Viable periodic solutions in state-dependent impulsive problems
type: Item
instance of: scholarly article
title: Viable periodic solutions in state-dependent impulsive problems
zbMATH Open document ID: 1341.34047
author: Grzegorz Gabor
published in: Collectanea Mathematica
publication date: 2015-09-16T00:00:00Z
review text: This nice paper presents a topological tool, the impulsive index, suitable for detecting periodic trajectories for state-dependent impulsive ODEs, i.e. impulses occur at variable times, under the additional requirement that these trajectories are viable in a prescribed closed subset \(K\). For the construction of the impulsive index the fixed point index on ANRs is applied to an induced discrete semidynamical system on a barrier, where jumps occur. The author shows in Section 2, that the impulsive index satisfies the main properties which one could expect: existence, additivity, homotopy. Section 2 is dedicated to present sufficient conditions for the homotopy property. Then, in Section 4, the Poincaré map technique is adapted to compute the impulsive index and an example is presented to illustrate it. Finally, in the last section of the paper, the author discusses advantages and disadvantages of an alternative construction of the impulsive index.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B37
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 6482868
zbMATH Keywords: state-dependent impulsive problem
zbMATH Keywords: periodic solution
zbMATH Keywords: viable solution
zbMATH Keywords: fixed point index
reviewed by: Daniel Franco
MaRDI profile type: Publication
Wikidata QID: Q59408420
full work available at URL: s13348-015-0139-x
OpenAlex ID: W1988036236
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DOI: 10.1007/S13348-015-0139-X
description: scientific article; zbMATH DE number 6482868