The Existence of Isolating Blocks for Multivalued Semiflows

Abstract

In this article, we show the existence of an isolating block, a special neighborhood of an isolated invariant set, for multivalued semiflows acting on metric spaces (not locally compact). Isolating blocks play an important role in Conley’s index theory for single-valued semiflows and are used to define the concepts of homology index. Although Conley’s index was generalized in the context of multivalued (semi) flows, the approaches skip the traditional construction made by Conley, and later, Rybakowski. Our aim is to present a theory of isolating blocks for multivalued semiflows in which we understand such a neighborhood of a weakly isolated invariant set in the same way as we understand it for invariant sets in the single-valued scenario. After that, we will apply this abstract result to a differential inclusion in order to show that we can construct isolating blocks for each equilibrium of the problem.