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November 19th, 2011, 12:13 PM   #1
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Applying the Contraction Mapping Principle

Problem:
Prove that there exists a continuous function such that:
.

This is what I have done:
Define the function by:
,
and observe that is and only if is a solution to the above. We next note that:







, where

Since is complete with the metric , we invoke the Contraction Mapping Principle to conclude that has a unique fixed point. In particular, there exists a function such that satisfies:
.
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November 19th, 2011, 01:23 PM   #2
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Re: Applying the Contraction Mapping Principle

It seems correct, except that the first line of your "align", it should be .
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