Pseudo-almost-periodic solutions to some semilinear differential equations

We study the existence and uniqueness of pseudo-almost-periodic solutions to semilinear differential equations of the form equation(∗)u′(t)+Au(t)=f(t,u(t)), where −A−A is the infinitesimal generator of an analytic semigroup acting on a (complex) Banach space XX, and f:R×X↦Xf:R×X↦X is a jointly continuous function. Under some additional assumptions on AA and ff, the existence and uniqueness of a pseudo-almost-periodic (classical) solution to (∗) is obtained by using both fractional powers of operators and the Banach’s fixed-point principle.