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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
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