Equilibria on \(L\)-retracts in Riemannian manifolds
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.
scientific article; zbMATH DE number 6700697
scientific article