Functional Differential Equations

Lyapunov’s second method gives sufficient conditions for stability and asymptotic stability. This method has been extended in several directions.7,8 One of the interesting extension of this method depends basically on the fact that a function satisfying the inequality $$m'(t)\leq w(t,m(t)) \;\;\;\;\;\;\; m(t_{0})=r_{0}$$ is majorized by the maximal solution of the equation $${\text{r' = w(t,r) r(}}{t_0}{\text{) = }}{{\text{r}}_0}$$ This comparision principle has been successfully employed to study a variety of problems in ordinary differential equations.