An averaging method for singularly perturbed systems of semilinear differential inclusions with analytic semigroups

We consider a system of two semilinear parabolic inclusions depending on a small parameter epsilon > 0 which is present both in front of the derivative in one of the two inclusions and in the nonlinear terms to model high-frequency inputs. The aim is to provide conditions in order to guarantee, for epsilon > 0 sufficiently small, the existence of periodic solutions and in order to study their behaviour as F tends to zero. Our assumptions permit the definition of upper semicontinuous, convex valued, compact vector operators whose fixed points represent the sought-after periodic solutions. The existence of fixed points is shown by using topological degree theory arguments. (C) 2003 Elsevier Science Ltd. All rights reserved.