Using closed orbits to bifurcate many periodic solutions for pendulum-type equations
type: Item
instance of: scholarly article
title: Using closed orbits to bifurcate many periodic solutions for pendulum-type equations
zbMATH Open document ID: 1068.34041
author: Antonio J. Ureña
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-02-22T00:00:00Z
review text: The paper deals with the boundary value problem for a pendulum-like equation of the type \[ u^{\prime\prime}+ cu^\prime+g(u)=\bar e+\tilde e(t)\,, \quad u(T)-u(0)=0\,,\quad u^\prime(T)-u^\prime(0)=0\,. \] Here, \(T>0\) is a real constant, \(g\in C(R/2\pi Z)\) is a continuous \(2\pi\)-periodic function with zero mean (i.e., \(\int_0^{2\pi} g(u)\,du=0\)), and the external force \(e=\bar e+\tilde e\in L_1[0,\,T]\) is decomposed as a sum of a constant \(\bar e\) and a function \(\tilde e\) with zero mean. A result of \textit{F. Donati} [C. R. Acad. Sci. Paris Sér. I Math. 317, 667--672 (1993; Zbl 0783.34033)] says that in the periodic problem for the conservative forced pendulum equation (i.e., with \(g(u)=\Lambda \sin(u)\)), it is always possible to find forcing terms \(\tilde e\) such that the problem has at least four geometrically different solutions (two solutions are called geometrically different if their difference is not a multiple of \(2\pi\)). This result has been extended by \textit{R. Ortega} [Rocky Mt. J. Math. 27, 861--876 (1997; Zbl 0899.34031)], who has shown that within the same framework it is possible to change \(4\) by any number. In the present paper it is shown how Ortega's theorem remains valid in the presence of friction, and for a bigger class of oscillating functions \(g\), namely, those which are restrictions to the real line of entire functions. The proof is based on topological tools.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 2136650
zbMATH Keywords: pendulum-like equation
zbMATH Keywords: periodic solutions
zbMATH Keywords: closed orbits
zbMATH Keywords: bifurcation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.11.057
OpenAlex ID: W2123133674
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
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DOI: 10.1016/J.JMAA.2003.11.057
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description: scientific article; zbMATH DE number 2136650