# Show That a Function is Contractive?

1. May 4, 2007

1. The problem statement, all variables and given/known data
Show that the following function is contractive on the indicated intervals. Determine the best values of [lamda] in Equation (2).

abs(x)^(2/3) on abs(x) < or = 1/3

2. Relevant equations
A mapping (or function) F is said to be contractive if there exists a number [lamda] less than 1 such that:

(Equation (2))
abs(F(x)-F(y)) < or = [lamda]*abs(x-y)

for all points x and y in the domain F.

3. The attempt at a solution
I'm not really sure what to do with this one or how to get [lamda] in this case.

2. May 4, 2007

### tim_lou

mean value theorem?

it seems that for the function that is given, such lamda doesn't exist. consider a interval extremely close to 0, let's say
$$(-\epsilon, \epsilon)$$

you see that
$$\frac{|F(x)-F(y)|}{|x-y|}$$
goes to infinity, as it approaches the derivative at 0.

are you sure you have the correct question?

3. May 5, 2007