A Note on Asymptotic Normal Structure and Close-to-Normal Structure

Tan, Kok-Keong

Canadian mathematical bulletin, Tome 25 (1982) no. 3, pp. 339-343

Résumé

A closed convex subset X of a Banach space E is said to have (i) asymptotic normal structure if for each bounded closed convex subset C of X containing more than one point and for each sequence in C satisfying ‖x n − x n + 1‖ → 0 as n → ∞, there is a point x ∈ C such that ; (ii) close-to-normal structure if for each bounded closed convex subset C of X containing more than one point, there is a point x ∈ C such that ‖x − y‖ < diam‖ ‖(C) for all y ∈ C While asymptotic normal structure and close-to-normal structure are both implied by normal structure, they are not related. The example that a reflexive Banach space which has asymptotic normal structure but not close-to normal structure provides us a non-empty weakly compact convex set which does not have close-to-normal structure. This answers an open question posed by Wong in [9] and hence also provides us a Kannan map defined on a weakly compact convex set which does not have a fixed point.

Détail

DOI : 10.4153/CMB-1982-047-3

Mots-clés : 46B20, 47H10, Asymptotic normal structure, close-to-normal structure, normal structure, nonexpansive mapping, Kannan map, fixed point