Exponential stability and inequalities of solutions of abstract functional differential equations

With the Lyapunov second method, we study the abstract functional differential equation, du dt = f (t,ut ). We obtain inequalities of solutions and exponential stability with conditions like: (i) W1(|u(t)|X) V (t,ut ) W2(D(t, ut ))+ t t−h L(s)W1(|u(s)|X)ds, (ii) V (1)(t, ut ) −η(t)W2(D(t, ut ))+ P(t). Applications in ordinary and partial functional differential equations are given. © 2006 Elsevier Inc. All rights reserved.