Note on the metric entropy for multivalued maps

TL;DR

This paper critically examines previous proofs of a variational principle for multivalued maps, providing counterexamples to some arguments and establishing the principle for a specific subclass of maps under additional assumptions.

Contribution

It identifies errors in recent proofs of the half variational principle and proves the full principle for a restricted class of multivalued maps with convex compact values.

Findings

Counterexamples show false arguments in previous proofs.

Full variational principle established for a subclass of maps.

Additional assumptions are necessary for the corrected results.

Abstract

The main aim of this note is to point out by means of counter-examples that some arguments of the proofs of two theorems about a "half variational principle" for multivalued maps, formulated recently by Vivas and Sirvent [Metric entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), pp. 6589-6604], are false and that our corrected versions require rather restrictive additional assumptions. Nevertheless, we will be able to establish the full variational principle for a special subclass of multivalued lower semicontinuous maps with convex compact values on a compact subset of a Banach space.