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

S.L. Singh, Department of Mathematics, Gurukula Kangri University, Hardwar, India, and 21, Govind Nagar, Rishikesh 249201, India; email: vedicmri@gmail.com

Rajendra Pant, Department of Mathematics, S.P.R.C. Post Graduate College, Rohalki-Kishanpur, Hardwar, India; email: pant.rajendra@gmail.com

Abstract/Résumé:

We obtain fixed and common point theorems generalizing fixed point theorems of W. A. Kirk and T. Suzuki for Banach and Meir–Keeler type asymptotic contractions.

Nous démontrons des théorèmes de points fixes et de points communs qui généralisent des théorèmes de points fixes du type de Banach et de Meir–Keeler pour les contractions asymptotiques.

Keywords: Banach contraction, Meir–Keeler type asymptotic contraction, asymptotic contraction, fixed point

AMS Subject Classification: Fixed-point and coincidence theorems 54H25

PDF(click to download): Remarks on some recent fixed point theorems

Simeon Reich, Department of Mathematics, The Technion–Israel Institute of Technology, 32000 Haifa, Israel; email: sreich@tx.technion.ac.il

Alexander J. Zaslavski, Department of Mathematics, The Technion–Israel Institute of Technology, 32000 Haifa, Israel; email: ajzasl@tx.technion.ac.il

Abstract/Résumé:

Let \(K\) be a bounded, closed and convex subset of a Banach space \(X\). We show that the iterates of a typical element (in the sense of Baire category) of a class of nonexpansive mappings which take \(K\) to \(X\) converge uniformly on \(K\) to the unique fixed point of this typical element.

Soit \(K\) un sous-ensemble borné, fermé et convexe d’un espace de Banach \(X\). Nous démontrons que les itérés d’un élément typique (au sens des catégories de Baire) d’une classe d’applications non-expansives de \(K\) dans \(X\) convergent uniformément sur \(K\) vers l’unique point fixe de cet élément typique.

Keywords: Banach space, approximate fixed point, complete metric space, fixed point, generic property, iteration, nonexpansive mapping, porous set, weakly inward

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

PDF(click to download): Convergence of iterates of typical nonexpansive mappings in Banach spaces

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