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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
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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
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