## Grant KJB100190701
1.1.2007 - 31.12.2009

Grantor: Grant Agency of Czech Academy of Sciences

### Asymptotics, oscillation, and quadratic functionals in theory of dynamic equations

This project is focused on the study of asymptotic and oscillatory properties of linear, half-linear, and nonlinear equations, symplectic and Hamiltonian systems, and associated quadratic functionals. The results will be connected with optimality conditions for nonlinear problems in the calculus of variations and optimal control, since the second variation of these problems is a quadratic functional. An important aspect of this project is the inclusion of the theory of dynamic equations on time scales, which allows to study differential and difference equations and systems within one theory, to explain and understand the discrepancies between these theories, and at the same time to generalize them to other time scales. We will focus e.g. on symplectic systems without normality or on the theory of conjugate and coupled points for these systems. Furthermore, we will study effective criteria for the existence of solutions with prescribed properties and open problems arising in this area.