On the minimal displacement of points under Lipschitzian mappings
Kazimierz Goebel · Pacific Journal of Mathematics · 1973
The aim of this paper is to study the evaluation of the quantity inf 11 x -Tx \ | when T is a Lipschitzian self mapping of a closed bounded and convex subset of a Banach space.It is proved that in an arbitrary Banach space there exists a function φ(k): 1, inf 11 x-Tx\\^L φ(k)r(X) where r(X) denotes the radius of the set X. Some precise formulas for φ(k) are obtained in certain spaces along with some general evaluations of it in arbitrary spaces.In particular, the case of Hubert space is considered and some evaluations for φ{k) are obtained in that setting.