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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
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