Show that the following mapping do not have a fixed point and explain why the Contraction Mapping Principle is not contradicted.
Problem: Let X=R and f(x) = x+1 for all x in X.
My Solution:
f do not have a fixed point because x does not equal to x+1 for any real number x.
d(f(x),f(y)) = |(x+1) - (y+1)| = |x-y|,
and d(x,y) = |x-y|.
So d(f(x),f(y)) = d(x,y), now I know I have to find a constant c with 0 <= C < 1 for the Contraction Mapping Principle to hold. But I can't find that c.
Please help.
I think I didn't word the question correctly, let me edit.
"Show that the following mapping do not have a fixed point and explain why the Contraction Mapping Principle is not contradicted."
So if I understand this question correctly, I have to show the mapping do not have a fixed point, but yet it is still a contraction. (I guess because it is not in a complete space?)
And yeah, I forgot to put the first part of my answer on there, it is added now.