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New results on critical oscillation constants depending on a graininess. (English) Zbl 1215.34115
Summary: We establish criteria of Hille-Nehari type for the half-linear second order dynamic equation $$(r(t)\Phi(y^\Delta))^\Delta +p(t)\Phi(y^\sigma) = 0$$, $$\Phi(u) = |u|(\alpha-1) \operatorname{sgn}u$$, $$\alpha > 1$$, on time scales, under the condition $$\int^\infty r^{1/(1-\alpha)}(s)\,\Delta s < \infty$$. As a particular important case, we get that there is a (non-improvable) critical oscillation constant which may be different from the one known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient $$r$$. Along with the results of the previous paper by the author, which dealt with the condition $$\int^\infty r^{1/(1-\alpha)}(s)\,\Delta s = \infty$$, a quite complete discussion on generalized Hille-Nehari type criteria involving the best possible constants is provided. To prove these criteria, appropriate modifications of the approaches known from the linear case $$(\alpha = 2)$$ or the continuous case $$(\mathbb T=\mathbb R)$$ cannot be used in general case, and, thus, we apply a new method.
As applications of the main results we state criteria for strong (non)oscillation, examine a generalized Euler type equation, and establish criteria of Kneser type. Examples from $$q$$-calculus and $$h$$-calculus, and a Hardy type inequality 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.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
##### Keywords:
Hille-Nehari type