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 $\mathbb{Q}\subset S⫋ℝ$.

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.

We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u\left(t\right)=U(t,s)u\left(s\right)+{\int }_s^{t}U(t,\xi )f(\xi ,u\left(\xi \right))d\xi$ when the evolution family ${\left(U(t,s)\right)}_{t\ge s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the solutions belonging...