## On a distribution of zeros of solutions of an iterated differential equation of the $$n$$-th order.(English)Zbl 0586.34030

This paper is concerned with an n-th order differential equation arising from a second order equation $(*)\quad y''(t)+q(t)y(t)=0\quad (t\in]- \infty,\infty [)$ by so-called iteration. In (*), $$q\in C^{n-2}$$ and q positive, moreover (*) is assumed to be oscillatory, that is every solution to (*) has an infinite number of zeros on the right and on the left of any real point. Let (u,v) be an arbitrary fundamental system of solutions to (*): then the general solution of the n-th order equation considered has the form $(**)\quad y(t)=\sum^{n}_{i=1}C_ iu(t)^{n-i}v(t)^{i-1}\quad (C_ i\in {\mathbb{R}}).$ The author discusses the classes $$S_ k$$ of solutions (**) satisfying $$C_ n=C_{n-1}=...=C_{k+1}=0$$, $$C_ k\neq 0$$ with a given k, $$1\leq k\leq n- 1$$. By definition, each $$y\in S_ k$$ has the zeros of u as (n-k)-fold zeros. It turns out that the number and multiplicity of further zeros of $$y\in S_ k$$ located between neighbouring zeros of u is completely characterized by the parameters $$C_ i$$. This leads to a full classification of all functions $$y\in S_ k$$ corresponding to their distribution of zeros.
Reviewer: E.Wagenführer

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A30 Linear ordinary differential equations and systems
Full Text:

### References:

 [1] Vlček V.: The first conjugate point of solution of the N-th order iterated differential equation. Acta Univ. Palackianae Olomucensis F. R. N., 1982, Tom 73. · Zbl 0522.34029 [2] Borůvka O.: Lineare Differentialtransformationen 2. Ordnung. VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. · Zbl 0153.11201 [3] Vlček V.: Conjugate points of solutions of an iterated differential equation of the N-th order. Acta Univ. Palackianae Olomucensis F. R. N., 1983, Tom 76. · Zbl 0545.34025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.