We establish the existence of solutions in a weak sense of
where t Є J = [0, T] and′ = d/dt. It is supposed that the unbounded, linear operators A(t) generate analytic and compact semigroups on a Hilbert space H and that B(t, x) are bounded linear operators. The function f(t, x) with values in H may have asymptotically sublinear growth.
We prove the existence of a periodic solution with the help of Schauder’s fixed point theorem.
Accordingly, we first verify that the corresponding linearized version of (0.1),
has a unique solution for each square integrable ψ(t), provided that the homogeneous problem has only the zero solution.