Eventual disconjugacy of  $ y^{ (n) } + \mu p(x) y = 0$ for every  $\mu $

Uri Elias

Address.
 Department of Mathematics, Technion,  Haifa 32000, Israel
 

E-mail. elias@tx.technion.ac.il

Abstract.
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.

AMSclassification. 34C10.

Keywords.  Eventual disconjugacy.