On the resonance problem for the \(\text{4}^{\text{th}}\) order ordinary differential equations, Fučík’s spectrum. (English) Zbl 1090.34521

Nečesal, Petr

Arch. Math., Brno 36, Suppl., 531-542 (2000).

Summary: We consider the boundary value problems for the fourth order nonlinear differential equation \(u^{\text{IV}} = f(x, u)\) 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 Fučík’s spectrum for the boundary value problems for the differential equation \(u^{\text{IV}} = \mu u^{+} - \nu u^{-}\), where \(u^+=\max \{u,0\}\) and \(u^-=\max \{-u,0\}\), together with our three boundary conditions.

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34L16	Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
65L15	Numerical solution of eigenvalue problems involving ordinary differential equations

Keywords:

Fučík’s spectrum; Landesman-Lazer type condition