New results on the existence of periodic solutions of periodically perturbed conservative systems
type: Item
instance of: scholarly article
title: New results on the existence of periodic solutions of periodically perturbed conservative systems
zbMATH Open document ID: 0874.34037
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-08-18T00:00:00Z
review text: The system of ordinary differential equations \[ \ddot{x}(t)+ \nabla G(u(t))=f(t),\tag{1} \] is considered, where \(f:\mathbb{R}^1 \rightarrow \mathbb{R}^n\) is a continuous function with period \(2\pi\) and \(G:\mathbb{R}^n \rightarrow \mathbb{R}^1\) has a continuous second partial derivative. Equation (1) is interpreted as the Newtonian equation of motion of a mechanical system subject to conservative internal forces and periodic external forces. The investigation which began in [\textit{Z. Shen} and \textit{M. A. Wolfe}, J. Math. Anal. Appl. 153, No. 1, 78-83 (1990; Zbl 0726.34033)] is continued. Weak sufficient conditions for the existence of a unique \(2\pi\)-periodic solution of (1) with boundary conditions are given. It is shown that some conditions in the previous works are unnecessary.
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1005588
zbMATH Keywords: periodic solutions
zbMATH Keywords: boundary value problem
zbMATH Keywords: system of ordinary differential equations
zbMATH Keywords: mechanical system
zbMATH Keywords: conservative internal forces
zbMATH Keywords: periodic external forces
author: Zuhe Shen
author: Michael A. Wolfe
MaRDI profile type: Publication
OpenAlex ID: W2083086360
cites work: An existence theorem for periodically perturbed conservative systems
cites work: Nonlinear boundary value problems and a global inverse function theorem
cites work: Periodically perturbed conservative systems and a global inverse function theorem
cites work: Q5336643
cites work: Q5569338
cites work: On periodically perturbed conservative systems
cites work: Application of a Lemma on Bilinear Forms to a Problem in Nonlinear Oscillations
cites work: On Poincaré's perturbation theorem and a theorem of W. S. Loud
cites work: Homeomorphisms Between Banach Spaces
cites work: On the periodic solution to the Newtonian equation of motion
cites work: On the existence of periodic solutions of periodically perturbed conservative systems
DOI: 10.1006/JMAA.1997.5204
description: scientific article; zbMATH DE number 1005588