Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications (Q651124)

Let \(X\) be a nonempty, geodesic convex and compact subset of a Hadamard manifold. Let \(A: X \multimap X\) be a multimap satisfying: \((1)\) for all \(x \in X\), \(x \notin \mathrm{Gco} A(x)\), where \(Gco\) denotes the geodesic convex hull of a set; \((2)\) \(A^{-1}(y)\) is open in \(X\) for each \(y \in X\). Then \(A\) has a maximal element in the sense that there exists \(x^* \in X\) with \(A(x^*) = \emptyset.\) The definition of essential stability of a maximal element is given. It is shown that most maximal element problems (in the sense of Baire category) are essential and that there exists at least one essential component of the solution set for each maximal element problem. As applications, existence and stability results for solutions of variational relations and equilibrium points of multi-objective games on Hadamard manifolds are considered.