On the extremal solutions of 𝑛th-order linear differential equations

Kim, W. J.

doi:10.1090/S0002-9939-1972-0294780-9

On the extremal solutions of $n$th-order linear differential equations
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Proc. Amer. Math. Soc. 33 (1972), 62-68 Request permission

Abstract:

Distribution of zeros of extremal solutions of linear nth-order differential equations is discussed. Existence and nonexistence of extremal solutions with certain zero distributions are established. For instance, it is proved that every extremal solution for $[\alpha , {\eta _1}(\alpha )]$ of the equation ${y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y = 0$ has a zero of order 2 at ${\eta _1}(\alpha )$ and has no more than $n - 2$ zeros on $[\alpha , {\eta _1}(\alpha ))\;{\text {if}}\;{p_i} \leqq 0,i = 0,1, \cdots , n - 2$.

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