Lecture15 - Lecture 15 Definition 1. Let M be a metric...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 15 Definition 1. Let M be a metric space. A contraction on M is a function : M M such that there exists a positive constant k < 1 with a,b M = d ( ( a ) , ( b )) kd ( a,b ) . Theorem 1. (Contraction Principle): Each contraction on a complete metric space M has a unique fixed point: there exists a unique z M such that ( z ) = z . Proof. Uniqueness is plain (without completeness). Existence is constructive. Choose any z M and for n N , define z n = n ( z ). (That is, induc- tively z n +1 = ( z n ).) Claim: ( z n ) n N is Cauchy. d ( z p ,z q ) d ( z p ,z p +1 ) + + d ( z q- 1 ,z q ) ( k p + + k q- 1 ) d ( z ,z 1 ) d ( z ,z 1 ) 1- k k p and we see that d ( z ,z 1 ) 1- k k p 0 as p . We say z n z (by completeness of M ). From z n +1 = ( z n ) and continu- ity of , it follows that z = ( z )....
View Full Document

Page1 / 3

Lecture15 - Lecture 15 Definition 1. Let M be a metric...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online