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 $\mathbb{Q}\subset S⫋ℝ$.

Let f(z) be a conformal mapping of an annulus A(R) = 1 < |z| < R and let f(A(R)) be a ring domain bounded by a circle and a k-circle. If R(φ) = w : arg w = φ, and l(φ) - 1 is the linear measure of f(A(R)) ∩ R(φ), then we determine the sharp lower bound of $l\left({\phi }_1\right)+l\left({\phi }_2\right)$ for fixed ${\phi }_1$ and ${\phi }_2$ $\left(0\le {\phi }_1\le {\phi }_2\le 2\pi \right)$.

It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\ge 1$ is an integer, then the $\gamma$-th order Cesàro mean $C_t^{\gamma }:=\gamma t^{-\gamma }{\int }_0^t{(t-s)}^{\gamma -1}T\left(s\right)ds$ and Abel mean $A_{\lambda }:=\lambda {\int }_0^{\infty }{e}^{-\lambda s}T\left(s\right)ds$ of the uniformly continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\ne a\in ℝ$, satisfy that (a) $\parallel C_t^{\gamma }\parallel \sim t^{k-\gamma }(t\to \infty )$ for all $0<\gamma \le k+1$; (b) $\parallel C_t^{\gamma }\parallel \sim t^{-1}(t\to \infty )$ for all $\gamma \ge k+1$; (c) $\parallel A_{\lambda }\parallel \sim \lambda (\lambda \downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function ${\left(C\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by ${(iaI+N)}^2$.