# 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 ?

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#### Solution Preview

Let be a compact metric space and let be contraction maps; that is there exist constants such that for all .

Define .

Let be any fixed points of the maps f and g respectively. Using the triangle ...

#### Solution Summary

This provides a proof regarding contraction maps and distance, as well as a deduction regarding fixed points and contraction maps.