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Unformatted text preview: Contracting Maps and an Application Math 508, Fall 2010 Jerry L. Kazdan One often effective way to show that an equation g ( x ) = b has a solution is to reduce the problem to find a fixed point x of a contracting map T , so Tx = x . For instance, assume V is a linear space and g : V V . Define a new map T : V V by T ( x ) : = x g ( x )+ b . Then clearly x is a fixed point of T if and only if it solves g ( x ) = b . Our setting is a metric space M with metric d ( x , y ) and a map T : M M . We say that T is a contracting map if there is some c with 0 < c < 1 such that d ( Tx , Ty ) cd ( x , y ) for all x , y M (1) so T contracts the distance between points. The following theorem was found by Banach who abstracted the essence of Picards existence theorem for ordinary differential equa tions. We will reverse the historical order and first prove Banachs version. Theorem 1 [P RINCIPLE OF C ONTRACTING M APS ] Let M be a complete metric space and T : M M a contracting map. Then T has a unique fixed point p M . P ROOF The uniqueness is short. Say p and q are fixed points. Then by the contracting condition d ( p , q ) = d ( T p , Tq ) cd ( p , q ) . Since c < 1, the only possibility is that d ( p , q ) = 0 so p = q . To prove the existence of a fixed point, pick any x M and inductively define the succes sive approximations x k by x k = Tx k 1 for k = 1 , 2 ,... . Then using (1) d ( x k + 1 , x k ) = d ( Tx k , Tx k 1 )...
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
 Fall '10
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 Math

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