Conjugate point properties for an even order linear differential equation
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- by George W. Johnson PDF
- Proc. Amer. Math. Soc. 45 (1974), 371-376 Request permission
Abstract:
Consider an even order equation ${D_n}y = 0$, where ${D_n}$ is defined recursively by ${D_0}y = y,{a_{k,k + 1}}{D_k}y = \{ ({D_{k - 1}}y)’ - \Sigma _{i = 1}^k{a_{ki}}{D_{i - 1}}y\}$, where ${a_{ij}}$ is continuous on $[0,\infty ),{a_{i,i + 1}} > 0,i = 1, \cdots ,n - 1,{a_{n,n + 1}} \equiv 1$, and ${a_{ij}} \equiv 0$ for $j > i + 1$. The $k$th conjugate point function ${\eta _k}(a)$ is defined and is shown to be a strictly increasing continuous function with domain $[0,b)$ or $[0,\infty )$. Extremal solutions are defined as nontrivial solutions with exactly $n - 1 + k$ zeros on $[a,{\eta _k}(a)]$, and are shown to exist, to have odd order zeros at $a$ and ${\eta _k}(a)$ and exactly $k - 1$ odd order zeros in $(a,{\eta _k}(a))$, thus establishing that ${\eta _k}(a) < {\eta _{k + 1}}(a)$ if ${\eta _k}(a) < \infty$.References
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Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 371-376
- MSC: Primary 34C10
- DOI: https://doi.org/10.1090/S0002-9939-1974-0348186-6
- MathSciNet review: 0348186