Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations (Q1882670)

The paper studies the existence of unique almost automorphic solutions to the equations \[ x'(t)=Ax(t)+f(t),\;x'(t)=Ax(t)+g \bigl(t,x(t)\bigr),\quad t\in\mathbb{R},\tag{1} \] when \(A\) generates an exponentially \(C_0\)-semigroup on a Banach space \(X\), and the functions \(f(t)\) and \(g(t,x)\) are almost automorphic in \(t\in\mathbb{R}\) for each \(x\in X\).