On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity
type: Item
instance of: scholarly article
title: On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity
zbMATH Open document ID: 1123.34033
author: Kurt Munk Andersen
author: Allan Sandqvist
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-04T00:00:00Z
review text: The scalar differential equation \(\dot{x} = f(t, x)\), where \(f(t, x)\) is continuous, \(T\)-periodic in \(t\) and weekly convex or concave in \(x\) has been investigated with respect to the number of \(T\)-periodic solutions by many authors. The general result is that there can be no, one or two \(T\)-periodic solutions or a connected band of \(T\)-periodic solutions. The last mentioned possibility can be excluded if \(f(t, x)\) is strictly convex or concave for some \(t\) in the period interval. The problem whether it is possible to determine the actual number of \(T\)-periodic solutions of a given equation of this type is treated in the present paper (assuming in addition a continuous derivative \(f_x'(t, x)\)) using the idea to view a periodic solution as a fixpoint of the successor function (the Poincaré map) \(S(\xi),\) i.e., as a zero of the deviation function \(P(\xi) = S(\xi) - \xi.\) It turns out that there are three cases. In each of these cases, using some monotonicity properties, the actual number of periodic solutions can be determined.
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5149557
zbMATH Keywords: closed solution
zbMATH Keywords: periodic solution
reviewed by: Bozhidar Cheshankov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.051
OpenAlex ID: W2026375647
Wikidata QID: Q115346256
cites work: Q3267606
cites work: Periodic Equilibria under Periodic Harvesting
cites work: The Number of Periodic Solutions of the Equation Ż=z N +p 1 (t )z
cites work: First order ordinary differential equations with several periodic solutions
cites work: Q3792992
cites work: Q3762900
cites work: On the number of solutions of the equation \(\sum^n_{j=0}a_j(t)x^j,0\leq t\leq 1\), for which \(x(0)=x(1)\)
cites work: Q5529795
cites work: Q5599348
cites work: On the number of closed solutions to an equation \(\dot x=f(t,x)\), where \(f_{x^ n}(t,x)\geq 0\) (n=1,2,or 3)
cites work: Qualitative behavior of the solutions of periodic first order scalar differential equations with weakly concave nonlinearity
cites work: Periodic solutions of polynomial first order differential equations
DOI: 10.1016/J.JMAA.2006.08.051
Recommended article: Q3129873
Recommended article: On the number of periodic solutions induced by a singular periodic solution
Recommended article: Closed form and multiplicity of characteristic numbers of periodic problems for differential equations
Recommended article: The number of periodic solutions of polynomial differential equations
Recommended article: Q3349207
Recommended article: Q4713710
Recommended article: On the close-to-convexity of entire solutions of differential equation
Recommended article: On the explicit determination of certain solutions of periodic differential equations
Recommended article: The numbers of periodic solutions of a polynomial differential equation
Recommended article: Q3808473
description: scientific article; zbMATH DE number 5149557