Almost automorphic functions in Frechet spaces and applications to differential equations (Q818956)

A Frechet space is a metrizable complete locally compact space \(X\). The concept of almost automorphy is a generalization of almost periodicity. In the paper under review a theory of almost automorphic functions with values in Frechet spaces is developed. In particular, it is shown, that if \(f:{\mathbb R}\rightarrow X\) is almost automorphic, then the derivative \(f'\) (if it exists and is uniformly continuous) and \(F(t)=\int_0^t f(s)ds\) are automorphic under some conditions. Some known concepts and results in Frechet spaces are given. The semilinear differential equation \(x'(t)=Ax(t)+f(t,x(t)), t\in {\mathbb R}\), in a Frechet space \(X\), where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup satisfying some conditions of exponential stability, is considered. Under some conditions on \(f\), the existence and uniqueness of an almost automorphic mild solution of the equation is proved.