Subharmonic and multiple subharmonic solutions for second order differential systems.
type: Item
instance of: scholarly article
title: Subharmonic and multiple subharmonic solutions for second order differential systems.
zbMATH Open document ID: 1039.34036
author: Norimichi Hirano
author: Naoki Shioji
published in: Differential and Integral Equations
publication date: 2003-11-17T00:00:00Z
review text: The authors seek for subharmonic solutions of a system of second-order differential equations in \(\mathbb R^N\) of the form \[ \ddot u(t)+G'(u(t))=f(t) \quad \text{for } t\in \mathbb R, \tag{1} \] where \(N\) is a natural number, \(f\in C(\mathbb R,\mathbb R^N)\) is \(T\)-periodic, \(G\in C^2(\mathbb R^N,\mathbb R)\) is nonconvex, \(\ddot u={d^2u\over dt^2}\), and \(G'=\text{grad} G\). Here, a subharmonic solution of (1) means a periodic solution of (1) which is \(kT\)-periodic but not \(T\)-periodic for some natural \(k\). The existence of subharmonic solutions is proved by means of a variational argument and a Morse index theorem for minimax-type critical points. The authors show that, if \(f\) is contained in some subset of the space of \(T\)-periodic, continuous functions from \(\mathbb R\) into \(\mathbb R^N\), then there exist at least two \(kT\)-periodic but not \(T\)-periodic solutions of (1) for every sufficiently large prime number \(k\).
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 2004891
zbMATH Keywords: existence
zbMATH Keywords: multiplicity
zbMATH Keywords: periodic
zbMATH Keywords: second-order
zbMATH Keywords: variational argument
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2004891