Berkeley MATH 104 - Lecture Notes on Contraction Mapping Theorem (3 pages)

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Lecture Notes on Contraction Mapping Theorem



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Lecture Notes on Contraction Mapping Theorem

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Lecture Notes


Pages:
3
School:
University of California, Berkeley
Course:
Math 104 - Introduction to Analysis
Introduction to Analysis Documents

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Math 104 Spring 2005 Anderson Lecture Notes on Contraction Mapping Theorem Definition 0 1 Let S d be a metric space A function f S S is a contraction if 0 1 x y S d f x f y d x y s is a fixed point of f if f s s Theorem 0 2 Contraction Mapping Theorem If S d is a complete metric space and f S S is a contraction then f has a unique fixed point Proof We first show that a fixed point exists Since S we may choose an arbitrary s0 S Consider the sequence sn defined by s1 f s0 s2 f s1 sn 1 f sn If s1 s0 then f s0 s1 s0 so s0 is a fixed point If s1 s0 we claim that d sn 1 sn n d s1 s0 The proof of the claim is by induction Note that d s2 s1 d f s1 f s0 d s1 s0 1 Now suppose that d sn 1 sn n d s1 s0 Then d sn 2 sn 1 d f sn 1 f sn d sn 1 sn n d s1 s0 n 1 d s1 s0 so the claim follows by induction Next we show that sn is a Cauchy sequence Let 0 Since 0 1 limn n 0 by Example 9 7 b Choose N such that 1 N d s1 s0 If m n N d sm sn d sm sm 1 d sn 1 sn m 1 d s1 s0 m 2 d s1 s0 n d s1 s0 1 m 1 n d s1 s0 by Exercise 9 18 1 1 n m d s1 s0 1 n d s1 s0 1 N d s1 s0 1 so sn is Cauchy Since S d is complete sn has a limit s S We will show that f s s Fix 0 There exists N1 such that n N1 d sn s 2 2 Since sn is Cauchy there exists N2 such that n N2 d sn 1 sn 2 Choose any n max N1 N2 Then d s f s d s sn 1 d sn 1 f s d s sn 1 d f sn f s d sn s 2 2 2 Since is arbitrary d s f s 0 so f s s Thus f has a fixed point To show the fixed point is unique suppose f s s and f t t Then d s t d f s f t d s t so 1 d s t 0 so d s t 0 Since d is a metric d s t 0 and thus s t so the fixed point is unique 3



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