1.Show that none of the following mappings have a fixed point and explain why the contraction mapping principle is not contradicted.
b. X = R and for x in X.
c. and for (x,y) in X
2. Define the function by
Show that this function has exactly one fixed point.
By a definition a point x in X is a fixed point for the mapping provided that
For (a) and (b) is it sufficient to show that and ?
For (c), the only fixed point would be , but
If this function has a fixed point, then
I should slap myself, because I can't figure a way to solve for x. First glance, I thought of using the quadratic formula, but I can't do that.
Thank you for your time.