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

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′(t) = Ax(t) + f(t, x(t)), t ∈ ℝ, where A is the infinitesimal generator of an exponentially stable C0‐semigroup. An application is also given to illustrate the abstract existence theorem.