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contracting-maps2

contracting-maps2 - Contracting Maps and an Application...

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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 Picard’s existence theorem for ordinary differential equa- tions. We will reverse the historical order and first prove Banach’s 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 0 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 ) cd ( x k , x k 1 ) so by induction d ( x k + 1 , x k ) c k d ( x 1 , x 0 ) . (2)
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