
Topology Topology Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 2nd, 2016, 06:47 AM  #1 
Newbie Joined: Dec 2016 From: Montreal Posts: 4 Thanks: 0  Banach's Contraction Mapping Theorem
As I understand the theorem, we Let (X,d) be a metric space and let f : X →X be a mapping. Then, a point x ∈ X is a fixed point of f if x = f (x), and f is called a contraction if there exists a fixed constant h < 1 such that d( f (x), f (y) ) ≤ hd(x,y), for all x,y ∈ X. I'm a little confused on how to approach problems using Banach's Contraction theorem though. For example, if f(x) = 3x − 4, g(x) = (1/2)sin x and q(x) = e^(x^2) d( f (x), f (y) ) = (3x4)  (3y4) ≤ hd(x,y) =  3x  3y  ≤ hd(x,y) = 3 x  y  ≤ hd(x,y) Would this be the proper approach f(x) = 3x − 4? 
December 2nd, 2016, 03:54 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,214 Thanks: 492 
Proper approach for what? Since h < 1, f(x) does not satisfy.

December 3rd, 2016, 08:29 AM  #3  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,353 Thanks: 591  Quote:
Then it would be obvious that d(f(x), f(y))= 3x y= 3 d(x, y). The "h" is at least 3 so this is not a "contraction map".  

Tags 
banach, contraction, mapping, theorem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Minkowski Functional used in Proof Hahn Banach Theorem  PeterPan  Real Analysis  3  March 12th, 2013 09:19 AM 
Applying the Contraction Mapping Principle  guynamedluis  Real Analysis  1  November 19th, 2011 01:23 PM 
Contraction Mapping and Diagonally Dominant Jacobian  needmath  Applied Math  0  August 14th, 2011 03:36 PM 
Contraction Mapping  six  Real Analysis  1  October 31st, 2010 07:19 PM 
contraction mapping theorem  aptx4869  Real Analysis  1  April 20th, 2007 10:10 PM 