On second order, periodic, symmetric, differential systems having subharmonics of all sufficiently large orders
type: Item
instance of: scholarly article
title: On second order, periodic, symmetric, differential systems having subharmonics of all sufficiently large orders
zbMATH Open document ID: 0853.34039
author: Nguyen Phuong Cac
author: Alan C. Lazer
published in: Journal of Differential Equations
publication date: 1997-01-06T00:00:00Z
review text: This article deals with the second order differential system (1) \(u'' (t) + g(u (t)) = e(t)\), where \(u : \mathbb{R} \to \mathbb{R}^N\), \(N \geq 1\), \(g = \nabla G\) with \(G \in C^1 (\mathbb{R}^N, \mathbb{R})\), and \(e : \mathbb{R} \to \mathbb{R}^N\) is a function which is continuous and periodic in \(t\), with minimal period \(2 \pi\). A periodic solution \(u\) of equation (1) is called a subharmonic of order \(k\) if \(2k \pi\) is the least positive period of \(u\). In preceding papers, some authors proved the existence of infinitely many subharmonics under suitable assumptions of subquadratic growth at infinity for the function \(G\). Some previous results of this kind (see, e.g. [\textit{A. Fonda} and \textit{A. C. Lazer}, Proc. Am. Math. Soc. 115, 183-190 (1992; Zbl 0752.34027)], or [\textit{A. Fonda} and \textit{M. Ramos}, J. Differential Equations 109, 354-372 (1994; Zbl 0798.34048)], ensured the existence of subharmonics of order \(p\), for each sufficiently large prime number \(p\). The same conclusion is achieved in Theorem 2.1 of this article, with respect to the existence of a sequence of large amplitude odd subharmonics, under the assumptions: \(G \) even, \(e (\cdot)\) odd, \(G (\xi) \to \infty\) and \(G(\xi)/ |\xi |^2 \to 0\) as \(|\xi |\to \infty\). In the main result of the paper (Theorem 3.1), the authors address their interest to the search of subharmonics of order \(k\), for each sufficiently large integer \(k\). This conclusion, which is usually much harder to prove, is obtained here under the supplementary condition \[ \exists \theta \in (0,2),\;\exists b : \theta G (\xi) \leq g (\xi) \cdot \xi + b,\;\forall \xi \in \mathbb{R}^N. \] The proof makes use of a variational approach, providing, for each \(k \geq 1\), a \(2k \pi\)-periodic solution \(u_k\) obtained as a minimum of the functional associated to (1) in a space of odd \(2k \pi\)-periodic functions. It is also proved that \(|u_k |_\infty \to \infty\) as \(k \to \infty\). The key poi…
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 885508
zbMATH Keywords: subharmonic of order \(k\)
zbMATH Keywords: second order differential system
zbMATH Keywords: periodic solution
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0076
OpenAlex ID: W2074547872
DOI: 10.1006/JDEQ.1996.0076
description: scientific article; zbMATH DE number 885508