On solutions of differential equations with ``common zero'' at infinity

Elbert, Árpád ; Vosmanský, Jaromír

Archivum mathematicum, Tome 33 (1997) no. 1-2, pp. 109-120.

Résumé

The zeros $c_k(\nu )$ of the solution $z(t, \nu )$ of the differential equation $z^{\prime \prime }+ q(t, \nu )\, z=0$ are investigated when $\lim \limits _{t\rightarrow \infty } q(t, \nu )=1$, $\int ^\infty | q(t, \nu )-1|\, dt \infty $ and $q(t, \nu )$ has some monotonicity properties as $t\rightarrow \infty $. The notion $c_\kappa (\nu )$ is introduced also for $\kappa $ real, too. We are particularly interested in solutions $z(t, \nu )$ which are “close" to the functions $\sin t$, $\cos t$ when $t$ is large. We derive a formula for $d c_\kappa (\nu )/d\nu $ and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair $J_\nu (t)$, $Y_\nu (t)$. We show the concavity of $c_\kappa (\nu )$ for $|\nu |\ge \frac{1}{2}$ and also for $|\nu |\frac{1}{2}$ under the restriction $c_\kappa (\nu )\ge \pi \nu ^2 (1-2\nu )$.

Détail

Classification : 33C10, 34A25, 34C10, 34M99
Keywords: common zeros; dependence on parameter; Bessel functions; higher monotonicity