A critical oscillation constant as a variable of time scales for half-linear dynamic equations

Year, pages: 2010, 237 - 256

We present criteria of Hille-Nehari type for the half-linear dynamic equation $(r(t)Φ(y\del))\del+p(t)Φ(y\sig)=0$ on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient $r$. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from $q$-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.

Řehák, P. 2010. A critical oscillation constant as a variable of time scales for half-linear dynamic equations. In Mathematica Slovaca, vol. 60, no.2, pp. 237-256. 0139-9918. DOI: https://doi.org/10.2478/s12175-010-0009-7

Řehák, P. (2010). A critical oscillation constant as a variable of time scales for half-linear dynamic equations. Mathematica Slovaca, 60(2), 237-256. 0139-9918. DOI: https://doi.org/10.2478/s12175-010-0009-7