Generic uniqueness of equilibrium points
Let \(X\) a convex compact subset of a normed linear space \(E\). Denote by \(M\) the class of all \(f:X\times X\to \mathbb R\) fulfilling (i) \(x\mapsto f(x,y)\) is usc on \(X\), \(\forall y\in X\), (ii) \(y\mapsto f(x,y)\) is convex on \(X\), \(\forall x\in X\), (iii) \(f(x,x)\geq 0\), \(\forall x\in X\), (iv) \(f(x,y)+f(y,x)\leq 0\), \(\forall x,y\in X\), (v) \(\sup\{|f(x,y)|; x,y\in X\}< \infty\). {Theorem.} There exists a dense residual subset \(Q\) of \(M\) such that, for each \(f\in Q\), the set of equilibrium points for \(f\) is a singleton. A non-compact (modulo \(X\)) version of this result is also given by the perturbation of the couples \((f,A)\), where \(f\) is as before and \(A\) is a convex compact part of \(X\).
scientific article; zbMATH DE number 5959689
scientific article