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 (∗) u ′ ( t ) + Au ( t ) = f ( t , u ( t ) ) , where − A is the infinitesimal generator of an analytic semigroup acting on a (complex) Banach space X , and f : R × X ↦ X is a jointly continuous function. Under some additional assumptions on A and f , 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.