Equilibria for Set-Valued Maps on Star-Shaped Domains

C. R. Math. Rep. Acad. Sci. Canada Vol. 47 (1) 2025, pp. 1–9
(Received: 2024-08-05 , Revised: 2024-09-11)

Hichem Ben-El-Mechaiekh, Department of Mathematics and Statistics, Brock University, St. Catharines, Ontario, Canada L2S 3A1; e-mail: hmechaie@brocku.ca

Abstract/Résumé:

We generalize the Ky Fan–Halpern equilibrium theorem for a Kakutani type (upper semicontinuous with non-empty closed convex values) set-valued map defined on convex compact domains in locally convex topological linear spaces [5] by replacing the convexity of the domain by its star-shapedness. The proof is simple and relies on an extension of the Ky Fan–Browder fixed point theorem to star-shaped compact domains due to the author [2].

Nous généralisons le théorème de Ky Fan–Halpern [5] sur l’existence d’un équilibre pour une application multivoque du type Kakutani (semi-continue supérieurement à valeurs d’ensembles non-vides, convexes et fermés) définie sur un domaine convexe et compact dans un espace vectoriel topologique localement convexe en remplaçant la convexité du domaine par la condition plus générale qu’il soit étoilé. La preuve est simple et basée sur une généralisation du théorème de point fixe de Ky Fan–Browder aux sous ensembles compacts et étoilés d’espaces topologiques vectoriels [2]

Keywords: Set-Valued Mapping, Star-Shaped Domain, System of Non-Linear Inequalities, Tangent and Normal Cones, equilibrium, fixed point

AMS Subject Classification: Fixed-point theorems, Applications in optimization; convex analysis; mathematical programming; economics, Convex sets in topological vector spaces, Fixed-point and coincidence theorems 47H10, 47N10, 52A07, 54H25

PDF(click to download): Equilibria for Set-Valued Maps on Star-Shaped Domains