Lipschitz stability for nonlinear Volterra integrodifferential systems
type: Item
instance of: scholarly article
title: Lipschitz stability for nonlinear Volterra integrodifferential systems
zbMATH Open document ID: 0653.45008
DOI: 10.1016/0096-3003(88)90001-X
author: M. Rama Mohana Rao
author: Saber Elaydi
published in: Applied Mathematics and Computation
publication date: 1988-01-01T00:00:00Z
review text: Consider the nonlinear integro-differential system \[ (1)\quad x'(t)=f(t,x(t))+\int^{t}_{t_ 0}g(t,s,x(s))ds,\quad x(t_ 0)=x_ 0,\quad t_ 0\geq 0, \] and the variational systems \[ y'(t)=f_ x(t,x(t,t_ 0,x_ 0))y(t)+\int^{t}_{t_ 0}g_ x(t,s,x(s,t_ 0,x_ 0))y(s)ds, \] where \(f\in C[J\times R^ n,R^ n]\), \(g\in C[J\times J\times R^ n,R^ n]\), \(f(t,0)=g(t,s,0)=0,\) \(J=[t_ 0,\infty)\), and f and g are continuously differentiable with respect to the components of x on \(J\times R^ n\) and \(J\times J\times R^ n\), respectively. By using the linearization technique and the technique of utilizing the variational system around an arbitrary solution, the authors investigate sufficient conditions for uniform Lipschitz stability of system (1) through their associated variational systems. Examples are given to demonstrate the relationship between the classical notions of stability and the new concept of Lipschitz stability.
Mathematics Subject Classification ID: 45M10
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 4065648
zbMATH Keywords: nonlinear integro-differential system
zbMATH Keywords: variational systems
zbMATH Keywords: uniform Lipschitz stability
reviewed by: Zhicheng Wang
MaRDI profile type: Publication
cites work: Perturbation theorems for nonlinear systems of ordinary different equations
cites work: Perturbations of nonlinear systems of differential equations. IV
cites work: Perturbations of nonlinear systems of differential equations. III
cites work: Lipschitz stability of nonlinear systems of differential equations
cites work: Perturbation theory for Volterra integrodifferential systems
cites work: Q5569338
cites work: Integro-differential equations and extension of Lyapunov's method
cites work: Asymptotic Behavior of Solutions of Volterra Integro-Differential Equations
full work available at URL: 0096-3003(88)90001-x
OpenAlex ID: W2000748514
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description: scientific article; zbMATH DE number 4065648