Answer to Question #5032 in Quantitative Methods for Michael

Question

Given an initial guess x(0) consider the general fixed point iteration x(n+1)=g(x(n))

Show that the fixed point iteration with g defined by g(x)= (5-e^x-2x)/4 cannot be guaranteed to converge to x(*), when x(0) exists in [0,1]

With f(x) defined as f(x) = e^x+6x-5 show that the fixed point iteration
x(n+1) = (5-e^(x(n)))/6, n=0,1,... must converge to x(*) with x(0) exist in [0,1]

I think that probably the solution is based on the contraction theorem!