Uniqueness and existence results for implicit impulsive differential equations
type: Item
instance of: scholarly article
title: Uniqueness and existence results for implicit impulsive differential equations
zbMATH Open document ID: 0969.34012
DOI: 10.1016/S0362-546X(98)00325-3
author: M. Kumpulainen
author: S. Heikkilä
author: S. Seikkala
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-09-18T00:00:00Z
review text: The authors deal with the implicit differential equation with impulses of the form \[ F(t,u(t),u'(t)+p_{1}(t)u(t))=0\;\text{ a.e. on } [t_0,t_1]:=J,\tag{*} \] with the conditions \(D(u(t_0),x_0)=0, u(\alpha+0)-u(\alpha)=Iu(\alpha), \alpha \in \Lambda,\) where \(\Lambda\) is a well-ordered subset of the interval \((t_0,t_1).\) The values of the functions are in a Banach space. First, the authors provide conditions which guarrantee the fact that equation (*) admits unique solutions in a prespecified set \(V.\) Next, the authors show the existence by transforming (*) into an infinite functional system of the form \(Lu_{n+1}=Nu_{n}, \) where the operators \(L,N\) are known. It is shown that the sequence \(u_n\) with \(u_0=0\) converges on \(J\) pointwise to a solution \(u\in V\) to problem (*). A special case is discussed in the last section of the paper.
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 34A09
zbMATH DE Number: 1491635
zbMATH Keywords: differential equations with impulses
zbMATH Keywords: implicit differential equations
reviewed by: George Karakostas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1491635