Comparison theorems for eigenvalue problems for 𝑛th order differential equations

Hankerson, Darrel; Peterson, Allan

doi:10.1090/S0002-9939-1988-0946624-9

Comparison theorems for eigenvalue problems for $n$th order differential equations
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by Darrel Hankerson and Allan Peterson PDF

Proc. Amer. Math. Soc. 104 (1988), 1204-1211 Request permission

Abstract:

We give a comparison theorem for eigenvalues for a $(k,n - k)$-conjugate boundary value problem for the systems ${( - 1)^{n - k}}Ly = \lambda P(t)y$ and ${( - 1)^{n - k}}Lz = \Lambda Q(t)z$, where $P(t)$ and $Q(t)$ are continuous $m \times m$ matrix functions. We assume that the corresponding scalar equation $Lx = 0$ is $(j,n - j)$-disconjugate for $k - 1 \leq j \leq n - 1$. A special case of this is when $Lx = 0$ is disconjugate; our results are new even in this case.

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