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Some analogues of Markov and Descartes systems for right disfocality


Authors: P. W. Eloe and Johnny Henderson
Journal: Proc. Amer. Math. Soc. 99 (1987), 543-548
MSC: Primary 34B05; Secondary 34B10, 34C10
DOI: https://doi.org/10.1090/S0002-9939-1987-0875394-7
MathSciNet review: 875394
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Abstract: A necessary and sufficient condition for the disconjagacy of the $ n$th order linear differential equation $ {y^{(n)}} + {a_1}(x){y^{(n - 1)}} + \cdots + {a_n}(x)y = 0$ on a compact interval $ I$ is that there exists a system of solutions $ {y_1}, \ldots ,{y_n}$ such that any one of the following is satisfied: (i) $ W({y_1}, \ldots ,{y_k}) > 0,1 \leq k \leq n$, on $ I$; (ii) $ W({y_i}_{_1}, \ldots ,{y_i}_{_k}) > 0,1 \leq {i_1} < \cdots < {i_k} \leq n,1 \leq k \leq n$, on $ I$; or (iii) $ W({y_i},{y_{i + 1}}, \ldots ,{y_i}_{ + k - 1}) > 0,1 \leq i \leq n - k + 1,1 \leq k \leq n$, on $ I$.

Necessary and sufficient criteria for the right disfocality of the linear differential equation on the compact interval $ I$ are established in terms of systems of solutions satisfying conditions which are analogous to those given in (i), (ii), (iii).


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DOI: https://doi.org/10.1090/S0002-9939-1987-0875394-7
Keywords: Disconjugate, right disfocal, Markov system, Descartes system, Fekete system
Article copyright: © Copyright 1987 American Mathematical Society