Asymptotic stability of the zero solution for degenerate retarded differential equations

In this paper, we mainly discuss relations between the asymptotic stability of the zero solution for retarded differential equations and real parts of all characteristic roots of characteristic equations. We obtain a necessary and sufficient condition for the asymptotic stability of the zero solution. We point out the mistake of [W.E. Brumley, On the asymptotic behavior of solutions of differential–difference equations of neutral type, Journal of Differential Equations 7 (1) (1970) 175–188], generalize the result of [H. Ren, Z. Zhen, Algebraic criteria of asymptotic stability of zero solution of neutral-type equation ẋ(t)+cẋ(t−τ)+ax(t)+bx(t−τ)=0, Acta Mathematica Sinica 42 (6) (1999) 1–11] and solve a problem posed in [Z.-X. Zheng, Introduction to Functional Differential Equations, Anhui Educational Press, 1992].