Equilibrium value and measure of systems of functions (Q1909709)

Let \(X\) be a compact Hausdorff space, and \(\{f_i\}\) and \(\{g_i\}\) two families of real-valued continuous functions defined on \(X\) and indexed by the same set \(I\). If a probability Radon measure \(\mu\) on \(X\) and \(\lambda\in R\) satisfy \(\int_Xg_id \mu= \lambda \int_Xf_id\mu\), \(i\in I\), then \(\lambda\) is called an equilibrium value and \(\mu\) an equilibrium measure of \(\{f_i\}\), \(\{g_i\}\). These concepts are associated with von Neumann's model of expanding economics. It was Ky Fan who has shown the existence and uniqueness of an equilibrium value and an equilibrium measure of systems \(\{f_1, \dots,f_n\}\), \(\{g_1, \dots, g_n\}\) for \(X\) being an \((n-1)\)-simplex under some convexity conditions. This paper generalizes this result to the case where \(X\) is a nonempty compact convex set in a topological vector space and \(I\) is a compact Hausdorff space.