Periodic solutions for some nonautonomous semilinear boundary evolution equations

Abstract

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 (0.1) { x ′ ( t ) = A m ( 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 ) = x 0 . 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.