at all points x ∈ Rⁿ. We also study the dependence of the solution set S_T(ξ) from the initial point ξ ∈ Rⁿ. In particular, we prove that, under our assumptions, the multifunction S_T admits a multivalued selection Φ which is upper semicontinuous with nonempty compact acyclic values.