This project deals with the oscillation theory for differential equations on hybrid time domains, including the continuous and discrete time. The principal aim is to explain the nature of oscillations on hybrid time domains (also called time scales), being an open problem in the theory of differential equations. We propose new approach to this problem by the investigation of the comparative index, which is a relatively new notion from matrix analysis and which was originally developed for the study of discrete oscillations. We also aim to develop related problems from the spectral theory or variational analysis on discrete and hybrid time domains, where the existence or nonexistence of oscillations plays a fundamental role, such as in the study of self-adjoint extensions of linear relations, spectral counting functions, or the optimality conditions in nonlinear optimization problems. We will also develop new applications of the methods and techniques from the oscillation theory in matrix analysis and other related fields (e.g. the Maslov index).

Publications