A nonresonance result with respect to the Fučik spectrum for a second order differential equations
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.
minimax characterization of the Fucik spectrum
scientific article; zbMATH DE number 840298