## Existence results for strong vector equilibrium problems and their applications

Giancarlo Bigi, Adela Capata and Gabor Kassay
### Summary

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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