Bounded and periodic solutions of infinite delay evolution equations. (Q1414227)

Evolution equations with infinite delay of the form \[ u^\prime (t)+A(t)u(t)=f(t,u(t),u_t),\quad t\geq0, \quad u(s)=\phi(s),\quad s\leq0, \tag \(*\) \] are considered in a general Banach space \(X\). Here, \(A(t)\) and \(f(t,x,y)\) are \(T\)-periodic in \(t\), the resolvent of the unbounded operator \(A(t)\) is compact, and \(u_t(s)=u(t+s)\), \(s\leq0.\) Under certain additional assumptions on the domain and the resolvent of the operator \(A(t)\), the authors deduce the existence of a \(T\)-periodic solution of equation \((*)\) from the boundedness and ultimate boundedness of all its solutions in a specially constructed Banach space \(C_g\subset X\). The boundedness is meant in usual sense, while the ultimate boundedness is a form of the eventual uniform boundedness. It is not clear if the trivial solution is excluded as a possibility for a periodic solution of equation \((*)\).