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Note on the metric entropy for multivalued maps

Jan Andres, Pavel Ludvík

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.

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.

Paper Structure

This paper contains 4 sections, 17 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Theorems of Vivas and Sirvent and their critical analysis
  4. Full-variational principle

Key Result

Proposition 1

Let $X$ be a compact metric space, $Y$ be a Banach space and $\varphi:X\multimap Y$ be an l.s.c. multivalued map with closed convex values. Then there exists a single-valued continuous selection $f\subset\varphi$ of $\varphi$, i.e. $f(x)\in\varphi(x)$, for every $x\in X$.

Theorems & Definitions (46)

  • Definition 1
  • Proposition 1: cf. Ma
  • Definition 2
  • Definition 3: cf. VS
  • Remark 1
  • Lemma 1
  • proof
  • Definition 4: cf. AL1
  • Remark 2
  • Lemma 2
  • ...and 36 more