A nonresonance result with respect to the Fučik spectrum for a second order differential equations
type: Item
instance of: scholarly article
title: A nonresonance result with respect to the Fučik spectrum for a second order differential equations
zbMATH Open document ID: 0840.35029
author: Mabel Cuesta Leon
published in: Differential and Integral Equations
publication date: 1996-07-08T00:00:00Z
review text: This paper deals with the existence of solutions to the following problem: \[ - u''= f(t, u(t)),\quad t\in [0, 2\pi],\quad u(0)= u(2\pi);\;u'(0)= u'(2\pi),\tag{1} \] where \(f: [0, 2\pi]\times \mathbb{R}\to \mathbb{R}\) is asymptotically linear and ``asymmetric'' i.e., the behaviour of \(f/s\) for \(s\) going to \(+ \infty\) may be different from what it is for \(s\) going to \(-\infty\) (``jumping'' nonlinearities). Fucik and Dancer showed that the solvability of (1) depends on the behaviour of \(\lim_{s\to +\infty} f(t, s)/s\), \(\lim_{s\to -\infty} f(t, s)/s\) with respect to the so-called Fucik spectrum. Gossez and Omari proved the existence of solutions of (1) under some conditions, using the degree theory of Leray-Schauder. Using a minimax characterization of the Fucik spectrum derived by Figueiredo and Ruf, the author improves the results of Gossez and Omari.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 840298
zbMATH Keywords: asymmetric nonlinearity
zbMATH Keywords: Fucik spectrum
zbMATH Keywords: minimax characterization of the Fucik spectrum
reviewed by: Maria Agostina Vivaldi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 840298