Generic uniqueness of solutions for a class of vector ky fan inequalities
Given a nonempty set \(X\), a Hausdorff topological vector space \(H\), a nonempty, closed, convex and pointed cone \(C\) in \(H\), and a function \(\phi:X\times X\rightarrow H\), a point \(x^{\ast}\in X\) is said to solve the vector Ky Fan inequality (often also called vector equilibrium problem) if \(\phi(x^{\ast},y)\in C\), for all \(y\in X\). This is a generalization of the Ky Fan inequality (equilibrium problem), where \(H=\mathbb{R}\) and \(C=\mathbb{R} _{+}\). In this paper, the generic uniqueness of solutions is studied, when \(X\) is a nonempty compact subset of a metric space, \(H\) is a Banach space and \(\text *{int}C\neq \emptyset\). In this setting, define \(M\) to be the set of all \(\phi:X\times X\rightarrow H\) which are monotone (i.e., for all \(x,y\in X\), \(x\neq y\), \(\phi(x,y)+\phi(y,x)\notin \text *{int}C\)), \(C\)-upper semicontinuous with respect to the second variable, \(\left \| \phi (\cdot,\cdot)\right \| \) is bounded on \(X\times X\), and such that the Ky Fan inequality has at least one solution. It is shown that \(M\), equipped with the metric \(\rho(\phi_{1},\phi_{2})=\sup \left | \phi_{1}-\phi_{2}\right | \), is a complete metric space, and there exists a dense residual subset \(Q_{1}\) of \(M\) such that for each \(\phi \in Q_{1}\), the set of solutions of the Ky Fan inequality is a singleton. Another result is also shown for the case of a noncompact set \(X\).
scientific article; zbMATH DE number 6132199
scientific article