Stability criterion for linear differential equations with a delayed argument

A semi-effective criterion for the stability of linear differential equations $\mathcal{L} x=f$ with retarded argument is proposed, the general solution of which is represented by the Cauchy formula
$$ x(t)=C(t,a)x(a)+\int\limits_a^tC(t,s) f(s) ds. $$
The Cauchy function satisfies the integral identity
$$ C(t,s) = U(t,s)U(s,s)^{-1} - \int\limits_s^tC(t,\varsigma)\mathcal{L}_s U(\cdot, s)(\varsigma)U(s,s)^{-1} d\varsigma, $$
where $\mathcal{L}_s$ is the contraction of the operator $\mathcal{L}$ by the interval $[s,\infty)$. Choosing the function $U$ so that the function is $\mathcal{L}_s U(\cdot, s) U(s,s)^{-1}$ is small enough, it is possible to obtain estimates of the Cauchy function $C(t,s)$, which guarantee the stability of the differential equation.