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

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.


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
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