Stability of solutions of linear delay differential equations

Abstract:

Consider the linear differential equation \[ (1)\quad \dot x(t) = \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i})} = 0,\quad t \geqslant {t_0},\] where ${p_i} \in C([{t_0},\infty ),{\mathbf {R}})$ and ${\tau _i} \geqslant 0$ for $i = 1,2, \ldots ,n$. By investigating the asymptotic behavior first of the nonoscillatory solutions of (1) and then of the oscillatory solutions we are led to new sufficient conditions for the asymptotic stability of the trivial solution of (1). When the coefficients of (1) are all of the same sign, we obtain a comparison result which shows that the nonoscillatory solutions of (1) dominate the growth of the oscillatory solutions.