Stability criterion for linear differential equations with a delayed argument (Q6132640)

In this paper, a linear autonomous differential equation of first order with a delayed argument is considered. First, necessary and sufficient conditions are established such that under those conditions the considered delay differential equation is uniformly stable, uniformly exponentially stable and stable. The proof of these outcomes are completed by help of the Cauchy formula, which denotes the general solution of the considered equation. In particular cases, several and different linear differential equations or systems of first order with constant delays are given as numerical applications via Proposition 1--Proposition 6. The next result of the paper, Theorem 2, includes necessary and sufficient conditions such that under those conditions the considered delay differential equation is Lyapunov stable. The proof of Theorem 2 is also done via the Cauchy formula. In particular cases, some examples of linear differential equations or systems of first order with variable delays are given as numerical applications of the given last theorem.