Petr Hasil

Oscillation of half-linear differential equations with asymptotically almost periodic coefficients

We investigate second-order half-linear differential equations with asymptotically almost periodic coefficients. For these equations, we explicitly find an oscillation constant. If the coefficients are replaced by constants, our main result (concerning the conditional oscillation) reduces to the classical one. We also mention examples and concluding remarks.MSC:34C10, 34C15.

Oscillation and Nonoscillation of Asymptotically Almost Periodic Half-Linear Difference Equations

We analyse half-linear difference equations with asymptotically almost periodic coefficients. Using the adapted Riccati transformation, we prove that these equations are conditionally oscillatory. We explicitly find a constant, determined by the coefficients of a given equation, which is the borderline between the oscillation and the nonoscillation of the equation. We also mention corollaries of our result with several examples.

Conditional Oscillation of Half-Linear Differential Equations with Coefficients Having Mean Values

We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

Critical Oscillation Constant for Difference Equations with Almost Periodic Coefficients

We investigate a type of the Sturm-Liouville difference equations with almost periodic coefficients. We prove that there exists a constant, which is the borderline between the oscillation and the nonoscillation of these equations. We compute this oscillation constant explicitly. If the almost periodic coefficients are replaced by constants, our result reduces to the well-known result about the discrete Euler equation.

Almost periodic transformable difference systems

Abstract We analyse almost periodic homogeneous linear difference systems whose coefficient matrices belong to the so-called transformable groups. We introduce the notion of weakly transformable groups and we use this notion to obtain generalizations of known results about non-almost periodic solutions of considered systems.

Limit periodic homogeneous linear difference systems

We study limit periodic homogeneous linear difference systems, where the coefficient matrices belong to a bounded group. We find groups of matrices with the property that the systems, which do not possess any non-zero asymptotically almost periodic solution, form a dense subset in the space of all considered systems. Analogously, we analyse almost periodic systems as well.

Critical higher order Sturm–Liouville difference operators

We investigate the so-called critical 2nth-order Sturm–Liouville difference operators and associated symmetric banded matrices. We show that arbitrarily small (in a certain sense) negative perturbation of a non-negative critical operator leads to an operator which is no longer non-negative.

Conjugacy of Self-Adjoint Difference Equations of Even Order

We study oscillation properties of 2n-order Sturm-Liouville difference equations. For these equations, we show a conjugacy criterion using the p-criticality (the existence of linear dependent recessive solutions at ∞ and -∞). We also show the equivalent condition of p-criticality for one term 2n-order equations.

Values of limit periodic sequences and functions

Abstract We consider limit periodic sequences and functions with values in pseudometric spaces. We construct limit periodic sequences and functions with given values. For any totally bounded countable set, we find a limit periodic sequence which attains each value from this set periodically. A corresponding result concerning such a construction of limit periodic functions is proved as well. For any totally bounded countable set which is dense in itself, we construct a limit periodic bijective map from the integers into this set. As corollaries, we obtain new results about non-almost periodic solutions of almost periodic transformable difference systems.

Friedrichs extension of operators defined by even order Sturm-Liouville equations on time scales

In this paper we characterize the Friedrichs extension of operators associated with the 2n-th order Sturm-Liouville dynamic equations on time scales with using the time reversed symplectic systems and its recessive system of solutions. A nontrivial example is also provided.