Consider the first order linear differential equation with several variable delays of the form $x^{\prime }\left(t\right)+\sum _{i=1}^m{p}_i\left(t\right)x(t-{\tau }_i\left(t\right))=0,t\ge t_0,$ where $p_i:[t_0,\infty )\to [0,\infty )$ and ${\tau }_i:[t_0,\infty )\to (0,\infty )$ are continuous functions, such that $t-{\tau }_i\left(t\right)\to \infty$ (as $t\to \infty$) for all $1\le i\le m$. Sufficient conditions are established under which some “$lim\; inf$-type” conditions for oscillation of all solutions remain valid if one replaces the $lim\; inf$ by $lim\; sup$. This improves and generalizes many existing results in the literature, which is illustrated via examples.

We consider linear differential equations with variable delay of the form $$x^{\prime }\left(t\right)+p\left(t\right)x(t-\tau \left(t\right))=0,t\ge t_0,$$ where $p:[t_0,\infty )\to [0,\infty )$ and $\tau :[t_0,\infty )\to (0,\infty )$ are continuous functions, such that $t-\tau \left(t\right)\to \infty$ (as $t\to \infty$ ). It is well-known that, for the oscillation of all solutions, it is necessary that $$B:=\underset{t\to \infty }{lim\; sup}A\left(t\right)\ge \frac{1}{e}\mathrm{holds},\mathrm{where}A:=\left(t\right){\int }_{t-\tau \left(t\right)}^{t}p\left(s\right)ds.$$ Our main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and $\tau$ , condition $B>1/e$ implies that all solutions of the above delay differential equation are oscillatory.

Scalar difference equations \(x_{k+1}=f(x_k,x_{k-d})\) with delay \(d\in {\mathbb {N}}\) are well-motivated from applications e.g. in the life sciences or discretizations of delay-differential equations. We investigate their global dynamics by providing a (nontrivial) Morse decomposition of the global attractor. Under an appropriate feedback condition on the second variable of f, our basic tool is an integer-valued Lyapunov functional.