Bounded and periodic solutions of infinite delay evolution equations

For A(t) and f (t, x, y) T -periodic in t , we consider the following evolution equation with infinite delay in a general Banach space X: u (t )+A(t)u(t)= f t,u(t),ut , t>0, u(s) = φ(s), s 0, (0.1) where the resolvent of the unbounded operator A(t) is compact, and ut (s) = u(t + s), s 0. By utilizing a recent asymptotic fixed point theorem of Hale and Lunel (1993) for condensing operators to a phase space Cg, we prove that if solutions of Eq. (0.1) are ultimate bounded, then Eq. (0.1) has a T -periodic solution. This extends and improves the study of deriving periodic solutions from boundedness and ultimate boundedness of solutions to infinite delay evolution equations in general Banach spaces; it also improves a corresponding result in J. Math. Anal. Appl. 247 (2000) 627–644 where the local strict boundedness is used.  2003 Elsevier Inc. All rights reserved.