Opuscula Math. 43, no. 2 (2023), 221-246
https://doi.org/10.7494/OpMath.2023.43.2.221
Opuscula Mathematica
Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations
Abstract. We consider the half-linear differential equation \[(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition \[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown that if certain additional conditions are 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.
Mathematics Subject Classification: 34C11, 34C10.
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- Manabu Naito
- Ehime University, Faculty of Science, Department of Mathematics, Matsuyama 790-8577, Japan
- Communicated by Josef Diblík.
- Received: 2022-12-15.
- Revised: 2023-02-14.
- Accepted: 2023-02-21.
- Published online: 2023-03-27.
