On well-posedness of an initial value problem for nonlinear second-order impulsive integro-differential equations of Volterra type in Banach spaces
type: Item
instance of: scholarly article
title: On well-posedness of an initial value problem for nonlinear second-order impulsive integro-differential equations of Volterra type in Banach spaces
zbMATH Open document ID: 1090.45007
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-04-06T00:00:00Z
review text: This paper studies an initial value problem of Volterra type in a real Banach space. At time zero the values of unkonwn function \(x\) and its derivative are given, and for \(t > 0\) the values of \(x''(t)\) are given as a function of \(t\) and \(x(s)\), \(0 \leq s \leq t\), except at some given time instances \(0 < t_1 < t_2 \cdots \) where \(x\) and \(x'\) have jump discontinuities, with values that depend on \(x(t_k)\) and \(x'(t_k)\). An existence and uniqueness theorem for this equation is developed, and an iterative scheme for the solution is developed, with an error estimate. The results are illustrated with two examples.
reviewed by: Olof Staffans
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45L05
zbMATH DE Number: 5017405
zbMATH Keywords: unique solution
zbMATH Keywords: nonlinear second-order impulsive integro-differential equation
zbMATH Keywords: initial value problem
zbMATH Keywords: generalized Banach fixed point theorem
zbMATH Keywords: Banach space
zbMATH Keywords: iterative scheme
zbMATH Keywords: error estimate
author: Lishan Liu
author: Yonghong Wu
author: Xinguang Zhang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.032
OpenAlex ID: W2015123784
cites work: Q5185568
cites work: Q4238355
cites work: Initial value problems for second order impulsive integro-differential equations in Banach spaces
cites work: Initial value problems for second-order integro-differential equations in Banach spaces
cites work: Q4716265
cites work: Q3732304
cites work: Q3995625
cites work: Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach spaces
cites work: Iterative method for solutions and coupled quasi-solutions of nonlinear Fredholm integral equations in ordered Banach spaces
cites work: Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications
cites work: A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces
cites work: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces
cites work: Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space
cites work: Fixed points for operators in a space of continuous functions and applications
cites work: Initial value problems for the second order mixed monotone type of impulsive differential equations in Banach spaces
cites work: Q4500032
cites work: Global solutions of nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces.
DOI: 10.1016/J.JMAA.2005.12.032
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description: scientific article; zbMATH DE number 5017405