Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
843015 | Nonlinear Analysis: Theory, Methods & Applications | 2009 | 7 Pages |
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].