The existence of isolating blocks for multivalued semiflows

TL;DR

This paper establishes the existence of isolating blocks for multivalued semiflows on metric spaces, extending Conley's index theory to more general, non-locally compact settings, and applies it to differential inclusions.

Contribution

It introduces a theory of isolating blocks for multivalued semiflows, aligning with the classical single-valued case, and demonstrates their construction for equilibria in differential inclusions.

Findings

Existence of isolating blocks for multivalued semiflows on metric spaces.

Application of the theory to construct isolating blocks for equilibria.

Extension of Conley's index theory to non-locally compact spaces.

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.