Fix a positive real number, let and define by
a. For what values of does the mapping have the property that ?
b. For what values of does the mapping have the property that and is a contraction?
a. Since , then the derivative is
Since , I need to find the maximum value of the function to be 1, since to be a subset.
Plug x = 1/2 back into,
Thus, since we are given prior that alpha is a positive real number then where
b. To be a contraction, then its Lipschitz constant C for the mapping is
I am having troubles with this inequality. Since , then and , how would I solve for the value of in ?
Thank you for your time.