hw4additional - d ( x, f ( x )) : x X } , show that it is a...

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Homework 4, Additional Problem. (1) Let ( X, d ) be a compact metric space and f : X X a mapping such that d ( f ( x ) , f ( y )) < d ( x, y ) for all x 6 = y . a. Show that there exists x 0 X such that f ( x 0 ) = x 0 . (Hint: Consider inf {
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Unformatted text preview: d ( x, f ( x )) : x X } , show that it is a minimum and that this minmum must be equal to zero.) b. Show that the xed point of part a) is unique. 1...
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This note was uploaded on 02/05/2012 for the course MATH 703 taught by Professor Schep during the Fall '11 term at South Carolina.

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