Multiple periodic solutions for discrete Hamiltonian systems
type: Item
instance of: scholarly article
title: Multiple periodic solutions for discrete Hamiltonian systems
zbMATH Open document ID: 1115.39017
author: Honghua Bin
author: Zhiming Guo
author: Jianshe Yu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-04-13T00:00:00Z
review text: This work deals with the discrete Hamiltonian system \[ \begin{cases} \Delta u_1(n)=-H_{u_2}(n,u_1(n+1),u_2(n)),\\ \Delta u_2(n)=H_{u_1}(n,u_1(n+1),u_2(n)), &n\in \mathbb Z,\end{cases}\tag{1} \] where \(u_1,\,u_2\in \mathbb R^N\) and \(\Delta u_i(n)=u_i(n+1)-u_i(n)\), \(i=1,2\). The function \(H\) is \(T\)-periodic in the first variable \(n\) (\(T>0\) is a given integer), it is of class \(C^2\) in the second variable \(u_1\) and the third variable \(u_2\), and the gradient of \(H\) with respect to the last two variables, \(\nabla H\) is asymptotically linear at zero. In addition, \(\nabla H\) is asymptotically linear at infinity or \(H(n,z)\) has superquadratic growth as \(| z| \to\infty\). Under the above assumptions, the authors prove the existence of multiple \(T\)-periodic solutions for the problem (1), by using some results from Morse theory.
reviewed by: Rodica Luca
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 37J05
zbMATH DE Number: 5142209
zbMATH Keywords: discrete Hamiltonian systems
zbMATH Keywords: Morse theory
zbMATH Keywords: asymptotically linear function
zbMATH Keywords: superquadratic growth
zbMATH Keywords: periodic solution
Wikidata QID: Q58005338
MaRDI profile type: Publication
full work available at URL: j.na.2006.01.029
OpenAlex ID: W2017314723
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DOI: 10.1016/J.NA.2006.01.029
description: scientific article; zbMATH DE number 5142209