Turning points and eigenvalue problems
type: Item
instance of: scholarly article
title: Turning points and eigenvalue problems
zbMATH Open document ID: 0853.34076
author: Eugene Stepanov
author: Sergey A. Vavilov
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-08-29T00:00:00Z
review text: The periodic boundary value problem for a linear O.D.E. \(\ddot x + \alpha \dot x + ({2 \pi \over T})^2x = \lambda f(t) x\) \((0 \leq t \leq T)\), \(x(0) = x(T)\), \(\dot x(0) = \dot x(T)\), with spectral parameter \(\lambda\) is studied. Here a damping coefficient \(\alpha \neq 0\) and a smooth \(T\)-periodic function \(f \neq 0\) are given, the latter being allowed to vanish on some part of \([0,T]\). Under suitable assumptions on the Fourier coefficients of \(f\) the authors prove the existence of characteristic values \(\lambda \in \mathbb{R}\) and corresponding nontrivial solutions \(x(t)\). Their approach uses certain nonlinear branching equations and topological degree theory. Applications to Hill's equation are discussed. In a final section the results are employed to investigate the stability of a cylindrical shell which is subject to the action of compression or tension forces.
reviewed by: Frank Mantlik
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 902569
zbMATH Keywords: periodic boundary value problem
zbMATH Keywords: linear O.D.E.
zbMATH Keywords: Hill's equation
zbMATH Keywords: stability of a cylindrical shell
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0170
OpenAlex ID: W2094009431
DOI: 10.1006/JMAA.1996.0170
description: scientific article; zbMATH DE number 902569