Almost Periodic Solutions of Differential Equations in Banach Spaces:
Some New Results and Methods
Toshiki Naito, Nguyen Van Minh, and Jong Son Shin
Abstract
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)u_{t} + f(t) , where in (*), A(t) is assumed
to generate a strongly continuous periodic evolutionary process, and in
(**), A is the generator of a C_{0} 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. |