Opuscula Mathematica
Opuscula Math. 41, no. 1 (2021), 71-94

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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About the authors ▼
Manabu Naito
Ehime University, Faculty of Science, Department of Mathematics, Matsuyama 790-8577, Japan
About the article ▼
Communicated by Petr Stehlík.
Received: 2020-07-29.
Revised: 2020-11-13.
Accepted: 2020-12-11.
Published online: 2021-02-08.
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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