The work characterizes when is the equation $ y^{ (n) } + \mu p(x) y = 0 $ eventually disconjugate for every value of $ \mu $ and gives an explicit necessary and sufficient integral criterion for it. For suitable integers $ q $, the eventually disconjugate (and disfocal) equation has 2-dimensional subspaces of solutions $ y $ such that $ y^{ (i) } > 0 $, $ i = 0, \ldots , q-1 $, $ (-1)^{i-q} y^{ (i) } > 0 $, $ i = q, \ldots , n $. We characterize the “smallest” of such solutions and conjecture the shape of the “largest” one. Examples demonstrate that the estimates are sharp.
@article{107900,
author = {Uri Elias},
title = {Eventual disconjugacy of $y^{(n)} + \mu p(x) y = 0$ for every $\mu $},
journal = {Archivum Mathematicum},
volume = {040},
year = {2004},
pages = {193-200},
zbl = {1116.34317},
mrnumber = {2068690},
language = {en},
url = {http://dml.mathdoc.fr/item/107900}
}
Elias, Uri. Eventual disconjugacy of $y^{(n)} + \mu p(x) y = 0$ for every $\mu $. Archivum Mathematicum, Tome 040 (2004) pp. 193-200. http://gdmltest.u-ga.fr/item/107900/
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