Note on the metric entropy for multivalued maps

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.