An averaging method for singularly perturbed systems of semilinear differential inclusions with \(C_0\)-semigroups
A system of two semilinear differential inclusions in separable Banach spaces with infinitesimal generators of \(C_{0}\)-semigroups and multivalued uppersemicontinuous perturbations with compact convex values is considered. They are of high frequence with respect to time and periodic with a specifed period. Moreover, the perturbations are condensing in the state variables with respect to a suitable measure of noncompactness. For sufficiently small \(\varepsilon > 0\), the existence of periodic solutions and their behaviour as \(\varepsilon \rightarrow 0\) are studied. The main tool is the topological degree theory for uppersemicontinuous condensing vector fields. The work complements the previous results of the authors where is assumed that the infinitesmal generators of analytic semigroups have a compact inverse.
scientific article; zbMATH DE number 1992540