lecture7 - Lecture 7 Contraction Mapping Theorem...

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Lecture 7 Contraction Mapping Theorem
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ECN/APEC 7240 VII - 2 Contraction Mappings Let ( X , d ) be a metric space and T : X X . We say that T is a contraction mapping iff for some β (0, 1), for all x , y X , We call the modulus of T . ). , ( )) ( ), ( ( y x d y T x T d
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ECN/APEC 7240 VII - 3 Operator Notation We will often view T as an operator on X . If x X , we write T ( x ) = Tx . For n N , define T n +1 = T n ° T , where T 1 = T . We also define T 0 to be the identity.
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ECN/APEC 7240 VII - 4 Contraction Mapping Theorem For any function f : X X , we say x * X is a fixed point of f iff f ( x *) = x *. Let ( X , d ) be a complete metric space and T : X X be a contraction with modulus β . 1. The mapping T has a unique fixed point x * X . 2. For any x 0 X , d ( T n ( x 0 ), x* ) n d ( x 0 , x *). Since all sequences generated by T converge to x *, x * is called an attractor of T .
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ECN/APEC 7240 VII - 5 Closed Subspaces Let ( X , d ) be a complete metric space and T : X X be a contraction with modulus β . Let
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lecture7 - Lecture 7 Contraction Mapping Theorem...

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