Characterization of the spectra of the Hill's equation coupled to different boundary value conditions and application to nonlinear boundary problems

Abstract

In this paper we will characterize the spectrum of the second order Hill's equation coupled to several boundary value conditions. More concisely, the idea consists of study the spectrum of the second-order differential Hill's equation coupled to Initial, Final, Neumann, Dirichlet, Periodic and Mixed boundary conditions, by applying the equality (10) proved by the authors in \cite{Cabada} and expressing the Green's function of the Hill's equation coupled to a given boundary condition as a combination of the Green's function related to another different boundary condition. These spectra are characterized as suitable sets of real values that verify an equality that depends on the Green's function of each case. We will also deduce some properties of these spectra and identities between Green's functions. The work continuous on the lines initiated on \cite{smic} and \cite{kkkk}. It is important to remark that the ideas and arguments used to deduce the comparison between the corresponding spectrum of the considered problems, and their characterization in many cases, are completely different to the ones used in \cite{kkkk}.