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A critical oscillation constant as a variable of time scales for half-linear dynamic equations

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Mathematica Slovaca

Abstract

We present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y Δ))Δ+p(t)Φ(y σ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study 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.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-61662, Brno, Czech Republic

    Pavel Řehák

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  1. Pavel Řehák
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Correspondence to Pavel Řehák.

Additional information

Communicated by Michal Fečkan

Supported by the Grants KJB100190701 of the Grant Agency of ASCR and 201/07/0145 of the Czech Grant Agency, and by the Institutional Research Plan AV0Z010190503.

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Řehák, P. A critical oscillation constant as a variable of time scales for half-linear dynamic equations. Math. Slovaca 60, 237–256 (2010). https://doi.org/10.2478/s12175-010-0009-7

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  • DOI: https://doi.org/10.2478/s12175-010-0009-7

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