Set of gemstone

The set of gem is a tenet of mathematical sub-region of the functional analysis . It goes back to a work by the mathematician Michael Edelstein from 1962 and deals with a fixed point property of certain non-expanding images . The theorem is related to Banach's Fixed Point Theorem and Browder-Göhde-Kirk's Fixed Point Theorem .

Formulation of the sentence

The set of gemstone can be summarized as follows:

Let be a non-empty subset of a, or in general a non-empty metric space , provided with a metric . ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle d}$

A strictly non-expanding mapping is given and its image set is compact in . ${\displaystyle f\colon X\to X}$ ${\displaystyle \Omega =f(X)}$ ${\displaystyle X}$

Then:

There is exactly one point with . ${\displaystyle x^{*}\in X}$ ${\displaystyle f(x^{*})=x^{*}}$

Here converges for each point the iterative sequence against this fixed point . ${\displaystyle x_{0}\in X}$ ${\displaystyle (f^{n}(x_{0}))_{n=0,1,\dots ,\infty }}$ ${\displaystyle x^{*}}$

Notes on the evidence

Following a thought by M. Kerin , one wins the existence of a fixed point because of the compactness of the image set directly by applying the principle of the minimum to the nonnegative real functional . This ensures that the minimum is assumed in a point , which must then be a fixed point. Because of the presupposed strict non-expansiveness of must apply, since it immediately followed and then , in contradiction to the minimum property of . ${\displaystyle \Omega }$ ${\displaystyle T\colon \Omega \to \mathbb {R} ^{\geq 0}\;,\;x\mapsto T(x)=d(f(x),x)}$ ${\displaystyle T}$ ${\displaystyle x^{*}\in \Omega }$ ${\displaystyle f}$ ${\displaystyle T(x^{*})=0}$ ${\displaystyle T(x^{*})>0}$ ${\displaystyle f(x^{*})\neq x^{*}}$ ${\displaystyle T(f(x^{*}))=d(f(f(x^{*})),f(x^{*}))<d(f(x^{*}),x^{*})=T(x^{*})}$ ${\displaystyle x^{*}}$

In addition, due to the strict non-expansiveness of , the uniqueness of the fixed point can also be inferred directly. Because for one of the different fixed points the contradicting inequality would immediately be deduced. ${\displaystyle f}$ ${\displaystyle x^{*}}$ ${\displaystyle x^{**}}$ ${\displaystyle d(x^{*},x^{**})=d(f(x^{*}),f(x^{**}))<d(x^{*},x^{**})}$

Sources and background literature

• M. Edelstein : On fixed and periodic points under contractive mappings . In: Journal of the London Mathematical Society . tape 37 , 1962, pp. 74-79 , doi : 10.1112 / jlms / s1-37.1.74 ( MR0133102 ).
• LW Kantorowitsch , GP Akilow : Functional analysis in standardized spaces . Edited in German by Prof. Dr. rer. nat. habil. P. Heinz Müller, Technical University of Dresden. Translated from Russian by Heinz Langer, Dresden, and Rolf Kühne, Dresden. Verlag Harri Deutsch , Thun, Frankfurt am Main 1978, ISBN 3-87144-327-1 , p. 512 ( MR0458199 ).
• 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 ).

Individual references and notes

1. ↑ Michael Edelstein: On fixed and periodic points under contractive mappings. In: J. London. Math. Soc. 37, p. 74 ff.
2. ^ JM Ortega, WC Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. 2000, p. 404 ff.
3. ↑ ^{a b c} L. W. Kantorowitsch, GP Akilow: functional analysis in standardized spaces. 1978, p. 512.
4. ^ Edelstein, op.cit, p. 74.
5. ↑ Ortega-Rheinboldt, op.cit, p. 404.
6. ↑ In the event that a subset is one , the metric should - as usual - be assumed to have been generated by a norm , for example by the Euclidean norm . ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle X}$