Ordinary differential equations and differential equations with delay: general properties and features

We consider the differential equation with delay
$$\dot{x}(t)=f\big(t,x(h(t))\big), \ \ t\geq 0, \ \ x(s)=\varphi(s), \ \ s<0,$$
with respect to an unknown function $x$ absolutely continuous on every finite interval. It is assumed that the function $f:\mathbb{R}_+ \times \mathbb{R} \to \mathbb{R}$ is superpositionally measurable, the functions $\varphi:(-\infty,0)\to \mathbb{R},$ $h:\mathbb{R}_+ \to \mathbb{R}$ are measurable, and $h(t)\leq t$ for a. e. $t\geq 0.$ If the more burdensome inequality $h(t)\leq t-\tau $ holds for some $\tau > 0,$ then the Cauchy problem for this equation is uniquely solvable and any solution can be extended to the semiaxis $\mathbb{R}_+ .$ At the same time, the Cauchy problem for the corresponding differential equation
$$\dot{x}(t)=f\big(t,x(t)\big), \ \ t\geq 0, $$
may have infinitely many solutions, and the maximum interval of existence of solutions may be finite. In the article, we investigate which of the listed properties a delay equation possesses (i.e. has a unique solution or infinitely many solutions, has finite or infinite maximum interval of existence of solutions), if the function $h$ has only one «critical» point $t_0 \geq 0,$ a point for which the measure of the set $\big\{t\in (t_0-\varepsilon, t_0+\varepsilon)\cap \mathbb{R}_+ :\, h(t)>t-\varepsilon \big\}$ is positive for any $\varepsilon >0.$ It turns out that for such a delay function, the properties of solutions are close to those of solutions of an ordinary differential equation. In addition, we consider the problem of the dependence of solutions of a delay equation on the function $h.$