Existence and uniqueness of nonlinear impulsive integro-differential equations
type: Item
instance of: scholarly article
title: Existence and uniqueness of nonlinear impulsive integro-differential equations
zbMATH Open document ID: 1112.45004
author: Tonny Paul
author: A. Anguraj
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2007-02-05T00:00:00Z
review text: The authors study the existence and uniqueness of mild and classical solutions for the following impulsive integro-differential equation \[ u'(t)=Au(t)+f\biggl(t,u(t),\int_{0}^{t}k(t,s)u(s)\,ds\biggr), \quad t\in [0,T_{0}], \;t\not=t_{i}, \] \[ u(0)=u_{0}, \] \[ \Delta u(t_{i})=I_{i}(u(t_{i})), \quad i=1,\ldots,p, \] where \(f:[0,T_{0}]\times X\times X\to X\) is a given function, \(A:D(A)\to X\) is an unbounded operator which generates a strongly continuous semigroup \(T(t), \;t\geq 0, \;k: [0,T_{0}]\times [0,T_{0}]\to R\) a continuous function, \(I_{i}:X\to X\), \(\Delta u(t_{i})=u(t_{i}^{+})-u(t_{i}^{-})\) represents the jump of the state \(u\) at time \(t_{i}\), and \(X\) is a Banach space. The proofs are based upon the semigroup theory and the Banach contraction principle. An example illustrating the abstract theory is missing.
reviewed by: Mouffak Benchohra
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5121915
zbMATH Keywords: evolution equations
zbMATH Keywords: impulsive conditions
zbMATH Keywords: semigroups
zbMATH Keywords: mild and classical solutions
zbMATH Keywords: Banach contraction principle
zbMATH Keywords: impulsive integro-differential equation
zbMATH Keywords: Banach space
MaRDI profile type: Publication
full work available at URL: dcdsb.2006.6.1191
OpenAlex ID: W2314665548
DOI: 10.3934/DCDSB.2006.6.1191
description: scientific article; zbMATH DE number 5121915