# Contraction Mapping Theorem
# Here we present an example of the contraction mapping theorem
# First we define an operator T acting on a function u(x)
> restart;
> T := u ->unapply(cos(2*Pi*x)+c*u(2*x),x);
# The degreee of contraction is governed by the constant c. We set it to some value < 1
> c:= .5;
# Lets apply this to the constant function g(x) = 1 several times.
> g:= unapply(sin(2*Pi*x),x);
> T(g);
> T(%);
> T(%);
> T(%);
# Now we'll define a sequence of functions f[j] = T(f[j-1]). We begin with some choice of f[0]
> c:=0.5;
> N:= 30;
> f[0] := x -> sin(4*Pi*x);
>  for i from 1 to N do
>    f[i] := T(f[i-1]):
> end do:
# Here is one of the functions in the sequence
> f[9](x);
# Now we plot some of the functions.
# 
> plot([f[2](x),f[3](x),f[5](x),f[8](x)],x=0..1, color=[blue,red,green,magenta,aquamarine]);
> plot(f[30](x),x=0..1, color=[blue], thickness=1, axesfont=[Times, Roman, 12], labelfont=[Times,Italic,14]);
> plot(f[15](x)-f[14](x),x=0..1);
> plot(f[30](x)-f[29](x),x=0..1);
> 
> 
#