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On conjugacy of high-order linear ordinary differential equations. (English) Zbl 0807.34041

The author considers the differential equation (1) \(u^{(n)}= p(t)u\), where \(n\geq 2\), \(p\in L_{\text{loc}}(I)\) and \(I\subset\mathbb{R}\) is an interval. Equation (1) is said to be conjugate in \(I\) if there exists a nontrivial solution of (1) with at least \(n\) zeros (each zero counted according to its multiplicity) in \(I\). Let \(l\in \{1,\dots,n- l\}\). Equation (1) is said to be \((l,n-l)\) conjugate in \(I\) if there exists a nontrivial solution \(u\) of (1) satisfying \(u^{(i)}(t_ 1)= 0\) for \(i= 1,\dots,l-1\), \(u^{(i)}(t_ 2)= 0\) for \(i= 0,\dots, n-l-1\), with \(t_ 1,t_ 2\in I\), \(t_ 1< t_ 2\). Sufficient conditions for conjugacy are given.

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:

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