Type: Article
Publication Date: 1979-04-01
Citations: 43
DOI: https://doi.org/10.2140/pjm.1979.81.379
A necessary and sufficient criterion for the (k,n — k)disfocality of the equation y{n) + p(x)y — 0, a α, such that the equation is not disconjuate on [α, t]. If the conjugate point of a exists, it is denoted by η(a). There exists a solution associated with the interval [α, 7}(a)]f which has a zero of multiplicity k at x = a and a zero of multiplicity at least n — k at x — η(a) for certain k, 1 ^ k ^ n — 1, and which does not vanish in (α, r]{a)). The subject of this paper is the disconjugacy of the equation (1) y{n) + p(x)y = 0 , where p(x) is of constant sign. For (1) we have further information about the solution associated with [α, ^(α)]. It has a zero exactly of multiplicity k at x — a and a zero exactly of multiplicity n — k at x = y](ri). Moreover, n — k is odd if p(x) ^ 0 and n — k is even if p(x) ^ 0 [16]. The distribution of the zeros of the solution associated with [α, rj(a)] suggests the following definition: Equation (1) is said to be (k, n — k)-disconjugate on an interval / if for every pair of points a, be I, a <b, there does not exist a nontrivial solution of (1) which satisfies