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<p>This note establishes an extension of the acute angle lemma (also known as the Hairy Ball Theorem or the Hedgehog Theorem in the case of single-valued mappings) to multifunctions with noncompact image sets. The main result establishes the existence of solutions for operator inclusions involving upper semi-continuous multifunctions with convex values. By relaxing the coercivity assumptions typically required in such analyses, we extend the applicability to scenarios where standard dissipation conditions do not hold. The introduced framework leverages the concept of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1220_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {K}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>-inf-compact support to ensure the existence of zeroes for multifunctions under less restrictive conditions. Applications to hemivariational inequalities and related variational problems are discussed. The examples and counterexamples that demonstrate the obtained generalizations are provided.</p>

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Acute angle lemma for noncompact image sets

  • 期刊论文

  • 发表日期：

    2025年7月18日

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  • Pavlo O. Kasyanov,
  • Liliia S. Paliichuk

摘要

This note establishes an extension of the acute angle lemma (also known as the Hairy Ball Theorem or the Hedgehog Theorem in the case of single-valued mappings) to multifunctions with noncompact image sets. The main result establishes the existence of solutions for operator inclusions involving upper semi-continuous multifunctions with convex values. By relaxing the coercivity assumptions typically required in such analyses, we extend the applicability to scenarios where standard dissipation conditions do not hold. The introduced framework leverages the concept of \({\mathbb {K}}\) K -inf-compact support to ensure the existence of zeroes for multifunctions under less restrictive conditions. Applications to hemivariational inequalities and related variational problems are discussed. The examples and counterexamples that demonstrate the obtained generalizations are provided.