Periodic solutions for some nonautonomous semilinear boundary evolution equations

In this work we study the Massera problem for the existence of a periodic mild solution of a class of nonautonomous semilinear boundary evolution equations equation(0.1){x′(t)=Am(t)x(t)+f(t,x(t)),t≥0,L(t)x(t)=Φ(t)x(t)+g(t,x(t)),t≥0,x(0)=x0. First, we prove the existence of a periodic solution for nonhomogeneous boundary evolution equations under the existence of a bounded solution on the right half real line. Next, by using a fixed point theorem, we investigate the existence of periodic solutions in the semilinear case. We end with an application to a periodic heat equation with semilinear boundary conditions.