104LectureNotesonContractionMappingTheorem

104LectureNotesonContractionMappingTheorem - Math 104Spring...

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Math 104–Spring 2005–Anderson Lecture Notes on Contraction Mapping Theorem Defnition 0.1 Let ( S, d ) be a metric space. A function f : S S is a contraction if α [0 , 1) x, y Sd ( f ( x ) ,f ( y )) αd ( x, y ) s is a fxed 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 fxed point. ProoF: We Frst show that a Fxed point exists. Since S 6 = , we may choose an arbitrary s 0 S . Consider the sequence ( s n ) deFned by s 1 = f ( s 0 ) s 2 = f ( s 1 ) . . . s n +1 = f ( s n ) If s 1 = s 0 ,then f ( s 0 )= s 1 = s 0 ,so s 0 is a Fxed point. If s 1 6 = s 0 , we claim that d ( s n +1 ,s n ) α n d ( s 1 ,s 0 ) The proof of the claim is by induction. Note that d ( s 2 ,s 1 )= d ( f ( s 1 ) ,f ( s 0 )) αd ( s 1 ,s 0 ) 1
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Now suppose that d ( s n +1 ,s n ) α n d ( s 1 ,s 0 ). Then d ( s n +2 ,s n +1 )= d ( f ( s n +1 ) ,f ( s n )) αd
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104LectureNotesonContractionMappingTheorem - Math 104Spring...

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