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How the constants in Hille-Nehari theorems depend on time scales. (English) Zbl 1139.39301
Summary: We present criteria of Hille-Nehari-type for the linear dynamic equation $$(r(t)y^\Delta)^\Delta+ p(t)y^\sigma=0$$, that is, the criteria in terms of the limit behavior of $$(\int_a^t 1/r(s)\Delta s) \int_t^\infty p(s)\Delta s$$ as $$t\to\infty$$. As a particular important case, we get that there is a (sharp) critical constant in those criteria which belongs to the interval $$[0,1/4]$$, and its value depends on the graininess $$\mu$$ and the coefficient $$r$$. Also we offer some applications, for example, criteria for strong (non-) oscillation and Kneser-type criteria, comparison with existing results (our theorems turn out to be new even in the discrete case as well as in many other situations), and comments with examples.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis
##### Keywords:
linear dynamic equation; oscillation; Kneser-type criteria
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##### References:
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