Existence of unbounded solutions of a linear homogenous system of differential equations with two delays

Abstract

Behavior of solutions of a linear homogeneous system of differential equations with deviating arguments in the form \begin{equation*} \dot y(t)=\beta(t)\left[y(t-\delta)-y(t-\tau)\right] \end{equation*} is discussed for $t\to\infty$. It is assumed that $y$ is an $n$-dimensional column vector, $n\geq 1$ is an integer, $\delta,\tau\in{\mathbb{R}}$, $\tau>\delta>0$, and $\beta(t)$ is an $n\times n$ matrix defined for $t\geq t_{0}$, $t_{0}\in\mathbb{R}$, and such that its elements are nonnegative, continuous functions and in every row of this matrix at least one element is nonzero. The existence of solutions in an exponential form under certain assumptions is proved. Sufficient conditions for the existence of unbounded solutions and the estimations for a solution are derived. A comparison with the known results and an illustrative example are given.