Problem:

1.Show that none of the following mappings have a fixed point and explain why the contraction mapping principle is not contradicted.

a. and

b. X = R and for x in X.

c. and for (x,y) in X

2. Define the function by

for

Show that this function has exactly one fixed point.

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Attempt:

Problem 1:

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

Problem 2:

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.