Approximate solutions of quasiequilibrium problems in Banach spaces

Abstract

In this note we show that a recent existence result on quasiequilibrium problems, which seems to improve deeply some well-known results, is not correct. We exhibit a counterexample and we furnish a generalization of a lemma about continuous ε-minimizers of quasiconvex functions depending on a parameter. This allows to establish an existence result of approximate solutions of quasiequilibrium problems.

A Ky Fan Minimax Inequality for Quasiequilibria on Finite-Dimensional Spaces

“…Unlike the equilibrium problems which have an extensive literature on results concerning existence of solutions, the study of quasiequilibrium problems to date is at the beginning even if the first seminal work in this area was in the seventies [4]. After that, the problem concerning existence of solutions has been developed in some papers [5,6,7,8,9,10,11]. Most of the results require either monotonicity assumptions on the equilibrium bifunction or upper semicontinuity of the set-valued map which describes the constraint.…”

Stability and Existence Results for Quasimonotone Quasivariational Inequalities in Finite Dimensional Spaces

“…In [19] Kien et al proposed an alternative result which allows to omit the assumption of lower semicontinuity. Recently, Castellani and Giuli [11] by using a parametric Borwein-Preiss variational principle [16] proved the existence of ε-solutions. All these results were established without using any monotonicity property of T .…”