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Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations

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Abstract

The purpose of this paper is to present new oscillation theorems and nonoscillation theorems for the nonlinear Euler differential equation t 2 x″ + g(x) = 0. Here we assume that xg(x) > 0 if x ≠ 0, but we do not necessarily require that g(x) be monotone increasing. The obtained results are best possible in a certain sense. To establish our results, we use Sturm’s comparison theorem for linear Euler differential equations and phase plane analysis for a nonlinear system of Liénard type.

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Authors and Affiliations

  1. Sophia University, Tokyo, Japan

    N. Yamaoka

  2. Shimane University, Matsue, Japan

    J. Sugie

Authors
  1. N. Yamaoka
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  2. J. Sugie
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1704–1714, December, 2006.

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Yamaoka, N., Sugie, J. Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations. Ukr Math J 58, 1935–1949 (2006). https://doi.org/10.1007/s11253-006-0178-2

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  • DOI: https://doi.org/10.1007/s11253-006-0178-2

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