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

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

Abstract/Résumé:

We extend the Ky Fan fixed point theorem for a set-valued map with convex non-empty values and open fibers defined on a star-shaped compact subset in a topological vector space. The proof is elementary and based on a matching theorem of Ky Fan for open covers of convex sets.

Nous généralisons le théorème de point fixe de Ky Fan pour une application à valeurs d’ensembles convexes non-vides et à images inverses ouvertes définie sur un domaine compact et étoilé dans un espace vectoriel topologique. La preuve est élémentaire et se base sur un théorème de “matching” de Ky Fan pour les recouvrements ouverts d’ensembles convexes.

Keywords:

AMS Subject Classification: Fixed-point theorems 47H10

PDF(click to download): The Ky Fan fixed point theorem on star-shaped domains

S. Reich / A.J. Zaslavki

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: contractive mapping, fixed point set, generic property, nonexpansive mapping, uniform space

AMS Subject Classification: Nonexpansive mappings; and their generalizations (ultimately compact mappings; measures of noncompactness and condensing mappings; $A$-proper mappings; $K$-set contractions; etc.), Fixed-point theorems, 47H09, 47H10, 58F99

PDF(click to download): Almost all nonexpansive mappings are contractive

S.L. Singh / S.N. Mishra

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: Coincidence point, fixed point, multi-valued maps, weak compatibility

AMS Subject Classification: Fixed-point theorems, Set-valued maps, Fixed-point and coincidence theorems 47H10, 54C60, 54H25

PDF(click to download): Some remarks on coincidences and fixed points