Existence results for strong vector equilibrium problems and their applications

Giancarlo Bigi, Adela Capata and Gabor Kassay


In the last years a great attention has been devoted to the so called equilibrium problem, which has been studied intensively by many researchers. It is well-known that its formulation contains, in particular, optimization problems, variational inequalities, saddle point problems, Nash equilibria, and other problems of interest in many practical applications. If the scalar equilibrium bifunction is replaced by a vector-valued bifunction and the image space is partially ordered by a convex cone with a nonempty interior, at least two different vector equilibrium problems can be considered: the so called [strong] vector equilibrium problem (VEP) and the weak vector equilibrium problem (WVEP). Vector equilibrium problems are natural extensions of several problems of practical interest like vector optimization problems or vector variational inequality problems. As far as we know, there are many existence results for (WVEP) and its particular cases, but not for (VEP).
Exploiting the idea developed by Kassay and Kolumban for the scalar equilibrium problem and relying on the Eidelheit's separation theorem in infinite dimensional spaces, new existence results for the strong vector equilibrium problem are given in Section 3. Furthermore, a result of Gong about the existence of solutions of (VEP) follows as a particular case of our existence results, using scalarization techniques and considering a parameterized strong vector equilibrium problem. Motivated by the lack of results for the existence of strong cone saddle points and strong vector variational inequalities, these particular cases of the strong equilibrium problem are studied. In particular, the same scalarization techniques are exploited in Section 4 in order to obtain an existence result for strong cone saddle-points under the same assumptions used in other papers for the weak case, while in Section 5 an existence result for a Stampacchia type strong vector variational inequality is finally given.

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