An existence theorem for periodic solutions in convex sets with applications
type: Item
instance of: scholarly article
title: An existence theorem for periodic solutions in convex sets with applications
zbMATH Open document ID: 0653.34029
DOI: 10.1007/BF03323213
published in: Results in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(C\subset \mathbb{R}^m\) be a closed convex set with nonempty interior and let \(G\subset C\) be bounded and open relatively to \(C\). We prove a continuation theorem which ensures the existence of a solution \(x(\cdot)\) of the boundary value problem \(\dot x=F(t,x)\), \(x(0)=x(T)\), such that \(x(t)\) remains in the closure of \(G\) relatively to \(C\), for all \(t\). To do this, a flow-invariance condition is combined with the functional-analytic framework of coincidence degree; moreover, the usual assumptions on the ``averaged'' vector field \(\bar f_0(x):=(1/T) \int^T_0 f(s,x;0)\, ds\) are replaced by a condition on the fixed point index of the map \(r(I+\bar f_0)\), where \(r: \mathbb R^m\to C\) is the canonical projection. Applications to a periodically perturbed differential system and to a Lotka-Volterra type equation are provided, too.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 4065460
zbMATH Keywords: flow-invariant set
zbMATH Keywords: fixed point index
zbMATH Keywords: continuation theorem
zbMATH Keywords: periodically perturbed differential system
zbMATH Keywords: Lotka-Volterra type equation
author: Anna Capietto
author: Fabio Zanolin
reviewed by: Anna Capietto
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03323213
OpenAlex ID: W2044261908
description: scientific article; zbMATH DE number 4065460