Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 755-770, 2018
Integral averaging technique for oscillation of damped half-linear oscillators
Yukihide Enaka, Masakazu Onitsuka
Received December 23, 2016. First published April 4, 2018.
Abstract: This paper is concerned with the oscillatory behavior of the damped half-linear oscillator $(a(t)\phi_p(x'))'+b(t)\phi_p(x')+c(t)\phi_p(x) = 0$, where $\phi_p(x) = |x|^{p-1}\mathop{\rm sgn} x$ for $x \in\mathbb{R}$ and $p > 1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young's inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if $p \neq2$ is presented.
Keywords: damped half-linear oscillator; integral averaging technique; Riccati technique; generalized Young inequality; oscillatory solution
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Affiliations: Yukihide Enaka, Masakazu Onitsuka (corresponding author), Department of Applied Mathematics, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama-shi 700-0005, Japan, e-mail: onitsuka@xmath.ous.ac.jp
