A qualitative result for higher-order discontinuous implicit differential equations

Abstract

Let n,k ∈ N , and let T > 0, Y ⊆ R n and ξ = (ξ 0 , ξ 1 ,..., ξ k -1 ) ∈ ( R n ) k . Given a function f :[0, T ]×( R n ) k × Y → R , we consider the Cauchy problem f ( t,u,u ′ ,..., u (k) ) = 0 in [0, T ], u (i) (0) = ξ i for every i = 0, 1,..., k −1. We prove an existence and qualitative result for the generalized solutions of the above problem. In particular, we prove that, under suitable assumptions, the solution set S T f ( ξ ) of the above problem is nonempty, and the multifunction ξ ∈ ( R n ) k → S T f ( ξ ) admits an upper semicontinuous multivalued selection, with nonempty, compact and connected values. The assumptions of our result do not require any kind of continuity for the function f (·,·, y ). In particular, a function f satisfying our assumptions could be discontinuous, with respect to the second variable, even at all points ξ ∈ ( R n ) k .