A qualitative result for higher-order discontinuous implicit differential equations

Paolo Cubiotti


Let n,kN, and let T > 0, YRn and ξ = (ξ0, ξ1,..., ξk-1) ∈ (Rn)k. Given a function f :[0, T]×(Rn)k×YR, we consider the Cauchy problem f(t,u,u,...,u(k)) = 0 in [0,T], u(i)(0) = ξi 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 STf(ξ) of the above problem is nonempty, and the multifunction ξ ∈ (Rn)kSTf(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.


Differential Equations; Differential Inclusions; Cauchy Problem; Generalized Solutions; Selections; Discontinuous Functions

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DOI: https://doi.org/10.1478/AAPP.991A2

Copyright (c) 2021 Paolo Cubiotti

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