Leray shudder alternative

The Leray-Schauder alternative is a mathematical statement from the field of non-linear functional analysis .

It was proven by the mathematicians Jean Leray and Juliusz Schauder and named after them. The Leray-Schauder alternative gives a sufficient condition for the existence of a fixed point. The central condition of the statement has an independent name and is called the Leray-Schauder boundary condition . The sentence has numerous corollaries that were known before the Leray-Schauder alternative was discovered and that have independent meanings.

Leray-Schauder boundary condition

Be a normalized space . The continuous mapping fulfills the Leray-Schauder boundary condition, if one exists, so that the inequality follows for all . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle r>0}$ ${\displaystyle \|x\|=r}$ ${\displaystyle f(x)\neq \lambda x}$ ${\displaystyle \lambda >1}$

Leray shudder alternative

Let there be a normalized space and a compact mapping that satisfies the Leray-Schauder boundary condition, then has at least one fixed point . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle f}$

The statement is called alternative because either the equation for one or the equation has a solution . However, the theorem does not offer any necessary conditions, so both equations can be fulfilled for certain ones. The central tool for proving the theorem is the Leray-Schauder degree of mapping . ${\displaystyle f(x)=\lambda x}$ ${\displaystyle \lambda >1}$ ${\displaystyle f(x)=x}$ ${\displaystyle f}$

Special cases

This section lists sufficient conditions for fixed points which have been proven by Altman, Krasnoselskii and others and which can be understood as special cases of the Leray-Schauder alternative. In the following we assume normalized space, a continuous function and a sphere with a radius . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle B_{r}=\{x:\|x\|<r\}\subset X}$ ${\displaystyle r}$

Altman's theorem

Be and apply ${\displaystyle x\in \partial B_{r}}$

${\displaystyle \|x-f(x)\|^{2}\geq \|f(x)\|^{2}+\|x\|^{2},}$

then has at least one fixed point. ${\displaystyle f}$

This statement was proven by Altman in 1957.

Petryshyn's theorem

Be and apply ${\displaystyle x\in \partial B_{r}}$

${\displaystyle \|x-f(x)\|\geq \|f(x)\|,}$

then has at least one fixed point. ${\displaystyle f}$

This statement was proven by Volodymyr Petryshyn in 1963.

Krasnoselskii's theorem

Be a pre-Hilbert space , is true and ${\displaystyle X}$ ${\displaystyle x\in \partial B_{r}}$

${\displaystyle \langle f(x),x\rangle \leq \|x\|^{2},}$

then has at least one fixed point. ${\displaystyle f}$

This statement was shown by Mark Krasnosel'skii in 1953. It can be understood as a special case of Altman's statement for prehilbert dreams.

Rothe's theorem

Be and apply ${\displaystyle x\in \partial B_{r}}$

${\displaystyle \|f(x)\|\leq \|x\|,}$

then has at least one fixed point. ${\displaystyle f}$

This statement was proven by Rothe in 1937.

swell

• Klaus Deimling: Nonlinear Functional Analysis . 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 , page 204.
• Robert F. Brown: A topological introduction to nonlinear analysis . Birkhäuser 2004, ISBN 0817632581 , page 27.
• Vasile I. Istratescu: Fixed Point Theory an Introduction . Springer Science & Business 2001, ISBN 9027712247 , page 166.