A qualitative result for higher-order discontinuous implicit differential equations

Abstract

Let n,k ∈ N, and let T > 0, Y ⊆ Rⁿ and ξ = (ξ₀, ξ₁,..., ξ_k-1) ∈ (Rⁿ)^k. Given a function f :[0, T]×(Rⁿ)^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 ∈ (Rⁿ)^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 ∈ (Rⁿ)^k.