On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument
type: Item
instance of: scholarly article
title: On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument
zbMATH Open document ID: 1061.34055
author: Rong Yuan
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-01-11T00:00:00Z
review text: It is shown that \[ x''(t)+ p(t) x(t)= q x([t])+ f(t)\tag{\(*\)} \] has a unique almost-periodic (ap) solution \(x\) on \(\mathbb{R}\), with Bohr spectrum \(\sigma_B(x)\subset \sigma_B(f)+ 2\pi\mathbb{Z}\), provided \(p:\mathbb{R}\to \mathbb{R}\) is continuous with period \(1\), \(0\neq q\) and real, \(f\) ap, \(q \int^1_0 v(s)\,ds\neq 1\) with \(v''+ pv= 0= v(0)\), \(v'(0)= 1\), and the roots \(z\) of a certain quadratic equation satisfy \(|z|\neq 1\). So, \(X\) is quasiperiodic if \(f\) is. If \(f\) has a rational period \(p/q\), then \(x\) has period \(q\). If \(f\) has an irrational period \(T\) with absolutely convergent Fourier series and \(f(-t)= -\overline{f(t)}\), then \(x\) is quasiperiodic with the basis \(1\), \(2\pi/T\). Similar results are obtained if in \((*)\) \(q x([t])\) is replaced by \(q x([t- 1])+ \mu g(t, x([t- 1,t])\) for small \(\mu\). Several explicit examples are discussed.
reviewed by: Hans Günzler
Mathematics Subject Classification ID: 34K14
Mathematics Subject Classification ID: 34K13
zbMATH DE Number: 2126033
zbMATH Keywords: almost-periodic solution
zbMATH Keywords: quasiperiodic solution
zbMATH Keywords: spectrum of solution
zbMATH Keywords: piecewise constant argument
zbMATH Keywords: delay equations
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.031
OpenAlex ID: W2085372545
cites work: Q3657815
cites work: Almost periodic differential equations and almost periodic flows
cites work: Q3975198
cites work: Q5563566
cites work: Almost periodic differential equations
cites work: Q3965074
cites work: On Almost Periodic Solutions of a Class of Differential Equations
cites work: Q4858399
cites work: A second-order delay differential equation with multiple periodic solutions
cites work: Excitability of a second-order delay differential equation
cites work: Complicated dynamics in a delay Klein–Gordon equation
cites work: On the existence of almost periodic solutions of a singularly perturbed differential equation with piecewise constant argument
cites work: On the regular solution of a second order delay differential equation
cites work: On the second-order differential equation with piecewise constant argument and almost periodic coefficients
cites work: ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
cites work: Uniform asymptotic stability of hybrid dynamical systems with delay
DOI: 10.1016/J.NA.2004.07.031
description: scientific article; zbMATH DE number 2126033