Is f(x) a Contraction Mapping?

[odck111]

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!

 

[dodo]

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]

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,

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.

 

[HallsofIvy]

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!

 

[blue_raver22]

I would like to clarify that the concept of a contraction mapping is typically used in the context of metric spaces and not for single-variable functions like f(x)=x. However, we can discuss the properties of this function and its behavior in the given interval.

Firstly, f(x)=x is not a contraction map in the interval [0,10] because it does not satisfy the necessary condition of |f(x)-f(y)|<k|x-y| for all x,y in the interval, where k is a constant less than 1. In fact, as you mentioned, f(x) approaches infinity as x approaches 0, which violates the definition of a contraction map.

To make modifications on the function or the interval in order to prove a contraction mapping result, we would need to find a k value that satisfies the necessary condition mentioned above. However, this may not be possible as the upper bound in the interval, 10, is important to maintain for the given function. Modifying the interval or the function itself may change its behavior and lead to different results.

In conclusion, f(x)=x is not a contraction map in the interval [0,10] and it may not be possible to modify the function or the interval to prove a contraction mapping result while maintaining the important upper bound. It is important to carefully consider the properties and limitations of a function when trying to prove certain results.