Abstract:The paper is devoted to the study of the properties of the Fu\v c\'\i k spectrum. In the first part, we analyse the Fu\v c\'\i k spectra of the problems with one second order ordinary differential equation with Dirichlet, Neumann and mixed boundary conditions and we present the explicit form of nontrivial solutions. Then, we discuss the problem with two second order differential equations with mixed boundary conditions. We show the relation between the Dirichlet boundary value problem and mixed boundary value problem; using results of E.~Massa and B.~Ruf, we derive some properties of the Fu\v c\'\i k spectrum of the mixed boundary value problem. Finally, we introduce a new proof of the closedness of the Fu\v c\'\i k spectrum and a lemma about convergence of the corresponding nontrivial solutions.
Keywords: Fu\v c\'\i k spectrum, system of ordinary differential equations of the second order, Dirichlet, Neumann and mixed boundary conditions
AMS Subject Classification: 34A34 34B15 47J10