A contribution to the problem of coexistence of two periodical solutions of the Hill's equation


Dragan Dimitrovski, Vladimir RajoviŠ


There are 17 theorems on characteristics of periodical solutions of the Hill's equation presented in this paper. These theorems were not presented either in the known monograph of Magnus and Winkler \cite{LIT01}, or in Kamke's monograph \cite{LIT02}, and \cite{LIT03}.\\ An elementary approach to the most important equation in the oscillations theory -- the Hill's equation, is given in this paper as opposed to the famous monograph [4] where the problem of periodicity of solutions is treated by means of the Floquet theorem. The approach is based on simple yet in the literature inadequately emphasized features of periodicity.\\ The particularly important question will be: when the integral of the only coefficient of the equation $\int b(x)dx$ is periodical, and when it is not. Depending on this, there are given various conditions for existence of entire and discontinuous solutions of the equation (\ref{eq.1}).\\ We will not deal with the particular case of (\ref{eq.1}), the equation $y^{\prime \prime }+\left( \lambda +Q(x)\right) y=0${\small , }$\lambda \neq 0$, where $b(x)=\lambda +Q(x)$, and $\int b(x)dx=\lambda x+\int Q(x)dx$ is not periodical, a case especially important for boundary problems solved by Floquet theorem in the way that there are two series of constants for $\lambda $ $\left\{\lambda _{n}\right\} $, and $\left\{ \lambda _{n}^{\prime}\right\} $, whose alternating combination is crucial for the stability of the solutions.\\ Therefore, the paper is more based on and related to quadratural aspects.