The aim of this paper is to
study the evaluation of the quantity inf∥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,∞) →⟨0,1)
such that for arbitrary T : X → X satisfying a Lipschitz condition constant
k > 1,inf∥x − Tx∥≦ 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 casc of
Hilbert space is considered and some evaluations for φ(k) are obtained in that
setting.
