Fixed point theorem by Krasnoselski

The fixed point theorem Krasnosel'skii ( English Krasnoselskii's fixed-point theorem ) is one of the many tenets that the Soviet mathematician Mark Krasnosel'skii the mathematical branch of nonlinear functional analysis contributed. The theorem goes back to a scientific publication by Krasnoselski in 1962 and deals with the question of the conditions under which a fixed point theorem applies to compact operators on Banach spaces . The theorem is related to Schauder's fixed point theorem .

Formulation of the sentence

Krasnoselski's Fixed Point Theorem can be stated as follows:

Let there be an orderly -Banachraum with norm and order cone . ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle K=\{x\in X\,\colon \,0\preceq x\}\subset X}$

The order cone is a closed subset of , which should not consist of the zero point alone, and the corresponding relation is a semi-order relation . ${\displaystyle K=\{x\in X\,\colon \,0\preceq x\}}$ ${\displaystyle X}$ ${\displaystyle \preceq }$

Let there be a compact operator and two different real numbers and , so that the two conditions ${\displaystyle K}$ ${\displaystyle \Psi \colon K\to K}$ ${\displaystyle R_{1}>0}$ ${\displaystyle R_{2}>0}$

(i) . ${\displaystyle \Psi (x){\not \preceq }x\,(\,\forall x\in {K\cap S_{R_{1}}(0)}\,)}$

(ii) . ${\displaystyle {x\not \preceq }\Psi (x)\,(\,\forall x\in {K\cap S_{R_{2}}(0)}\,)}$

are fulfilled.

Then:

${\displaystyle \Psi }$has a fixed point , which is also the relationship ${\displaystyle x_{0}=\Psi (x_{0})\in K}$

${\displaystyle \min \left(R_{1}\;,\;R_{2}\right)<\|x_{0}\|<\max \left(R_{1}\;,\;R_{2}\right)}$

enough.

Explanations

• Mean the above conditions (i) and (ii) that, for having always applicable and with always . ${\displaystyle x\in K}$ ${\displaystyle \|x\|=R_{1}}$ ${\displaystyle {x-\Psi (x)}\not \in K}$ ${\displaystyle x\in K}$ ${\displaystyle \|x\|=R_{2}}$ ${\displaystyle {\Psi (x)-x}\not \in K}$
• If the above conditions (i) and (ii) are fulfilled, one speaks (in the English technical language) of a cone compression , for a cone expansion . ${\displaystyle R_{1}<R_{2}}$ ${\displaystyle R_{1}>R_{2}}$

background

The derivation of Krasnoselski's Fixed Point Theorem uses the following important theorem by the American mathematician James Dugundji from 1951:

In a Banach space every non-empty , closed and convex subset is a retract of . ${\displaystyle X}$ ${\displaystyle C\subseteq X}$ ${\displaystyle X}$

Inference

With Krasnoselski's fixed point theorem it is possible, under certain circumstances, to infer the existence of very many fixed points. Namely, it entails the following corollary:

If above conditions (i) and (ii) apply even to a whole sequence of number pairs with positive real numbers and if the two number sequences and both converge to , then the compact operator has countably infinite fixed points ${\displaystyle \left(R_{1}(n)\;,\;R_{2}(n)\right)_{n=1,2,3,\ldots }}$ ${\displaystyle R_{1}(n)\neq R_{2}(n)\;(n\in \mathbb {N} )}$ ${\displaystyle \left(R_{1}(n)\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \left(R_{2}(n)\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \infty }$ ${\displaystyle \Psi \colon K\to K}$ .

literature

• Herbert Amann : Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces . In: SIAM Review . tape 18 , 1976, p. 620-709 ( MR0415432 ).
• Philippe G. Ciarlet : Linear and Nonlinear Functional Analysis with Applications . Society for Industrial and Applied Mathematics , Philadelphia, PA 2013, ISBN 978-1-61197-258-0 ( MR3136903 ).
• J. Dugundji: An extension of Tietze's theorem . In: Pacific Journal of Mathematics . tape 1 , 1951, p. 353-367 ( MR0044116 ).
• M. Krasnoselskii: Positive Solutions of Operator Equations (Russian. Published in English under the title "Positive Solutions of Operator Equations" by Leo F. Boron, Noordhoff, Groningen) . State publishing house for physical and mathematical literature, Moscow 1962 ( MR0181881 ).
• Eberhard Zeidler : Lectures on nonlinear functional analysis I: Fixed point theorems . BG Teubner Verlagsgesellschaft , Leipzig 1976 ( MR0473927 ).
• 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. ↑ ^{a b c} Eberhard Zeidler: Lectures on nonlinear functional analysis I 1976, p. 154
2. ↑ ^{a b c d e} Eberhard Zeidler: Nonlinear Functional Analysis and its Applications I 1986, p. 562
3. ^ Philippe G. Ciarlet: Linear and Nonlinear Functional Analysis with Applications. 2013, p. 736
4. ↑ For is here the - sphere . ${\displaystyle r>0}$ ${\displaystyle S_{r}(0)}$ ${\displaystyle r}$
5. ^ Herbert Amann: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 18, p. 657
6. ^ Zeidler (1986), p. 563