Oscillation and nonoscillation in higher order neutral equations. (English) Zbl 0766.34050
Sufficient conditions are obtained, which guarantee that for the scalar neutral equation of the form (1) \(x^{(n)}(t)-cx^{(n)}(t-\tau)+px(t- \sigma)=0\) either all the solutions are oscillatory or there exists a nonoscillatory solution. The following formulations give an idea about the obtained results:
(Theorem 3.1.) Let \(n\) be an odd positive integer and suppose \(p(t)\geq p_ 0>0\) for \(t\geq 0\). Assume that \(p_ 0\left({e\sigma\over n}\right)^ n+cr\left({p_ 0\over 1-c}\right)^{1/n}>1-c\) holds. Then every nontrivial solution if (1) is oscillatory.
(Theorem 3.2.) Let \(n\) be an odd positive integer and let \(p\in C(R_ +,R_ +)\), \(0<p\leq p_ 0\), be nondecreasing. Suppose there exists a positive number \(\mu\) satisfying \(ce^{\mu\tau}+{p_ 0e\mu\tau\over\mu^ n}\leq 0\). Then there exists a nonoscillatory solution, which tends to zero as \(t\to\infty\).
(Theorem 3.1.) Let \(n\) be an odd positive integer and suppose \(p(t)\geq p_ 0>0\) for \(t\geq 0\). Assume that \(p_ 0\left({e\sigma\over n}\right)^ n+cr\left({p_ 0\over 1-c}\right)^{1/n}>1-c\) holds. Then every nontrivial solution if (1) is oscillatory.
(Theorem 3.2.) Let \(n\) be an odd positive integer and let \(p\in C(R_ +,R_ +)\), \(0<p\leq p_ 0\), be nondecreasing. Suppose there exists a positive number \(\mu\) satisfying \(ce^{\mu\tau}+{p_ 0e\mu\tau\over\mu^ n}\leq 0\). Then there exists a nonoscillatory solution, which tends to zero as \(t\to\infty\).
Reviewer: I.Ginchev (Varna)
MSC:
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34K40 | Neutral functional-differential equations |
