Foliations, associated reductions and lower and upper solutions
type: Item
instance of: scholarly article
title: Foliations, associated reductions and lower and upper solutions
zbMATH Open document ID: 1026.34018
author: C. De Coster
author: Massimo Tarallo
published in: Calculus of Variations and Partial Differential Equations
publication date: 2002-12-01T00:00:00Z
review text: In this interesting paper, the authors answer \textit{J. Mawhin}'s question [Boll. Unione Mat. Ital., VI. Ser., A 3, 229-238 (1984; Zbl 0547.34032)] in establishing the existence of a solution to the periodic boundary value problem \[ u''(t) + g(t,u(t)) = 0, \quad u(0)=u(T), \quad u'(0)=u'(T), \] for \(g\) asymptotically linear and satisfying a nonresonance condition to the left of the eigenvalue \((2\pi/T)^2\) as well as an Ahmad-Lazer-Paul condition to the right of the eigenvalue \(0\); that is \[ \eta \leq \frac{g(t,s)}{s} \leq \gamma < (\frac{2\pi}{T})^2 \quad \text{for \(t \in [0,T]\) and \(|s|\geq R\)}, \] and \[ \int_0^T\int_0^s g(t,r) dr ds \to \infty \text{ as \(|s|\to \infty\)}. \] In fact, a more general result is presented taking into account the Fučik spectrum. The approach is mixed. It uses variational and topological methods with upper and lower solutions. Variational methods permit to deduce the existence of upper and lower solutions which are used to obtain a priori bounds of a suitable problem. It is worth to mention that the lower and upper solutions are not well ordered. Therefore, the solution does not lie between them.
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1837033
zbMATH Keywords: foliations
zbMATH Keywords: reductions
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: Fučik spectrum
zbMATH Keywords: existence
reviewed by: Marlène Frigon
MaRDI profile type: Publication
full work available at URL: s005260100116
OpenAlex ID: W326215264
DOI: 10.1007/S005260100116
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description: scientific article; zbMATH DE number 1837033