Ordinary differential equations and systems with time-dependent discontinuity sets

Abstract

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

Viability theory applied to discontinuous ordinary differential equations

“…In this case, we say that the discontinuity set Km is "resolvent". We finish this section with another result in [1] about the existence of extremal solutions of (1.1) in the scalar case that we will need in the next section.…”

“…This work was developed by J.A. Cid and R. L. Pouso in [ 1 ] and it is based in the idea of asking the discontinuity sets of / to be either unviables or resolvents, in a sense that we will specify later. 1.…”

Positive Solutions for Discontinuous Systems via a Multivalued Vector Version of Krasnosel’skiĭ’s Fixed Point Theorem in Cones

“…The following definition introduces some curves where we allow the functions f i to be discontinuous in each variable. The idea of using such curves can be found in some recent papers for second-order discontinuous scalar problems [1][2][3] and, in some sense, it recalls the notion of time-depending discontinuity sets from [9]. Definition 1.…”

“…Nevertheless, the regularization is usually made in the nonlinearities transforming the problem into a differential inclusion and the solutions are often given in the sense of the set-valued analysis (Krasovskij and Filippov solutions [5,6]), see e.g., [7,8]. Similar ideas are also used in the papers [5,9] where there are provided some sufficient conditions for the Krasovskij solutions to be Carathéodory solutions. Recently, second-order scalar discontinuous problems have been investigated by using variational methods [10][11][12].…”

A Survey of Recent Results for the Generalizations of Ordinary Differential Equations

“…Biles and Schechter posed the open problem of proving an existence result for discontinuous systems of differential equations lacking a quasimonotonicity property; see [7, page 3352]. Motivated by that question, Cid and Pouso [8] explored an alternative approach to discontinuous equations which consisted, roughly speaking, of inserting the differential equation into a semicontinuous differential inclusion for which existence results were available, and then positing assumptions on the discontinuities of the former differential equation so that every solution of the inclusion is a solution of the equation. Besides getting an existence result for nonquasimonotone discontinuous systems, the approach in [8] came to unify and extend previous similar results for autonomous equations proven in [9] and for nonautonomous equations proven in [10].…”