Fundamental solution and asymptotic stability of linear delay differential equations

In this paper we formulate sufficient conditions for the asymptotic stability of linear delay systems of the form x(k)(t) = - (l=0)Sigma(m) (j=1)Sigma(n) a(kj)((l))x(j)(t - tau((l))(kj)), k = 1,...,n, t >= 0, where a(kj)((0)), a(kj)((l)) epsilon R, tau((0))(kj) = 0, tau((l))(kj) >= 0, k j = 1,...,n,l = 1,...,m. In order to apply our results, we give estimates for the integral integral(infinity)(0) vertical bar v(t)vertical bar dt, where v is the fundamental solution of certain associated scalar linear delay differential equations with multiple delays.