Almost automorphic mild solutions to some semi-linear abstract differential equations with deviated argument (Q2496950)

The author considers the semilinear differential equation with deviated argument \(x'(t)= Ax(t)+ f(t, x(t))\), \(x[\alpha(x(t), t)]\), \(t\in\mathbb{R}\), in a Banach space \((X,\|\cdot\|)\) where \(A\) is the infinitesimal generator of an exponentially stable \(C_0\)-semigroup. Sufficient conditions for the existence and uniqueness of an almost automorphic mild solution of the equation are given.