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Unformatted text preview: Lecture 15 Definition 1. Let M be a metric space. A contraction on M is a function : M M such that there exists a positive constant k < 1 with a,b M = d ( ( a ) , ( b )) kd ( a,b ) . Theorem 1. (Contraction Principle): Each contraction on a complete metric space M has a unique fixed point: there exists a unique z M such that ( z ) = z . Proof. Uniqueness is plain (without completeness). Existence is constructive. Choose any z M and for n N , define z n = n ( z ). (That is, induc- tively z n +1 = ( z n ).) Claim: ( z n ) n N is Cauchy. d ( z p ,z q ) d ( z p ,z p +1 ) + + d ( z q- 1 ,z q ) ( k p + + k q- 1 ) d ( z ,z 1 ) d ( z ,z 1 ) 1- k k p and we see that d ( z ,z 1 ) 1- k k p 0 as p . We say z n z (by completeness of M ). From z n +1 = ( z n ) and continu- ity of , it follows that z = ( z )....
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- Summer '10