Boundedness and almost periodicity of solutions of partial functional differential equations (Q1614721)

Necessary and sufficient conditions are studied for the abstract functional-differential equation \[ x'= Ax(t)+ Fx_t+ f(t) \] to have almost-periodic solutions with the same structure of spectrum as \(f\). Here, it is assumed that \(t\) is a real variable and \(x= x(t)\) takes values in a Banach space \(X\). Moreover, \(A\) is the infinitesimal generator of a strongly continuous semigroup, \(x_t(s)= x(t+ s)\) and \(x_t\in C([-r, 0], X)\), where \(r\) is a positive real number. The operator \(F\) is defined as \(F\varphi= \int^0_{-r} d\eta(s) \varphi(s)\).