On the Resonance Problem for the 4$^{\mathrm{th}}$ Order Ordinary Differential Equations, Fucik's spectrum

Petr Necesal

Address. Centre of Applied Mathematics, Faculty of Applied Sciences, University of West Bohemia, P.O. Box 314,
                306 14 Pilsen, Czech Republic

E-mail: necesal@kma.zcu.cz

Abstract. We consider the boundary value problems for the fourth order nonlinear differential equation $u^{\mathrm{IV}} = f(x, u)$ together with three different boundary conditions (the Dirichlet, the periodic and the Navier boundary conditions). We discuss the existence results for these boundary value problems at resonance. Our results contain the Landesman--Lazer type conditions. We also show some numerical results concerning {\em Fucik's spectrum} for the boundary value problems for the differential equation $u^{\mathrm{IV}} = \mu u^{+} - \nu u^{-}$, where $u^+=\max\{u,0\}$ and $u^-=\max\{-u,0\}$, together with our three boundary conditions.

AMSclassification. 34B15, 34L16, 65L15

Keywords. Fucik's spectrum, Landesman-Lazer type condition