Solution Found!
Let g C1[a, b] and p be in (a, b) with g( p) = p and
Chapter 2, Problem 24(choose chapter or problem)
Let g C1[a, b] and p be in (a, b) with g( p) = p and |g ( p)| > 1. Show that there exists a > 0 such that if 0 < |p0 p| < , then |p0 p| < |p1 p| . Thus, no matter how close the initial approximation p0 is to p, the next iterate p1 is farther away, so the fixed-point iteration does not converge if p0 = p.
Questions & Answers
QUESTION:
Let g C1[a, b] and p be in (a, b) with g( p) = p and |g ( p)| > 1. Show that there exists a > 0 such that if 0 < |p0 p| < , then |p0 p| < |p1 p| . Thus, no matter how close the initial approximation p0 is to p, the next iterate p1 is farther away, so the fixed-point iteration does not converge if p0 = p.
ANSWER:Step 1 of 3
First, prove that 
If 
Therefore, for every 
Simplify further.
