A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces
type: Item
instance of: scholarly article
title: A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces
zbMATH Open document ID: 1014.45007
DOI: 10.1016/S0022-247X(02)00366-9
author: Lishan Liu
author: Congxin Wu
author: Fei Guo
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The authors investigate the following initial value problem for the first-order nonlinear impulsive integro-differential equation of mixed type in a real Banach space E: \[ x'=f(t,x,Tx,Sx) \quad\text{for} \quad t\in J,\;t\neq t_k, \tag{1} \] \[ \Delta x|_{t=t_k}= I_k\bigl(x(t_k)\bigr) \quad \text{for} \quad k=1,2,\dots,p, \tag{2} \] where \(f\in C(J\times E\times E \times E,E)\), \(J=[0,a]\), \(h\in C(E,E)\), \(0<t_1<t_2 \cdots <t_p<a\), \(x_0\in E\) and \[ (Tx)(t)= \int^t_0k(t,s) x(s)ds, \quad (Sx)(t)=\int^a_0 h(t,s)x(s)ds. \] Under some hypotheses formulated in terms of upper or lower solutions with help of an order determined by a cone in \(E\) a result on the existence and uniqueness of a solution of the problem (1)--(3) is obtained.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1866690
zbMATH Keywords: initial value problem
zbMATH Keywords: first-order nonlinear impulsive integro-differential equation
zbMATH Keywords: Banach space
zbMATH Keywords: upper or lower solutions
zbMATH Keywords: cone
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
reviewed by: Józef Banaś
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1866690