An existence and qualitative result for discontinuous implicit differential equations

Abstract

Let T > 0 and Y ⊆ R n . Given a function f :[0, T ] × R n × 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 ∈ R n . We also study the dependence of the solution set S T ( ξ ) from the initial point ξ ∈ R n . 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.