Almost automorphic mild solutions to some semilinear abstract differential equations with deviated argument in Fréchet spaces

Abstract

In this paper we consider the semilinear differential equation with deviated argument in a Fréchet space $x^{\prime}(t) = A x(t) + f(t, x(t), x[\alpha(x(t),t)]),$ $t \in {\mathbb{R}}$, where $A$ is the infinitesimal (bounded) generator of a $C_{0}$-semigroup satisfying some conditions of exponential stability. Under suitable conditions on the functions $f$ and $\alpha$ we prove the existence and uniqueness of an almost automorphic mild solution to the equation.