### 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*) = 0 in [0,^{(k)}*T*],*u*(0) = ξ^{(i)}_{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}.#### Keywords

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

#### Full Text:

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

Copyright (c) 2021 Paolo Cubiotti

This work is licensed under a Creative Commons Attribution 4.0 International License.