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

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


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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