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

Authors:
Mikhail Kamenskii;Paolo Nistri
Affiliations:
Department of Mathematics, Voronezh State University, Voronezh, Russia;Dipartimento di Ingegneria dell'Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy
Venue:
Nonlinear Analysis: Theory, Methods & Applications
Year:
2003

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Abstract

We consider a system of two semilinear parabolic inclusions depending on a small parameter ε 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 ε 0 sufficiently small, the existence of periodic solutions and in order to study their behaviour as ε 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.