Existence of a pair of periodic solutions of an O.D.E. Generalizing a problem in nonlinear elasticity, via variational methods
type: Item
instance of: scholarly article
title: Existence of a pair of periodic solutions of an O.D.E. Generalizing a problem in nonlinear elasticity, via variational methods
zbMATH Open document ID: 0672.34030
DOI: 10.1016/0022-247X(88)90022-4
author: Sergio Gaete
author: Raúl Manásevich
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: It is known that the radial oscillations of an elastic spherical membrane subjected to an internal pressure p: \({\mathbb{R}}\to (0,+\infty)\), continuous and T-periocic, are governed by the scalar equation (1) \(\ddot x=\tilde f(t,x),\) where \(\tilde f:{\mathbb{R}}\times (0,+\infty)\to {\mathbb{R}}\) is given by (2) \(\tilde f(t,x)=p(t)x^ 2-x+x^{-5}.\) Here x(t) is the ratio of the deformed radius of the membrane at time t with respect to the undeformed radius. A more general equation than (1), i.e. (3) \(\ddot x=g(t,x),\) where \(g: {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}}\) is continuous and T-periodic in t is considered in the article. The existence of two T- periodic continuous functions \(z_ i(i=1,2): {\mathbb{R}}\to {\mathbb{R}}\) such that \((i)\quad \max_{t\in {\mathbb{R}}}z_ 1(t)<\min_{t\in {\mathbb{R}}}z_ 2(t),\) (ii) if \(x\in (z_ 1(t),z_ 2(t))\) then \(g(t,x)<0\), and if \(g(t,x)\leq 0\) than \(x\in [z_ 1(t),z_ 2(t)]\) was assumed. Under these conditions the existence of at least two T-periodic solutions of the equation (3) by means of a variational method is proved.
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 4099716
zbMATH Keywords: radial oscillations
zbMATH Keywords: elastic spherical membrane
MaRDI profile type: Publication
cites work: Dual variational methods in critical point theory and applications
cites work: Q4748596
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Q4040206
cites work: A Note on the Topological Degree at a Critical Point of Mountainpass-Type
full work available at URL: 0022-247x(88)90022-4
OpenAlex ID: W1983950961
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description: scientific article; zbMATH DE number 4099716