Fourier multipliers and periodic solutions of delay equations in Banach spaces (Q852809)

The aim of this paper is the study of the equation \[ u'(t)= Au(t)+ F(u_t)+ f(t),\quad t\in\mathbb{R},\tag{\(*\)} \] where \((A, D(A))\) is a (unbounded) linear operator on a Banach space \(X\), \(u_t(\cdot)= u(t+\cdot)\) on \([-r, 0]\) and the delay operator \(F\in L(L^p((-r, 0),X),X)\) for some \(1\leq p<\infty\). Existence and uniqueness of a periodic solution of \((*)\) and maximal regularity results for strong solutions are established. The conditions are obtained in terms of \(R\)-boundedness of linear operators determined by the equations and \(L^p\)-Fourier multipliers. Periodic mild solutions are also studied.