Weissinger's Fixed Point Theorem

The fixed point theorem Weissinger is a fixed-point theorem in the analysis . He generalizes Banach's fixed point theorem .

The theorem was established and proven by Johannes Weissinger in 1952.

statement

Be a Banach space and be closed and not empty as well as a self-image for the ${\displaystyle ({\mathcal {B}},\|\cdot \|)}$ ${\displaystyle U\subset {\mathcal {B}}}$ ${\displaystyle T:U\to U}$

${\displaystyle \|T^{j}u-T^{j}v\|\leq \alpha _{j}\|u-v\|,\quad \forall u,v\in U}$

holds with numbers such that . Then has exactly one fixed point in , namely ${\displaystyle \alpha _{j}\geq 0}$ ${\displaystyle \textstyle \sum _{j=1}^{\infty }\alpha _{j}<\infty }$ ${\displaystyle T}$ ${\displaystyle U}$

${\displaystyle u=\lim _{n\to \infty }T^{n}u_{0}}$

with any . The error estimate applies ${\displaystyle u_{0}\in U}$

${\displaystyle \|u-u_{n}\|\leq \sum _{j=n}^{\infty }\alpha _{j}\|u_{1}-u_{0}\|}$

with . ${\displaystyle u_{j}=T^{j}u_{0}}$

Remarks

• The condition is obviously sufficient, because in this case one can choose. ${\displaystyle \sum _{j=1}^{\infty }\|T^{j}\|<\infty }$ ${\displaystyle \alpha _{j}=\|T^{j}\|}$
• The proof of this fixed point theorem essentially agrees with the classical proof of Banach's fixed point theorem. The Banach fixed-point theorem follows with the replacement of a constant than Lipschitz constant figure . ${\displaystyle \alpha _{j}=k^{j}}$ ${\displaystyle k}$ ${\displaystyle T}$
• Weissinger's fixed point theorem serves as the basis for proof of existence and uniqueness in the theory of differential equations . In particular, Picard-Lindelof's theorem follows from it .

Individual evidence

1. ↑ Johannes Weissinger: On the theory and application of the iteration method . In: Mathematical News . tape 8 , p. 193-212 .