Bound sets approach to boundary value problems for vector second-order differential inclusions
type: Item
instance of: scholarly article
title: Bound sets approach to boundary value problems for vector second-order differential inclusions
zbMATH Open document ID: 1177.34015
author: Jan Andres
author: Martina Kožušníková
author: Luisa Malaguti
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-06-09T00:00:00Z
review text: The paper deals with the second-order boundary value problem \[ \ddot x(t)\in F(t,x(t),\dot x(t))\quad \text{for a.a.}\;t\in J, x\in S, \leqno(1) \] where \(J=[t_0,t_1]\) is a compact interval, \(F: J\times \mathbb{R}^n\times \mathbb{R}^n \multimap \mathbb{R}^n\) is an upper-Carathéodory mapping and \(S\) is a subset of \(AC^1(J,\mathbb{R}^n)\). The authors develop a continuation principle for the solvability of (1) using fixed point index arguments. The main assumption which yields the possibility to apply the continuation principle is the transversality condition which is guaranted here by means of Liapunov-like bounding functions. In addition, for the Floquet semi-linear problem \[ \ddot x(t)+A(t)\dot x(t)+B(t)x(t)\in F(t,x(t),\dot x(t))\quad \text{for a.a.}\;t\in J, \] \[ x(t_1)=Mx(t_0),\quad \dot x(t_1)=N\dot x(t_0), \] where \(A,B:J\to \mathbb{R}^n\times \mathbb{R}^n\) are integrable matrix functions and \(M\) and \(N\) are real \(n\times n\) matrices with \(M\) non-singular, a viability result will be obtained by means of a bound sets technique.
reviewed by: Irena Rachůnková
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5563005
zbMATH Keywords: upper-Carathéodory differential incusions
zbMATH Keywords: Floquet problem
zbMATH Keywords: viability result
MaRDI profile type: Publication
full work available at URL: j.na.2008.10.028
OpenAlex ID: W2078909574
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DOI: 10.1016/J.NA.2008.10.028
description: scientific article; zbMATH DE number 5563005