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 ${\displaystyle f\colon X\to Y}$ ${\displaystyle x_{1},x_{2}\in X}$ ${\displaystyle x_{1}\neq x_{2}}$

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

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 .

literature

• Felix E. Browder: Nonexpansive nonlinear operators in a Banach space . In: Proceedings of the National Academy of Sciences . tape 54 , 1965, pp. 1041-1044 , JSTOR : 73047 ( MR0187120 ).
• Kazimierz Goebel: An elementary proof of the fixed-point theorem of Browder and Kirk . In: Michigan Mathematical Journal . tape 16 , 1969, p. 381-383 , doi : 10.1307 / mmj / 1029000322 ( MR0251604 ).
• Dietrich Göhde: On the principle of contractive mapping . In: Mathematical News . tape 30 , 1965, pp. 251-258 , doi : 10.1002 / mana.19650300312 ( MR0190718 ).
• Vasile I. Istrățescu : Fixed Point Theory . An Introduction. With a Preface by Michiel Hazewinkel (=  Mathematics and its Application . Volume 7 ). D. Reidel Publishing Company , Dordrecht, Bosto, London 1981, ISBN 90-277-1224-7 ( MR0620639 ).
• Jacek Jachymski: Another proof of the Browder-Göhde-Kirk theorem via ordering argument . In: Bulletin of the Australian Mathematical Society . tape 65 , 2002, pp. 105-107 , doi : 10.1017 / S0004972700020104 ( MR1889383 ).
• WA Kirk: A fixed point theorem for mappings which do not increase distances . In: American Mathematical Monthly . tape 72 , 1965, pp. 1004-1006 , JSTOR : 2313345 ( MR0189009 ).
• James M. Ortega , WC Rheinboldt : Iterative Solution of Nonlinear Equations in Several Variables . (Unabridged republication of the work first published by Academic Press, New York and London, 1970) (=  Classics in Applied Mathematics . Volume 30 ). Society for Industrial and Applied Mathematics, Philadelphia 2000, ISBN 0-89871-461-3 , pp. 404-407 ( MR1744713 ).
• Albrecht Pietsch : History of Banach Spaces and Linear Operators . Birkhäuser Verlag, Boston / Basel / Berlin 2007, ISBN 0-8176-4367-2 ( MR2300779 ).
• Dirk Werner : Functional Analysis . 6th, corrected edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .
• Eberhard Zeidler : Nonlinear Functional Analysis and its Applications . I: Fixed-Point Theorems. Translated by Peter R. Wadsack. Springer Verlag, New York / Berlin / Heidelberg / Tokyo 1986, ISBN 0-387-90914-1 ( MR0816732 ).

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