Existence of Nash Equilibria for Generalized Games without Upper Semicontinuity

Quasi-equilibrium problems with non-self constraint map

“…As a corollary of Theorem 4.1, Proposition 4.1 and Remark 4, we recover the following existence result due to Cubiotti [14] by considering K to be a self-map.…”

“…In a similar way to [14,11], we now show the existence of projected solutions for quasi-equilibrium problems without upper semicontinuity of the constraint map by using Corollary 2.3. But before that, we need to introduce a few definitions.…”

“…[14, Theorem 2.1] Let C be a non-empty compact convex subset of R n , K : C ⇒ C be a set-valued map, and f : C × C → R be a bifunction. Assume that 1.…”

Approximate solutions of quasiequilibrium problems in Banach spaces

“…An interesting existence result for quasiequilibrium problems on R n was established in [5] where the upper semicontinuity of the set-value map K is not assumed. The proof of the theorem depends heavily on the finite dimensionality of the space.…”

“…The proof of the theorem depends heavily on the finite dimensionality of the space. In a recent paper [6], the authors established an existence result of approximate solutions of quasiequilibrium problem defined on a Banach space and, applying the Maximum Theorem [7], they extended the result in [5] to infinite dimensional spaces. Unfortunately the first result is incorrect and the mentioned improvement fails.…”