## Differential and Integral Equations

### Some remarks about the Fucik spectrum and application to equations with jumping nonlinearities

Chakib Abchir

#### Abstract

Let $L$ be a selfadjoint operator with compact resolvent and $\lambda$ an eigenvalue of $L$. When $\lambda$ is simple, it is well known that the Fucik spectrum $\Sigma$ near $\lambda$ consists of two nonincreasing curves. In this paper, we show that when $\lambda$ is not simple, $\Sigma$ contains two nonincreasing curves such that all points above or under both curves are not in $\Sigma$. After that, we give some existence results of solutions of the equation $Lu=\alpha u^{+}-\beta u^{-}+g(.,u)$ where $u^{\pm }=\max (0,\pm u)$.

#### Article information

Source
Differential Integral Equations, Volume 15, Number 9 (2002), 1045-1060.

Dates
First available in Project Euclid: 21 December 2012