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

We consider the differential equation with delay

$x ̇\left(t\right)=f(t,x(h\left(t\right) ) ), t\ge 0, x\left(s\right)=\phi \left(s\right), s<0$,

with respect to an unknown function $f:R_{+}\times R\to R$ absolutely continuous on every finite interval. It is assumed that the function  is superpositionally measurable, the functions $\phi :(-\infty ,0)\to R$, $h:R_{+}\to R$ are measurable, and $h\left(t\right)\le t$ for a. e. $t\ge 0$. If the more burdensome inequality $h\left(t\right)\le t-\tau$ holds for some $\tau >0$, then the Cauchy problem for this equation is uniquely solvable and any solution can be extended to the semiaxis $R_{+}$. At the same time, the Cauchy problem for the corresponding differential equation

$x ̇\left(t\right)=f(t,x(t) ), t\ge 0$,

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  has only one “critical’’ point $t_0\ge 0$, a point for which the measure of the set $t\in (t_0-\epsilon ,t_0+\epsilon )\cap R_{+}:h\left(t\right)>t-\epsilon$ is positive for any $\epsilon >0$. 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 $h$.