Pseudo almost periodic solutions for some differential equations in a Banach space

A pseudo almost periodic (PAP) function f(t) in a Banach space is a function that can be expressed as the sum of an almost periodic function g(t) and a continuous function e(t) which vanishes as t approaches infinity, and moreover is bounded and has a zero asymptotic mean value. The main theorem on PAP solutions of a pseudo almost periodic nonlinear perturbation of a linear autonomous ordinary differential equation is proven. The extension of the theorem to nonlinear equations is also discussed.