Affine-periodic solutions for higher order differential equations
type: Item
instance of: scholarly article
title: Affine-periodic solutions for higher order differential equations
zbMATH Open document ID: 1445.34067
author: Yanyan Li
published in: Applied Mathematics Letters
publication date: 2020-04-07T00:00:00Z
review text: The authors consider nonautonomos differential equations of \(n^{\mbox{th}}\) order \[ \mathcal{P}(D)(x) = f(t,x), \] where \(D\) stands for the derivative of \(x(t)\) with respect to \(t\) and \(\mathcal{P}\) is a monic polynomial with no constant term. The function of \(f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}\) in addition being continuous is assumed to be an \(a\)-affine \(T\)-periodic function, i.e., \[ f(t+T,x)=af(t,a^{-1}x)\ \ \forall t \in \mathbb{R}, \] where \(a\in\mathbb{R}\setminus\{0\}\), \(T>0\). The main goal of the paper is to establish the existence of \(a\)-affine \(T\)-periodic solutions of the equation under consideration, where \(a\)-affine \(T\)-periodic solution is defined as the one satisfying the condition \[ x(t+T) = ax(t)\ \ \forall t \in \mathbb{R}. \] As I have never heard before about affine-periodic functions, the first thing I did was a quick search. According to MathSciNet and Google the affine-periodicity was introduced by \textit{Y. Zhang} et al. [Abstr. Appl. Anal. 2013, Article ID 157140, 4 p. (2013; Zbl 1303.34033)]. All publications and related work with exception of one paper published 2018 in an obscure Turkish journal, lead to the same research group in People's republic of China. I read \(4\)-page of [loc. cit.] and the physical motivation behind the notion of affine-periodicity is lacking. The mathematics in the paper under review and [loc. cit.] seems correct but I could not bear to read more without being able to convince myself that the concept of affine-periodicity has any significance.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A45
zbMATH DE Number: 7187453
zbMATH Keywords: affine-periodic solutions
zbMATH Keywords: higher-order differential equations
zbMATH Keywords: extremum principle
zbMATH Keywords: lower and upper solutions method
zbMATH Keywords: monotone iterative technique
reviewed by: Predrag Punoševac
MaRDI profile type: Publication
full work available at URL: j.aml.2020.106341
OpenAlex ID: W3011780719
cites work: Affine-periodic solutions for dissipative systems
cites work: A Disorienting Look at Euler's Theorem on the Axis of a Rotation
cites work: Lattice Symmetry-Breaking Perturbations for Spiral Waves
cites work: On the asymptotic wavenumber of spiral waves in $\lambda -\omega $ systems
cites work: Bifurcation of nonlinear Bloch waves from the spectrum in the Gross-Pitaevskii equation
cites work: Dynamic stability of vortex solutions of Ginzburg-Landau and nonlinear Schrödinger equations
cites work: Levinson’s Problem on Affine-Periodic Solutions
cites work: Rotating periodic solutions of second order dissipative dynamical systems
cites work: LaSalle type stationary oscillation theorems for affine-periodic systems
cites work: EXISTENCE OF DISSIPATIVE-AFFINE-PERIODIC SOLUTIONS FOR DISSIPATIVE-AFFINE-PERIODIC SYSTEMS
cites work: Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems
cites work: Rotating periodic solutions for super-linear second order Hamiltonian systems
cites work: Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems
cites work: Affine-periodic solutions by averaging methods
cites work: Rotating periodic solutions for convex Hamiltonian systems
cites work: Existence of affine-periodic solutions to Newton affine-periodic systems
cites work: Q3989729
cites work: An abstract monotone iterative technique
DOI: 10.1016/J.AML.2020.106341
description: scientific article; zbMATH DE number 7187453