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The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient ${\lambda }^{2}q\left(s\right),s\in [s_0,\infty )$ is investigated, where $\lambda \in ℝ$ and $q\left(s\right)$ is a nondecreasing step function tending to $\infty$ as $s\to \infty$. Let $S$ denote the set of those $\lambda$’s for which the corresponding differential equation has a solution not tending to 0. It is proved that $S$ is an additive group. Four examples are given with $S=\left\{0\right\}$, $S=\mathbb{Z}$, $S=𝔻$ (i.e. the set of dyadic numbers), and $\mathbb{Q}\subset S⫋ℝ$.

The zeros $c_k\left(\nu \right)$ of the solution $z(t,\nu )$ of the differential equation $z^{\text{'}\text{'}}+q(t,\nu )z=0$ are investigated when $\underset{t\to \infty }{lim}q(t,\nu )=1$, ${\int }^{\infty }|q(t,\nu )-1|dt<\infty$ and $q(t,\nu )$ has some monotonicity properties as $t\to \infty$. The notion $c_{\kappa }\left(\nu \right)$ is introduced also for $\kappa$ real, too. We are particularly interested in solutions $z(t,\nu )$ which are “close" to the functions $sint$, $cost$ when $t$ is large. We derive a formula for $d{c}_{\kappa }\left(\nu \right)/d\nu$ and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair $J_{\nu }\left(t\right)$, $Y_{\nu }\left(t\right)$. We show the concavity of $c_{\kappa }\left(\nu \right)$ for $\left|\nu \right|\ge \frac{1}{2}$ and also...

A second-order half-linear ordinary differential equation of the type $$\left(\right|y^{\text{'}}{|}^{\alpha -1}{y}^{\text{'}}{)}^{\text{'}}+\alpha q\left(t\right){\left|y\right|}^{\alpha -1}y=0\left(1\right)$$ is considered on an unbounded interval. A simple oscillation condition for (1) is given in such a way that an explicit asymptotic formula for the distribution of zeros of its solutions can also be established.

We consider linear differential equations of the form $${\left(p\left(t\right)x^{\text{'}}\right)}^{\text{'}}+\lambda q\left(t\right)x=0(p\left(t\right)>0,q\left(t\right)>0)\left(\mathrm{A}\right)$$ on an infinite interval $[a,\infty )$ and study the problem of finding those values of $\lambda$ for which () has principal solutions $x_0(t;\lambda )$ vanishing at $t=a$. This problem may well be called a singular eigenvalue problem, since requiring $x_0(t;\lambda )$ to be a principal solution can be considered as a boundary condition at $t=\infty$. Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence $\left\{{\lambda }_n\right\}$ of eigenvalues such...