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<p>The first goal of this note is to highlight several discrepancies in the proofs of the recent paper by Vivas and Sirvent (Discrete Contin Dyn Syst Ser B 27(11):6589–6604, 2022. <a href="https://doi.org/10.3934/dcdsb.2022010">https://doi.org/10.3934/dcdsb.2022010</a>). The results in question, which relate metric and topological entropies for multivalued maps and can be referred to as “half variational principles” for multivalued maps, are pioneering in this field. Our second goal is to provide revised statements accompanied by correct proofs. By employing the concept of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1245_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(h_+\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation> entropy for multivalued maps, developed in our earlier papers, we were able to partially strengthen the results of Vivas and Sirvent and 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. However, our approach requires two rather restrictive additional assumptions, which were also used by Vivas and Sirvent in another related context.</p>

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Note on the Metric Entropy for Multivalued Maps

  • 期刊论文

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  • 发表日期：

    2025年2月27日

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  • Jan Andres,
  • Pavel Ludvík

摘要

The first goal of this note is to highlight several discrepancies in the proofs of the recent paper by Vivas and Sirvent (Discrete Contin Dyn Syst Ser B 27(11):6589–6604, 2022. https://doi.org/10.3934/dcdsb.2022010). The results in question, which relate metric and topological entropies for multivalued maps and can be referred to as “half variational principles” for multivalued maps, are pioneering in this field. Our second goal is to provide revised statements accompanied by correct proofs. By employing the concept of \(h_+\) h + entropy for multivalued maps, developed in our earlier papers, we were able to partially strengthen the results of Vivas and Sirvent and 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. However, our approach requires two rather restrictive additional assumptions, which were also used by Vivas and Sirvent in another related context.