Ondřej Došlý

A necessary and sufficient condition for oscillation of the Sturm--Liouville dynamic equation on time scales

In this paper, we investigate oscillatory properties of the second order Sturm-Liouville dynamic equation on a time scale (r(t)xΔ)Δ + c(t)xσ = 0. (*) Using the so-called trigonometric transformation we establish a necessary and sufficient condition for oscillation of (*). This condition is then used to derive an explicit oscillation criterion for this equation.

Sturmian and spectral theory for discrete symplectic systems

We consider symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most n. In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.

Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales

In this paper we study qualitative properties of the socalled symplectic dynamic system (S) zδA =Stz on an arbitrary time scale T, providing a unified theory for discrete symplectic systems and differential linear Hamiltonian systems . We define dis-conjugacy (no focal points) for conjoined bases of (S) and prove, under a certain minimal normality assumption, that disconjugacy of (S) on the interval under consideration is equival ent to the positivity of the associated quadratic functional. Such statement is commonly called Jacobi condition. We discuss also the solvability of the corresponding Riccati matrix equation and transformations. This work may be regarded as a generalization of the results recently obtained by the second author for linear Hamiltonian systems on time scales.

Oscillation theorems for symplectic difference systems

We consider symplectic difference systems involving a spectral parameter, together with the Dirichlet boundary conditions. The main result of the paper is a discrete version of the so-called oscillation theorem which relates the number of finite eigenvalues less than a given number to the number of focal points of the principal solution of the symplectic system. In two recent papers the same problem was treated and an essential ingredient was to establish the concept of the multiplicity of a focal point. But there was still a rather restrictive condition needed, which is eliminated here by using the concept of finite eigenvalues (or zeros) from the theory of matrix pencils.

Symplectic Dynamic Systems

This chapter continues from [86, Chapter 7] the study of symplectic dynamic systems of the form (S)

Positive Semidefiniteness of Discrete Quadratic Functionals

z^\Delta = S(t)z

Oscillation and spectral theory for symplectic difference systems with separated boundary conditions

on time scales. In particular, we investigate the relationship between the nonoscillatory properties (no focal points) of certain conjoined bases of (S), the solvability of the corresponding Riccati matrix dynamic equation, and the positivity of the associated quadratic functional. Furthermore, we establish Sturmian separation and comparison theorems. As applications of the transformation theory of symplectic dynamic systems, we study trigonometric and hyperbolic symplectic systems, and the Prufer transformation.

Nonexistence of Positive Solutions of PDE's with p-Laplacian

Abstract Nonoscillation criteria for the half-linear second-order difference equation are established. These criteria are derived using the Riccati and variational technique.