Equilibria on \(L\)-retracts in Riemannian manifolds
type: Item
instance of: scholarly article
title: Equilibria on \(L\)-retracts in Riemannian manifolds
zbMATH Open document ID: 1369.58007
DOI: 10.12775/TMNA.2016.017
author: S. Hosseini
author: M. R. Pouryayevali
published in: Topological Methods in Nonlinear Analysis
publication date: 2017-04-04T00:00:00Z
review text: Let \(M\) be a Riemannian manifold equipped with a Riemannian metric \(d\); a set \(S \subset M\) is called an \(L\)-retract provided there exist a neighborhood \(U\) of \(S\) in \(M\), a retraction \(r: U \to S\), and a constant \(L>0\) such that \[ d(x,r(x)) \leq L\,\operatorname{dist} (x,S) \] for all \(x \in U.\) By \(TM\) and \(T_xM\) are denoted the tangent bundle and the tangent space to \(M\) at a point \(x \in M\) respectively. Let \(S\) be a compact \(L\)-retract on \(M\), \(\Omega\) a nonempty open set in \(S\). The authors define the topological degree for a upper semicontinuous multimap \(\Phi: \Omega \multimap TM\) with compact convex values such that \(\Phi (x) \subset T_xM\) for all \(x \in \Omega\) under assumption that \(Z(\Phi) = \{x \in \Omega: 0 \in \Phi (x)\}\) is compact and \(\Phi (x) \cap T_S(x) \neq \emptyset\) for all \(x \in \Omega\), where \(T_S(x)\) is the Clarke tangent cone to \(S\) at \(x.\) As the immediate consequence the theorem on the existence of an equilibrium \(x_\star \in S\), \(0 \in \Phi (x_\star)\) for a multimap \(\Phi : S \multimap TM\) of such type and an \(L\)-retract \(S\) with a nontrivial Euler characteristic is obtained.
reviewed by: Valerii V. Obukhovskij
Mathematics Subject Classification ID: 58C30
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 58C06
zbMATH DE Number: 6700697
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: \(L\)-retract
zbMATH Keywords: set-valued map
zbMATH Keywords: topological degree
zbMATH Keywords: equilibrium
zbMATH Keywords: Euler characteristic
MaRDI profile type: Publication
Wikidata QID: Q115239565
full work available at URL: tmna.2016.017
OpenAlex ID: W2309228592
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description: scientific article; zbMATH DE number 6700697