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!