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    Compact metric space and contraction maps

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    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.