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

Abstract

Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued maps which holds in finite dimensional spaces. Furthermore this result allows one to locate the position of a solution. Sufficient conditions, which are easier to verify, may be obtained by imposing restrictions either on the domain or on the bifunction. These facts make it possible to yield various existence results which reduce to the well known Ky Fan minimax inequality when the constraint map is constant and the quasiequilibrium problem coincides with an equilibrium problem. Lastly, a comparison with other results from the literature is discussed.

Quasi-equilibrium problems with non-self constraint 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.…”

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.…”

A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces

“…The sublevel set {y ∈ [0, 5] : f (x, y) < 0} is not convex for all x ∈ (0, 5) and [12, Theorem 2.1] does not apply. Anyway, the map coF(x) = ⎧ ⎨ if x ∈ [0, 2), [3, 4] if x ∈ [2, 3) ∪ (4, 5], (x, 4] if x ∈ [3, 4],has not fixed points, and all the assumptions of Corollary 1 hold.Corollary 1 is strongly related to Theorem 3.1 in[8] although not equivalent. Actually, the results are not comparable.…”

“…Actually, the results are not comparable. Example 6 describes a case where all the assumptions of Corollary 1 hold but {y ∈ [0, 5] : f (x, y) < 0} is not convex for all x ∈ fix K =[3,4].Vice versa consider the quasiequilibrium problem (2) associated withC = [0, 2], K (x) = {2 − x} and f (x, y) = −1 if x ∈ [0, 2) and y = 2 0 otherwiseThen, fix K = {1}, the assumptions of Theorem 3.1 in[8] are trivially satisfied, but the set{(x, y) ∈ fix K × C : f (x, y) ≥ 0} = {1} × [0, 2) is not closed.More recently, assumption (i ) has been used in[10, Theorem 2.3] in the context of a locally convex topological vector space. Nevertheless, their result requires the closedness and the convexity of the values of K which is both upper and lower semicontinuous, and the convexity of the level set {x ∈ C : f (x, y) ≥ 0}, for each y ∈ C. Take C = [−2, 2], f (x, y) = x 4 − y 4 − 2x 2 + 2y 2 and…”

“…Example 8 Let C = [−2, 2], g 1 , g 2 : C × C → R defined as follows: g 1 (x, y) = 2y 4 − x 4 , g 2 (x, y) = min{1 − x 2 + y 2 , 0}…”