Is f(x) a Contraction Mapping?

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Discussion Overview

The discussion revolves around the contraction mapping property of the function f(x) = 10/x + (1/20)x^2, specifically for the interval 0≤x≤10. Participants explore whether this function can be classified as a contraction mapping and discuss potential modifications to either the function or the interval to achieve this property.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that f(x) is not a contraction map, providing specific evaluations of f(1) and f(2) to illustrate that the distances do not satisfy the contraction condition.
  • Another participant questions the definition of the interval, suggesting that 0
  • There is a concern raised about the function mapping onto itself within the specified interval, noting that f(x) approaches infinity as x approaches 0.
  • Some participants propose exploring modifications to the function or the interval to potentially establish a contraction mapping result.
  • One participant mentions verifying that |f'(x)|≤0.9<1, indicating a partial understanding of the contraction mapping criteria but expresses difficulty in demonstrating the function's self-mapping property.

Areas of Agreement / Disagreement

Participants generally agree that the function is not a contraction mapping as defined. However, there is disagreement regarding the appropriate definition of the interval and whether modifications can lead to a contraction mapping result.

Contextual Notes

Participants note the importance of the upper bound in the interval and express uncertainty about the implications of the function approaching infinity as x approaches 0. There are also unresolved questions about the exact nature of the interval and its impact on the analysis.

odck111
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Hello All:
I am working on a function given as f(x) = 10/x + (1/20)x^2 for x such that 0≤x≤10. What can be said about the contraction mapping property of f(x)=x? If it is not a contraction map, is there any way to make modifications on the function or the interval and prove a contraction mapping result? The upper bound in the interval is important to keep.. Attempt on problem: I can verify that |f'(x)|≤0.9<1 but I am stuck when it comes to showing that the function maps onto itself in the given interval. It is indeed not true since f(x)→infinity when T=0.

Thanks very much!
 
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Correct me if I'm wrong, but I believe 0≤x≤10 is not a region; 0<x<10 is. You may want to check the problem statement.
 
I don't think it will matter but I removed "region". There is no problem statement actually, this is something I am trying to solve for my research.
Thanks,

dodo said:
Correct me if I'm wrong, but I believe 0≤x≤10 is not a region; 0<x<10 is. You may want to check the problem statement.
 
odck111 said:
Hello All:
I am working on a function given as f(x) = 10/x + (1/20)x^2 for x such that 0≤x≤10. What can be said about the contraction mapping property of f(x)=x?
It is clearly not a contraction map.
f(1)= 10+ 1/20= 10.05 and f(2)= 5+ 4/20= 5.2
The distance from 5.2 to 10.05 is definitely NOT less than the distance from 1 to 2.


If it is not a contraction map, is there any way to make modifications on the function or the interval and prove a contraction mapping result? The upper bound in the interval is important to keep..


Attempt on problem: I can verify that |f'(x)|≤0.9<1 but I am stuck when it comes to showing that the function maps onto itself in the given interval. It is indeed not true since f(x)→infinity when T=0.

Thanks very much!
 

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