Pseudo-almost-periodic solutions for delayed differential equations with integrable dichotomies and bi-almost-periodic Green functions

Abstract

© 2017 John Wiley & Sons, Ltd. Using the existence of integrable bi-almost-periodic Green functions of linear homogeneous differential equations and the contraction fixed point, we are able to prove the existence of almost and pseudo-almost-periodic mild solutions under quite general hypotheses for the differential equation with constant delay x(t)=A(t)x(t)+f(t,x(t),x(t-τ)),t∈R, in a Banach space X, where τ>0 is a fixed constant. The results extend the corresponding ones in the case of exponential dichotomy. Some examples illustrate the importance of the concepts.