Fixed Point Theory: Lipschitz or Contraction?

Click For Summary

Discussion Overview

The discussion revolves around the relationship between Lipschitz continuous mappings and contraction mappings within the context of fixed point theory. Participants explore the definitions and implications of these classes of mappings.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests that if a mapping is a contraction, it is also contractive, nonexpensive, and Lipschitz, prompting a question about which class is more general.
  • Another participant states that contraction is a special case of Lipschitz continuous functions, specifically when the Lipschitz constant is less than 1.
  • Some participants propose that Lipschitz mappings are indeed more general than contraction mappings.

Areas of Agreement / Disagreement

There appears to be agreement among participants that Lipschitz mappings are more general than contraction mappings, although the discussion does not delve into the implications of this assertion.

Contextual Notes

The discussion does not clarify the specific definitions of "contractive" or "nonexpensive," nor does it address any assumptions regarding the mappings being discussed.

ozkan12
Messages
145
Reaction score
0
I see that if a mapping is contraction then it is contractive then it is nonexpensive and then it is lipschtiz...so, which class of mapping is general ? lipschitz or contraction ? which one ? thank you for your attention :)
 
Physics news on Phys.org
Contraction is a special case of Lipschitz continuous functions, namely, with Lipschitz constant $K$ satisfying $0\le K<1$.
 
so, lipschitz mappings are more general than contraction ?
 
ozkan12 said:
so, lipschitz mappings are more general than contraction ?
Yes.
 
ok. thanks a lot :)
 

Similar threads

MHB Lipschitz Continuity .... and Continuity in R^n ....
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Does Ricci Flow Contract a 3-Sphere to Its Center?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Fixed Point Theorem & Contractive Maps
  • · Replies 2 ·
Replies
2
Views
2K
MHB Contraction constant in banach contraction principle
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad No structure in ##(d,x)## in ##\textbf{Met*}(X)## is admissible
  • · Replies 0 ·
Replies
0
Views
2K
MHB Why is the contraction constant important in the Banach contraction principle?
  • · Replies 4 ·
Replies
4
Views
3K
MHB Contractive Mappings: Unique Fixed Point & Notation Questions
  • · Replies 1 ·
Replies
1
Views
2K
MHB Continuous mapping and fixed point
  • · Replies 6 ·
Replies
6
Views
2K
MHB Linear Mappings are Lipschitz Continuous .... D&K Example 1.8.14 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Proof of Contraction Mapping for $f\left(x\right)={e}^{{-e}^{-x}}$
  • · Replies 12 ·
Replies
12
Views
3K