# Non-expanding figure

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: ${\ displaystyle f \ colon X \ to Y}$${\ displaystyle (X, d_ {X})}$${\ displaystyle (Y, d_ {Y})}$

${\ displaystyle d_ {Y} (f (x_ {1}), f (x_ {2})) \ leq d_ {X} (x_ {1}, x_ {2}) \; (\ forall x_ {1} , x_ {2} \ in X)}$

Does such a mapping even always satisfy the strict inequality for with${\ displaystyle f \ colon X \ to Y}$${\ displaystyle x_ {1}, x_ {2} \ in X}$${\ displaystyle x_ {1} \ neq x_ {2}}$

${\ displaystyle d_ {Y} (f (x_ {1}), f (x_ {2}))   ,

so one calls strictly non-expanding . ${\ displaystyle f}$

## 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 .${\ displaystyle E}$ ${\ displaystyle X \ subseteq E}$
Continue to be a non-expanding map, i.e. such that the inequality is always satisfied.${\ displaystyle f \ colon X \ to X}$${\ displaystyle \ | {f (x_ {1}) - f (x_ {2})} \ | _ {E} \ leq \ | {x_ {1} -x_ {2}} \ | _ {E} \ ; (\ forall x_ {1}, x_ {2} \ in X)}$
Then:
The fixed point set is a non-empty, closed and convex subset of .${\ displaystyle \ operatorname {Fix} (f) = \ {x \ in X \ colon f (x) = x \}}$${\ displaystyle X}$
In particular, there is one with .${\ displaystyle x_ {0} \ in X}$${\ displaystyle f (x_ {0}) = x_ {0}}$

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 .${\ displaystyle f}$${\ displaystyle \ operatorname {Lip} _ {f} \ leq 1}$
• The theorem of Edelstein deals with the fixed point properties of certain strictly non-expanding images .

## 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