Integral averaging technique for oscillation of damped half-linear oscillators

Enaka, Yukihide ; Onitsuka, Masakazu

Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 755-770.

Résumé

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 $p \neq 2$ is presented.

Détail

DOI : 10.21136/CMJ.2018.0645-16

Classification : 34C10, 34C15
Keywords: damped half-linear oscillator; integral averaging technique; Riccati technique; generalized Young inequality; oscillatory solution