Vol. 45, No. 1, 1973

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On the minimal displacement of points under Lipschitzian mappings

Kazimierz Goebel

Vol. 45 (1973), No. 1, 151–163

The aim of this paper is to study the evaluation of the quantity inf x Txwhen 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 Txφ(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.

Mathematical Subject Classification 2000
Primary: 47H99
Received: 23 November 1971
Published: 1 March 1973
Kazimierz Goebel