Stability conditions for systems of linear nonautonomous delay differential equations (Q1822263)

Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution for systems of the form \[ y_ i'(t)=\sum^{n}_{k=1}\int^{T}_{0}\delta_{ik}(t,s)y_ k(t- s)d\eta_{ik}(s)-c_ i(t)y_ i(t), \] where \(t\geq 0\), \(0\leq T<\infty\), \(i=1,2,...,n\). In the present paper, not only sufficient, but also necessary stability conditions are derived. The main result is found by a monotone technique and the additional criteria by using the method of Lyapunov functionals. A few examples are given to illustrate the usefulness for the criteria.