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