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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.

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
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