Periodic solutions of differential inclusions on compact subsets of \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: Periodic solutions of differential inclusions on compact subsets of \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0705.34040
DOI: 10.1016/0022-247X(90)90038-H
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: Let M be a compact subset of \({\mathbb{R}}^ n\) such that there exists an open set U (M\(\subset U)\) and a continuous function r: \(U\to M\) such that \(dist(x,M)=| x-r(x)|\) for every \(x\in U\). It is shown that the Bouligand contingent cone valued map \(T_ M: M\to {\mathbb{R}}^ n\) is lower semi-continuous. Suppose that the Euler characteristic of M is not equal to zero and a Carathéodory type convex valued map F: \([0,\infty)\times M\to {\mathbb{R}}^ n\) satisfies \(F(t,x)\cap T_ M(x)\neq \emptyset\) for all (t,x)\(\in [0,\infty)\times M\). If the map F is periodic with respect to the first variable then the differential inclusion \(x'(t)\in F(t,x(t))\) possesses a periodic solution. The problem of existence of periodic solutions is reduced to the problem of existence of fixed points of the set valued translation operator along trajectories of the differential inclusion.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 4156961
zbMATH Keywords: tangent cones
zbMATH Keywords: metric retraction
zbMATH Keywords: Bouligand contingent cone valued map
author: Slawomir Plaskacz
reviewed by: Slawomir Plaskacz
MaRDI profile type: Publication
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Wikidata QID: Q131316921
description: scientific article; zbMATH DE number 4156961