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Integral averaging technique for oscillation of damped half-linear oscillators

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Abstract

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)ϕp(x′))′ + b(t)ϕp(x′) + c(t)ϕp(x) = 0, where ϕp(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.

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Authors and Affiliations

  1. Department of Applied Mathematics, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama-shi, 700-0005, Japan

    Yukihide Enaka & Masakazu Onitsuka

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  1. Yukihide Enaka
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  2. Masakazu Onitsuka
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Correspondence to Masakazu Onitsuka.

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Enaka, Y., Onitsuka, M. Integral averaging technique for oscillation of damped half-linear oscillators. Czech Math J 68, 755–770 (2018). https://doi.org/10.21136/CMJ.2018.0645-16

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  • DOI: https://doi.org/10.21136/CMJ.2018.0645-16

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