Integral averaging technique for oscillation of damped half-linear oscillators

Yukihide Enaka; Masakazu Onitsuka

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 3, page 755-770
  • ISSN: 0011-4642

Abstract

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This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ( a ( t ) φ p ( x ' ) ) ' + b ( t ) φ p ( x ' ) + c ( t ) φ p ( x ) = 0 , where φ p ( x ) = | x | p - 1 sgn x for x ∈ 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 ≠ 2 is presented.

How to cite

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Enaka, Yukihide, and Onitsuka, Masakazu. "Integral averaging technique for oscillation of damped half-linear oscillators." Czechoslovak Mathematical Journal 68.3 (2018): 755-770. <http://eudml.org/doc/294404>.

@article{Enaka2018,
abstract = {This paper is concerned with the oscillatory behavior of the damped half-linear oscillator $(a(t)\phi _p(x^\{\prime \}))^\{\prime \}+b(t)\phi _p(x^\{\prime \})+c(t)\phi _p(x) = 0$, 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 $p \ne 2$ is presented.},
author = {Enaka, Yukihide, Onitsuka, Masakazu},
journal = {Czechoslovak Mathematical Journal},
keywords = {damped half-linear oscillator; integral averaging technique; Riccati technique; generalized Young inequality; oscillatory solution},
language = {eng},
number = {3},
pages = {755-770},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Integral averaging technique for oscillation of damped half-linear oscillators},
url = {http://eudml.org/doc/294404},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Enaka, Yukihide
AU - Onitsuka, Masakazu
TI - Integral averaging technique for oscillation of damped half-linear oscillators
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 755
EP - 770
AB - This paper is concerned with the oscillatory behavior of the damped half-linear oscillator $(a(t)\phi _p(x^{\prime }))^{\prime }+b(t)\phi _p(x^{\prime })+c(t)\phi _p(x) = 0$, 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 $p \ne 2$ is presented.
LA - eng
KW - damped half-linear oscillator; integral averaging technique; Riccati technique; generalized Young inequality; oscillatory solution
UR - http://eudml.org/doc/294404
ER -

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