Opuscula Math. 41, no. 1 (2021), 71-94
https://doi.org/10.7494/OpMath.2021.41.1.71

 
Opuscula Mathematica

Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I

Manabu Naito

Abstract. We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\).

Keywords: asymptotic behavior, nonoscillatory solution, half-linear differential equation, Hardy-type inequality.

Mathematics Subject Classification: 34C11, 34C10, 26D10.

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  • Manabu Naito
  • Ehime University, Faculty of Science, Department of Mathematics, Matsuyama 790-8577, Japan
  • Communicated by Petr Stehlík.
  • Received: 2020-07-29.
  • Revised: 2020-11-13.
  • Accepted: 2020-12-11.
  • Published online: 2021-02-08.
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Cite this article as:
Manabu Naito, Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I, Opuscula Math. 41, no. 1 (2021), 71-94, https://doi.org/10.7494/OpMath.2021.41.1.71

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