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November 19th, 2011, 12:13 PM  #1 
Member Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0  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: . 
November 19th, 2011, 01:23 PM  #2 
Member Joined: Nov 2009 From: France Posts: 98 Thanks: 0  Re: Applying the Contraction Mapping Principle
It seems correct, except that the first line of your "align", it should be .


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