On existence of positive periodic solutions
type: Item
instance of: scholarly article
title: On existence of positive periodic solutions
zbMATH Open document ID: 0899.34028
DOI: 10.1007/BF01305348
author: Klaudiusz Wójcik
published in: Monatshefte für Mathematik
publication date: 1998-11-10T00:00:00Z
full work available at URL: https://eudml.org/doc/178819
review text: Let \(\Omega\) be an open convex subset of \(\mathbb{R}^n\). A differential equation \[ \dot x=f(t,x) \tag{1} \] on \(\mathbb{R}\times \text{cl} \Omega\) is considered. It is assumed that the right-hand side of (1) is \(T\)-periodic in \(t\). The main theorem asserts the existence of a fixed point of the Poincaré map associated to (1) in \(\Omega\) provided there exist two so-called periodic isolated segments in \([0,T]\times \text{cl} \Omega\) satisfying some topological conditions and properly located with respect to the equation. The theorem is applied to results on planar \(T\)-periodic nonautonomous Lotka-Volterra-type systems \[ \dot x=x(a(t)-b(t)y),\qquad \dot y=y(c(t)-d(t)x) \tag{2} \] on \([0,\infty)^2\) and \[ \dot x=a(t)x+b(t,y),\qquad \dot y=yc(t,y) \tag{3} \] on \(\mathbb{R}\times [0,\infty)\). The system (2) has a \(T\)-periodic solution in \((0,\infty)^2\) provided the maps \(a\), \(b\), \(c\), and \(d\) have positive values. If \(c(t,y)\) is positive for each \(y\geq 0\) small enough and every \(t\), and negative for some positive \(y\) and every \(t\) then (3) has a \(T\)-periodic solution in \(\mathbb{R}\times (0,\infty)\) provided, for every \((t,y)\), \(a(t)>0\) and \(b(t,y)\geq 0\) (or \(b(t,y)\leq 0\)).
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 1156714
zbMATH Keywords: periodic solutions
zbMATH Keywords: isolating block
zbMATH Keywords: Lotka-Volterra systems
reviewed by: Roman Srzednicki
MaRDI profile type: Publication
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OpenAlex ID: W2066425483
description: scientific article; zbMATH DE number 1156714