Functional evolution equations

Of concern is the functional evolution equation du dt ∈ Bu+ F(t,ut ), t > 0, u0 = φ(s) ∈ C1 [−r, 0];X , where ut (s) = u(t +s), r >0, s ∈ [−r, 0], and X is a real Banach space. It is shown by the method of lines combined with the Crandall–Pazy theorem, that for the initial data u0 in a generalized domain, this equation has a limit solution, which is Lipschitz continuous in t , and that this limit solution is a unique strong one if further assumptions on B are imposed, and that the zero solution is asymptotically stable. An application is given to a class of partial functional differential boundary value problems.  2003 Elsevier Inc. All rights reserved.