Oscillation criteria for a special class of 2n-order ordinary differential equations. (English) Zbl 0644.34024

Fiedler, Frank

Math. Nachr. 131, 205-218 (1987).

The author considers the differential equation \((1)\quad (-1)^ n(x^{\alpha}u^{(n)}(x))^{(n)}+q(x)u(x)=0,\) \(x>0\), where the function q is continuous. Equation (1) is said to be oscillatory if for each positive number a there exist positive numbers b and c with \(a<b<c\) such that the equation (1) has a nontrivial solution u satisfying \(u^{(m)}(b)=u^{(m)}(c)=0\), \(m=0,1,...,n-1\). Sufficient conditions for the oscillation of equation (1) are given for \(\alpha =1,3,5,...,2n-1\). They improve some previous results of the author in special cases.

Reviewer: J.Ohriska

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

oscillation of solutons

References:

[1]	Ahlbrandt, Canad. J. Math. 33 pp 229– (1981) · Zbl 0462.47030
[2]	und , Methoden der Mathematischen Physik, Springer-Verlag, Berlin–Heidelberg–New York 1968
[3]	Fiedler, J. Diff. Equations 42 pp 155– (1981)
[4]	Fiedler, Nachr. 107 pp 187– (1982)
[5]	Direct methods of qualitative spectral analysis of singular differential operators, Jerusalem 1965
[6]	Reid, Math. 13 pp 665– (1963)