Can anyone help me with the following proof please:
Let A>reals and Φ: A>reals Є C1.
Then the smallest k with
| Φ(x) – Φ(y)| ≤ |x-y| for all x,y in A is given by
k = max |Φ’(ε)| with ε ЄA.
Show that the contraction constant k is given by k = max |Φ’(ε)| with ε ЄA.
Thanks to anyone who is able to help.
Did you try ? That's less difficult than it looks like. Starting from
for all ,if we let tend to we get that for all ,
What is the LHS (think about the definition of the derivative) ? What is the RHS ?
That sounds like "I don't have the solution so I can't do it"...because we havent been shown
I can't think of a proof that doesn't use this theorem.nor do i know what the mean value theorem is.