Non-expanding figure

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The concept of non-expanding mapping comes from functional analysis , one of the sub-areas of mathematics . The non-expanding maps are among the Lipschitz continuous mappings between metric spaces . Among other things, they are significant in connection with fixed point sentences .

definition

A mapping for two metric spaces and is called non-expanding if the following inequality is always satisfied:

Does such a mapping even always satisfy the strict inequality for with

  ,

so one calls strictly non-expanding .

Fixed point theorem by Browder-Göhde-Kirk

The non-expanding images of metric spaces in themselves also include the contractive images . As with the latter, the question of the existence of fixed points also arises for the former . Browder-Göhde-Kirk's Fixed Point Theorem provides an answer to this question . It is related to the fixed point theorems of Banach and Schauder and goes back to works by Felix Earl Browder , Dietrich Göhde and William A. Kirk from the 1960s.

Browder-Göhde-Kirk's fixed point theorem can be summarized as follows:

Let a uniformly convex Banach space be given and in it a non-empty , closed , bounded and convex subset .
Continue to be a non-expanding map, i.e. such that the inequality is always satisfied.
Then:
The fixed point set is a non-empty, closed and convex subset of .
In particular, there is one with .

Browder-Göhde-Kirk's fixed-point theorem gave rise to a number of follow-up investigations which led to various proof variants and generalizations.

Remarks

  • The non-expanding mappings are precisely those Lipschitz continuous mappings between metric spaces which have the Lipschitz constant .
  • The theorem of Edelstein deals with the fixed point properties of certain strictly non-expanding images .

literature

Individual evidence

  1. ^ Eberhard Zeidler: Nonlinear Functional Analysis and its Applications I 1986, p. 478
  2. Dirk Werner: Functional Analysis. 2007, p. 173
  3. ^ Albrecht Pietsch: History of Banach Spaces and Linear Operators. 2007, p. 244