Electron. J. Diff. Equ., Vol. 2016 (2016), No. 95, pp. 1-17.

Nonoscillation criteria and energy functional for even-order half-linear two-term differential equations

Ondrej Dosly, Vojtech Ruzicka

Abstract:
We investigate oscillatory properties of even-order half-linear differential equations and conditions for negativity of the associated energy functional. First, using the relationship between positivity of the functional and nonoscillation of the investigated equation, we prove Hille-Nehari type nonoscillation criteria which extend criteria known in the linear case. In the second part of the paper, we present conditions which guarantee that the energy functional attains a negative value, i.e., it is unbounded below.

Submitted July 27, 2015. Published April 12, 2016.
Math Subject Classifications: 34C10.
Key Words: Even-order half-linear differential equation; energy functional; nonoscillation, Wirtinger inequality.

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Ondrej Dosly
Department of Mathematics and Statistics
Masaryk University
Kotlarska 2, CZ-611 37 Brno, Czech Republic
email: dosly@math.muni.cz
Vojtech Ruzicka
Department of Mathematics and Statistics
Masaryk University
Kotlarska 2, CZ-611 37 Brno, Czech Republic
email: 211444@mail.muni.cz

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