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.
• Revised: 2020-11-13.
• Accepted: 2020-12-11.
• Published online: 2021-02-08.