Stability Criterion for Linear Differential Equations with a Delayed Argument

Abstract

A semieffective criterion for the stability of linear differential equations $\mathcal{L}x = f$ with delayed argument is proposed, the general solution of which can be represented by the Cauchy formula

$$x(t) = C(t,a)x(a) + \int\limits_a^t C(t,s)f(s){\kern 1pt} ds.$$

The Cauchy function satisfies the integral identity

$$C(t,s) = U(t,s)U{{(s,s)}^{{ - 1}}} - \int\limits_s^t C(t,\varsigma ){{\mathcal{L}}_{s}}U( \cdot ,s)(\varsigma )U{{(s,s)}^{{ - 1}}}{\kern 1pt} d\varsigma ,$$

where ${{\mathcal{L}}_{s}}$ is the contraction of the operator $\mathcal{L}$ to the interval $[s,\infty )$. Choosing function $U$ so that function ${{\mathcal{L}}_{s}}U( \cdot ,s)U{{(s,s)}^{{ - 1}}}$ is small enough, one can obtain estimates of the Cauchy function $C(t,s)$, guaranteeing the stability of the differential equation.