Asymptotics for third-order nonlinear differential equations: non-oscillatory and oscillatory cases. (English) Zbl 1516.34084
In this paper, the authors study the third-order differential equation
\[
(a(t)w'')'+wA(w^2,t)=0, \quad t\geq 0,
\]
where \(a(t)>0\), \(a'(t)\geq 0\), and the nonlinear function \(A(\cdot,\cdot)\) satisfies \(A(z,t)>0\) for \(z>0\) and certain monotone property. By comparing it with first-order retarded differential equation and differential inequality and using the Kusano-Naito’s and Philos’ approaches, they obtain criteria for the asymptotic behavior of non-oscillatory solutions. This leads to sufficient conditions for the existence of (non-)oscillatory solutions which complement the results in the literature.
Reviewer: Qingkai Kong (DeKalb)
MSC:
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
Keywords:
nonlinear differential equation; oscillation and non-oscillation; asymptotic behavior; comparison technique; third-order differential equationReferences:
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