Some analogues of Markov and Descartes systems for right disfocality

Eloe, P. W.; Henderson, Johnny

doi:10.1090/S0002-9939-1987-0875394-7

Some analogues of Markov and Descartes systems for right disfocality
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by P. W. Eloe and Johnny Henderson PDF

Proc. Amer. Math. Soc. 99 (1987), 543-548 Request permission

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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