Uniform asymptotic stability in functional differential equations

The classical uniform asymptotic stability result for a system of functional differential equations (l) X' = F(t, xt) calls for a Liapunov functional VQ, 0) satisfying W(14,(0)1) < V(t, 0) < WI(I4O(O)I) + W2(I 01411), V ') < W3(l4(0)I), and If(Q, xt)l bounded for Illxtlll bounded. We show that it is not necessary to require If (, xt)I bounded. Here, I I I is the L -norm.