Problem:
Compact metric space and contraction maps
Here is the question.
Let (X,d) be a compact metric space, and let Con(X) denote the set of contraction maps on X. We shall define the distance between two maps f,g which belongs to Con(X) as follows :
d_c (f,g) = sup d ( f(x) , g(x) ) for any x that belongs to X.
a) Show that
d(y_f,y_g) <= d_c(f,g)/(1-min(cf,cg)) (where <= mean smaller or equal)
Where y_f and y_g are the fixed points of f and g, respectively, and cf and cg are their contraction factors.
b) What property regarding fixed points and contraction maps can be deduced from this result ?