Geodesic
Uniformly Lipschitzian Families of Transformations in Banach Spaces
Canadian journal of mathematics, Tome 26 (1974) no. 5, pp. 1245-1256

Voir la notice de l'article provenant de la source Cambridge University Press

The observations of this paper evolved from the concept of 'asymptotic nonexpansiveness' introduced by two of the writers in a previous paper [10]. Let X be a Banach space and K ⊆ X. A mapping T : K → K is called asymptotically nonexpansive if for each x, y ∊ K where {ki } is a fixed sequence of real numbers such that ki →1 as i → ∞ . It is proved in [10] that if K is a bounded closed and convex subset of a uniformly convex space X then every asymptotically nonexpansive mapping T : K → K has a fixed point. This theorem generalizes the fixed point theorem of Browder-Göhde-Kirk [2 ; 12 ; 16] for nonexpansive mappings (mappings T for which ||T(x) — T(y)|| ≦ ||x — y||, x, y ∊ K) in a uniformly convex space. (A generalization along similar lines also has been obtained by Edelstein [4].) 
Goebel, K.; Kirk, W. A. Uniformly Lipschitzian Families of Transformations in Banach Spaces. Canadian journal of mathematics, Tome 26 (1974) no. 5, pp. 1245-1256. doi: 10.4153/CJM-1974-119-9
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