Prove the function $K$ is defined by $K(x) = 4x(1-x)$ maps the interval $[0,1]$ into itself and it is not a contraction. Prove that it has a fixed point. Why does this not contradict the Contraction Mapping Theorem?

I understand how to show that it maps the interval into itself, how to show it has a fixed point, and that it is not a contraction. I am confused about the Contraction Mapping Theorem part.


It's really logic:

Contraction mapping principle: If $f : [0,1] \to [0,1]$ is a contraction, then it has a fixed point.

Your example just showed the converse of the above statement:

If $f : [0,1] \to [0,1]$ has a fixed point, then it is a contraction.

is not true. As the converse of a statement is not the same as that statement, your example does not contradict the contraction mapping principle.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.