On properties of third-order differential equations via comparison principles. (English) Zbl 1254.34108
Summary: The objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equation
\[
[r(t)[x'(t)]^\gamma]'' + p(t)x(\tau(t)) = 0,
\]
where \(r,\tau,p\in C([t_0,\infty))\), \(\gamma\) is the ratio of two positive integers, \(r(t), p(t)>0\), \(\lim_{t\to\infty}\tau(t)=\infty\), and \(\int^\infty_{t_0} r^{-1/\gamma}(s)\,\text{d}s = \infty\) holds. Employing the trench theory of canonical operators, we deduce properties of the solutions via new comparison theorems. The results obtained essentially improve and complement earlier ones.
MSC:
| 34K25 | Asymptotic theory of functional-differential equations |
| 34K11 | Oscillation theory of functional-differential equations |
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