Quasi-equilibrium problems with non-self constraint map

Abstract

In 2016 Aussel, Sultana and Vetrivel developed the concept of projected solution for quasi-variational inequality problems and projected Nash equilibrium. We introduce a new concept of solution for quasi-equilibrium problems and we study the existence of such solutions. Additionally, as a consequence of our results, we give existence results of projected solutions for quasi-optimization problems, quasi-variational inequalities problems and generalized Nash equilibrium problems.

Inverse maximum theorems and some consequences

“…However, this kind of problem has other names in the literature like pseudo-games or coupled constraint equilibrium problems, but here we prefer the name generalized Nash games according with [22,14]. Additionally, it is known that generalized Nash games can be reformulate as quasi-variational inequality problems or quasi-equilibrium problems, see for instance [8,10,9,5] and references therein.…”

An Existence Result for Quasi-equilibrium Problems via Ekeland’s Variational Principle

“…[20] and its references therein. Recent works on the existence of solutions for this kind of problem involving convexity assumptions are given in [21][22][23][24][25]. In [26] an existence result was provided for quasi-equilibrium problems, without any convexity condition, via Ekeland's variational principle.…”

Coerciveness condition for quasi-equilibrium problems

“…Many existence results for (QEP) involve the compactness of C, see for instance [2,8,9,[11][12][13]17]. In this compact context, Lassonde, in [17], proposed an existence result without any lower semi-continuity assumption on the constraint set-valued map.…”

“…On the other hand, Cubiotti in [13] gave a version without upper semi-continuity assumption of the constraints. Later in [9,12], the authors used Cubiotti's idea, in order to generalize the famous minimax inequality due to Ky Fan. Although in [12] the authors deal with a non-compact set C, they still consider constraint maps having compact values.…”