Abstract
In this paper, we introduce symmetric variational relation problems and establish the existence theorem of solutions of symmetric variational relation problems. As the special cases, symmetric (vector) quasi-equilibrium problems and symmetric variational inclusion problems are obtained. Further, we study the notion of essential stability of equilibria of symmetric variational relation problems. We prove that most of symmetric variational relation problems (in the sense of Baire category) are essential and, for any symmetric variational relation problem, there exists at least one essential component of its solution set.MSC:49J53, 49J40.
Highlights
Let X and Y be real locally convex Hausdorff spaces, and let C and D be nonempty subsets of X and Y, respectively
The equilibrium problem contains as special cases, for instance, optimization problems, problems of Nash equilibria, fixed point problems, variational inequalities, and complementarity problems
Motivated and inspired by research works mentioned above, in this paper, we introduce symmetric variational relation problems, and study the existence and essential stability of solutions of symmetric variational relation problems
Summary
Let X and Y be real locally convex Hausdorff spaces, and let C and D be nonempty subsets of X and Y , respectively. Let P be the set-valued mapping from Z to Z such that for every z ∈ Z, P(z) is a pointed, closed and convex cone of Z with a nonempty interior int P(z) and θ : X −→ Z, η : Y −→ Z. The stability of the solution set of variational relation problems was studied in [ , ].
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