Weighted Pseudo Almost-Periodic Functions and Applications to Semilinear Evolution Equations

Abstract

We first give a solution to a key problem concerning the completeness of the space of weighted pseudo almost-periodic functions and then establish a new composition theorem with respect to these functions. Some important remarks with concrete examples are also presented. Moreover, we prove an existence theorem for the weighted pseudo almost-periodic mild solution to the semilinear evolution equation: $x^{\prime }\left(t\right)=Ax\left(t\right)+f\left(t,x\left(t\right)\right)$, $t\in \mathrm{\backslash Bbb\; R}$, where $A$ is the infinitesimal generator of an exponentially stable $C_0$-semigroup. An application is also given to illustrate the abstract existence theorem.