Almost Periodic Solutions of Differential Equations in Banach Spaces: Some New Results and Methods

Toshiki Naito, Nguyen Van Minh, and Jong Son Shin

In this paper we give an overview of recent results on almost periodic solutions of differential equations and functional differential equations in Banach spaces of the form (*)  u'(t)=A(t)u(t)+f(t) and (**) u'(t) = Au(t) + F(t)ut + f(t) , where in (*), A(t) is assumed to generate a strongly continuous periodic evolutionary process, and in (**),  A is the generator of a C0 semigroup, F(t)  is a bounded linear operator, periodic in t. The results surveyed are focused on the conditions so that the equations have (unique) almost periodic solutions with structure of spectrum as the one of  f. The conditions are stated in terms of the spectral properties of either the operator-coefficients in the case of autonomous equations or the monodromy operators in the case of periodic equations.