Existence of global solutions for second order impulsive abstract partial differential equations
type: Item
instance of: scholarly article
title: Existence of global solutions for second order impulsive abstract partial differential equations
zbMATH Open document ID: 1237.34136
author: Juan-Miguel Gracia
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-10-17T00:00:00Z
review text: The authors consider second order impulsive differential equations of the form: \[ x'(t)=Ax(t)+f(t, x(t),x(\rho(t))),\;t\in J, \] \[ \begin{aligned} \Delta x(t_{k})&=I_{k}(x(t_{k}^{-})), \quad k\in \mathbb{F}\subset\mathbb N,\\ \Delta x'(t_{k})&=\bar I_{k}(x(t_{k}^{-})),\quad k\in \mathbb{F}\subset\mathbb N,\end{aligned} \] \[ x(0)=x_0\in E, \] where \(J:=[0,b]\) or\ \(J=[0,\infty)\), \(A\) is the infinitesimal generator of a family of cosine operators. \(f: J\times E\times E\to E\) is a Carathédory function, \(\rho: J\to J\) is a continuous function. The functions \(I_k, \bar I_k\) characterize the jump of the solutions at impulse points \(t_k\) (\(k\in\mathbb{F}\)). Existence results are proved, when the right-hand side is a Lipschitz or Carathédory function. The proofs are based on a nonlinear alternative of Leray-Schauder type. One example is presented.
reviewed by: Abdelghani Ouahab
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35R12
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 34K45
zbMATH DE Number: 5959724
zbMATH Keywords: second order abstract Cauchy problem
zbMATH Keywords: family of cosine operators
zbMATH Keywords: impulsive systems
zbMATH Keywords: global solution
MaRDI profile type: Publication
full work available at URL: j.na.2011.06.054
OpenAlex ID: W2041897180
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DOI: 10.1016/J.NA.2011.06.054
description: scientific article; zbMATH DE number 5959724