Abstract
Sharp estimation for the solutions of inhomogeneous delay differential and Halanay type inequalities
Highlights
Halanay [11] proved an upper estimation for the nonnegative solutions of an autonomous continuous time delay differential inequality with maxima
This, so called Halanay inequality, and its generalizations became a powerful tool in the stability theory of delay differential equations
In this paper we study these inequalities and the inhomogeneous linear delay differential equation x (t) = −α (t) x (t) + β (t) x (t − τ (t)) + (t), t ≥ t0
Summary
Halanay [11] proved an upper estimation for the nonnegative solutions of an autonomous continuous time delay differential inequality with maxima. There are almost no papers (see [6] and [12]) which have been devoted to the asymptotic analysis of the nonnegative solutions of the inhomogeneous Halanay-type differential inequality x (t) ≤ −α (t) x (t) + β (t) sup x (s) + (t) , t ≥ t0, t−τ(t)≤s≤t (1.1). In this paper we study these inequalities and the inhomogeneous linear delay differential equation x (t) = −α (t) x (t) + β (t) x (t − τ (t)) + (t) , t ≥ t0. It is worth to note that in the literature and in our paper just the nonnegative solutions of the Halanay-type inequality (1.1) are investigated, because they give estimation for the norm of the solutions of more complicated systems of delay differential equations.
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