Conditional oscillation of half-linear Euler-type dynamic equations on time scales

Abstract

We investigate second-order half-linear Euler-type dynamic equations on time scales with positive periodic coefficients. We show that these equations are conditionally oscillatory, i.e., there exists a sharp borderline (a constant given by the coefficients of the given equation) between oscillation and non-oscillation of these equations. In addition, we explicitly find this so-called critical constant. In the cases that the time scale is $\mathbb{R}$ or $\mathbb{Z}$, our result corresponds to the classical results as well as in the case that the coefficients are replaced by constants and we take into account the linear equations. An example and corollaries are provided as well.

Keywords

• time scale
• dynamic equation
• oscillation theory
• conditional oscillation
• oscillation constant
• euler equation
• riccati technique
• half-linear equation