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

Abstract

We consider the differential equation with delay x ̇(t)=f(t,x(h(t) ) ), t≥0, x(s)=φ(s), s<0, with respect to an unknown function x absolutely continuous on every finite interval. It is assumed that the function f:R_+×R→R is superpositionally measurable, the functions φ:(-∞,0)→R, h:R_+→R are measurable, and h(t)≤t for a. e. t≥0. If the more burdensome inequality h(t)≤t-τ holds for some τ>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 ̇(t)=f(t,x(t) ), t≥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≥0, a point for which the measure of the set {t∈(t_0-ε,t_0+ε)∩R_+:h(t)>t-ε} is positive for any ε>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.