Lyapunov-Bohl exponent and Green's function for some class of integro-differential equations
type: Item
instance of: scholarly article
title: Lyapunov-Bohl exponent and Green's function for some class of integro-differential equations
zbMATH Open document ID: 0842.45007
DOI: 10.4171/ZAA/655
author: E. A. Ermolova
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1996-07-24T00:00:00Z
review text: The aim of this paper is to study the solutions of the following nonlinear integro-differential equation of Barbashin type: \[ {{\partial x (t,s)} \over {\partial t}}= c(t, s) x(t, s)+ \int^b_a k(t, s, \sigma, x(t, \sigma)) d\sigma, \tag{1} \] and the linear integro-differential equation \[ {{\partial x (t, s)} \over {\partial t}}= c(t, s) x(t, s)+ \int^b_a k(t, s, \sigma) x(t, \sigma) d\sigma. \tag{2} \] The equation (1) can be written as a differential equation \[ {{dx} \over {dt}}= c(t) x+ k(t,x), \tag{3} \] with multiplication operator \((c(t) x) (s)= c(t, s) x(s)\) and integral operator \[ k(t, x) (s)= \int^b_a k(t, s, \sigma, x(\sigma)) d\sigma. \] The author discusses the upper estimates for Lyapunov-Bohl exponent of (1) and (2), and gives necessary and sufficient conditions for the exponential dichotomy of the differential equation (3).
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 856798
zbMATH Keywords: differential equation in Banach spaces
zbMATH Keywords: nonlinear integro-differential equation of Barbashin type
zbMATH Keywords: Lyapunov-Bohl exponent
zbMATH Keywords: exponential dichotomy
MaRDI profile type: Publication
cites work: Q5546685
full work available at URL: https://doi.org/10.4171/zaa/655
OpenAlex ID: W2121811066
description: scientific article; zbMATH DE number 856798