Periodic solutions of semilinear equations of evolution of compact type

AbstractThe existence of solutions in a weak sense of x′ + (A + B(t, x))x = f(t, x), x(0) = x(T) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H; B(t,x) is a bounded linear operator and f(t, x) a function with values in H; for each square integrable ϑ(t) the problem with B(t, ϑ(t)) and f(t, ϑ(t)) in place of B(t, x) and f(t, x) has a unique solution; and B and f satisfy certain boundedness and continuity conditions