From deterministic to stochastic equilibrium problems

  • Mohamed Ait Mansour Polydisciplinary faculty, Safi, Cadi Ayyad University Morocco
  • R. A. Elakri Faculté des Sciences, Université Chouaib Doukkali, El Jadida, Morocco
  • M. Laghdir Faculté des Sciences, Université Chouaib Doukkali, El Jadida, Morocco


In this paper, by means of Castaing representation Theorem, we extend
the results by M. Ait Mansour, R-.A. Elakri and M. Laghdir [Equilibrium
and quasi-equilibrium problems under j-quasimonotonicity and
j-quasiconvexity, Existence, stability and applications, Minimax Theory
and its Applications, 2 (2) (2017), 175–229] to random equilibrium problems
wherein the objective data are subject to a random perturbation. We
obtain deterministic as well as random (or measurable) and integral solutions
to random equilibrium problems. The case when the random parameter
has a probability of realization suggests us to introduce stochastic
formulations of equilibria covering the expected value approach and the
almost sure method. Thus, we prove the existence of two further kinds of
equilibria under uncertainty: almost sure solutions and expected stochastic
equilibrium points. Finally, we present an application to stochastic
quasiconvex programming for which we establish the existence of almost
sure minimizers.