Contractive Mappings: Unique Fixed Point & Notation Questions

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SUMMARY

In compact metric spaces, contractive mappings are established to have a unique fixed point, which is a fundamental concept in fixed-point theory. The notation $card{F}_{T} \le 1$ is used to denote the cardinality of the set of fixed points associated with a contractive mapping, indicating that there is at most one fixed point. The discussion raises questions about the appropriateness of this notation, suggesting that $card{F}_{T} = 1$ may be more accurate under certain conditions. Clarification on the definitions of $F_T$ and the notation used is also sought, emphasizing the need for precise mathematical language.

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ozkan12
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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 ?
 
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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?
 

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