On first order ordinary differential equations with infinitely many state dependent impulses
type: Item
instance of: scholarly article
title: On first order ordinary differential equations with infinitely many state dependent impulses
zbMATH Open document ID: 0889.34015
author: F. Dubeau
published in: Differential Equations and Dynamical Systems
publication date: 1998-06-22T00:00:00Z
review text: The following system of differential equations with infinitely many state-dependent impulses is considered \[ x'(t)= f(x(t), t)+ \sum_{j\in J}\alpha_j(x(\tau_j)) \delta(t- \tau_j)\quad (t\in (0,T)),\tag{1} \] (2) \(x(0)= x_0\), where \(T>0\), \(x_0\in\mathbb{R}^n\), \(x: [0,T]\to \mathbb{R}^n\) is an unknown function, \(f:\mathbb{R}^n\times [0,T]\to \mathbb{R}^n\) is a given mapping, \(\delta\) is the Dirac delta function, \(J\) is a countable set of indices, \(\alpha_j: \mathbb{R}^n\to \mathbb{R}^n\) is a given mapping and \(\tau_j\) is a number from the interval \([0,T]\) for each \(j\in J\). It is assumed that the sum of the absolute values of all impulses is finite. Using the method of successive approximation it is shown that the problem (1)--(2) has a unique solution in the space of vector functions with bounded variation on the interval \([0,T]\).
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 1098675
zbMATH Keywords: infinite
zbMATH Keywords: state-dependent impulses
zbMATH Keywords: Dirac delta function
zbMATH Keywords: successive approximation
reviewed by: Józef Banaś
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1098675