MHB Contractive Mappings: Unique Fixed Point & Notation Questions

  • Thread starter Thread starter ozkan12
  • Start date Start date
Click For Summary
In compact metric spaces, contractive mappings are known to have a unique fixed point, leading to discussions about the notation used for cardinality. The notation $card{F}_{T} \le 1$ raises questions, with some suggesting it should be $card{F}_{T} = 1$ instead. Clarification is sought on the meaning of this notation, particularly regarding its definition and the nature of $F_T$. Participants are encouraged to provide insights on the notation and its implications in the context of contractive mappings. Understanding these concepts is essential for further discussions on fixed points in metric spaces.
ozkan12
Messages
145
Reaction score
0
İn compact metric space, we know that contractive mappings have a unique fixed point...And some book I see that $card{F}_{T}\le 1$...Why we use this notation ? İn my opinion we must use $card{F}_{T}=1$ ? Have you any suggestions ? Can you help me ?
 
Physics news on Phys.org
Couple comments:

1. Try using the
Code: 
\text{}
wrapper around things that want to look like regular text. So, for example, you'd type
Code: 
\text{card} \, F_T \le 1
to obtain $\text{card} \, F_T \le 1$.

2. Second of all, could you please define this notation for us? I've never seen it before. Is it some sort of cardinality? And what is $F_T$? An operator? A contractive mapping?
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
View full post »

Similar threads

Undergrad No structure in ##(d,x)## in ##\textbf{Met*}(X)## is admissible
  • · Replies 0 ·
Replies
0
Views
2K
Undergrad Proving a Fixed Point Theorem for Shrinking Maps on Compact Spaces
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Question about lifting of a function
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
955
MHB Continuous mapping and fixed point
  • · Replies 6 ·
Replies
6
Views
2K
MHB Generalization of Banach contraction principle
  • · Replies 1 ·
Replies
1
Views
2K
Does T have a unique fixed point in X?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Contraction and contractive mapping
  • · Replies 4 ·
Replies
4
Views
2K
MHB Contraction constant in banach contraction principle
  • · Replies 3 ·
Replies
3
Views
2K
MHB Kannan mapping and quasi-nonexpansive mapping
  • · Replies 7 ·
Replies
7
Views
2K