\(T\)-periodic solutions and a priori bounds
type: Item
instance of: scholarly article
title: \(T\)-periodic solutions and a priori bounds
zbMATH Open document ID: 0981.39005
DOI: 10.1016/S0895-7177(00)00161-8
published in: Mathematical and Computer Modelling
publication date: 2001-03-19T00:00:00Z
review text: In view of Schaefer's theorem, in the case of a normed linear space \((B,\parallel \cdot \parallel)\) and of a continuous mapping \(H\) of \(B\) into \(B\) which is compact on each bounded subset of \(B\), we have either (i) the equation \(x=\lambda Hx\) has a solution for \(\lambda =1\) or (ii) the set of all such solutions, for \(0<\lambda <1\), is unbounded. Using this fixed-point theorem and a Lyapunov functional, the author studies the existence of a \(T\)-periodic solution of the difference equation \[ \Delta x(n)=Dx(n)+\sum_{j=-\infty }^nC(n-j)x(j)+g(n) \] where \(D\) and \(C\) are given \(k\times k\) real matrices with \(\sum_{u=0}^\infty |C(u)|<\infty \), and \(x(n)\), \(g(n)\) are \(k\times 1\) vectors with \(g(n+T)=g(n)\). An example which gives a necessary condition for the existence of periodic solution of this equation, and a sufficient condition concerning the simpler case of \(\Delta x(n)=Dx(n)+g(n)\) are included.
reviewed by: Nicolae Cotfas
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1545784
zbMATH Keywords: difference equations
zbMATH Keywords: periodic solutions
zbMATH Keywords: a priori bound
zbMATH Keywords: Schaefer's fixed point theorem
zbMATH Keywords: Lyapunov functional
author: Youssef N. Raffoul
MaRDI profile type: Publication
cites work: Periodic Solutions of Linear Integro‐Differential Equations
cites work: Periodicity and stability of linear Volterra difference systems
cites work: Über die Methode der a priori-Schranken
cites work: Q3862642
cites work: Volterra integral and differential equations
cites work: Liapunov functionals of convolution type
description: scientific article; zbMATH DE number 1545784