Bounded and periodic solutions of infinite delay evolution equations

Abstract

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: (0.1) u′(t)+A(t)u(t)=f t,u(t),u t , t>0, u(s)=φ(s), s⩽0, where the resolvent of the unbounded operator A( t) is compact, and u t ( 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 C g , 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.