Solutions of $(ry^{(n)})^{(n)} + qy = 0$ of class $\mathcal {L}_{p}[0, \infty )$

Abstract:

For a certain class of ordinary differential operators L, this paper determines the maximum number m of linearly independent solutions of class ${\mathcal {L}_p}[0,\infty )$ of $L(y) = 0$. For $L(y) = {(r{y^{(n)}})^{(n)}} + qy$, and $p = 2$, the principal result is that if $\smallint _0^t|q{|^2}\;d\tau = O(t)$ as $t \to \infty$, then $m \leqq n$.