Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces (Q927278)

Let \(X\) be a Banach space. The authors prove that the space of vector-valued pseudo almost automorphic functions endowed with the supremum norm \((PAA(\mathbb R;X),\| \cdot\| _0)\) is complete. As a consequence, they obtain an existence and uniqueness theorem of mild pseudo almost automorphic solutions to the semilinear differential equation \[ x'(t)=Ax(t)+f(t,x(t)),\quad t\in\mathbb R, \] where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup on \(X\) and \(f:\mathbb R\times X\rightarrow X\) is a pseudo almost automorphic function. Finally, the authors provide an example as an application of this theorem.