Abstract

Let \(X\) be a complex Banach space, let \(C\) be an injection on \(X\) and let \(A\) be the generator of a \(C\)-semigroup. If the range \(R(C)\) of \(C\) is closed, the author proves that the equation \[ {\dot u}(t) = Au(t) + f(t) \] has a unique \(1\)-periodic mild solution \(u\) with \(u(t)\in R(C)\), for every \(1\)-periodic \(f \in C(\mathbb{R}, R(C))\) provided that \(1 \in \varrho_C(T(1))\).