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Fixed point of function on compact metric space

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Assume that (X, d) is a compact metric space, and let f: X -> X be a function such that the inequality d(f(x), f(y)) < d(x, y) holds for all distinct elements x, y in X. Show that f has a unique fixed point.

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Solution Summary

The following is proved in detail: If the indicated type of function has fixed points "a" and "b", then a = b.

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To prove that f has a unique fixed point, we will show that if f has fixed points "a" and "b", then a = b.

So let "a" and "b" be fixed points of f.

Since "a" is a fixed point of f, we have f(a) = a.

Since "b" is a fixed point of f, we have f(b) = ...

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Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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