Let
X be a
complete metric space, and let
f: X->X be a
contracting function (we shall be using the
definitions and
symbols of my
writeup in that
node, so you might want to read it). Our aim is to show that
f has a
unique fixed point. The
proof gives a taste of
constructive methods in complete metric spaces; non-constructive methods also abound, but have a vastly different character. In particular, we shall give a method which
converges to the fixed point of
f; this method is in fact used, e.g. to draw
fractals or solve
differential equations.
Uniqueness is easy: suppose x,y are 2 fixed points of f. Then
d(x,y) =
d(f(x),f(y)) <=
c d(x,y)
with
c<1, so
d(
x,y)=0 and
x=y --
f has at most one fixed point.
For existence, first note that any contracting function must be continuous. Pick any x0 in X, and define xn=f(xn-1).
If we can prove that this sequence converges, by applying f and using its continuity we see that
f(lim xn) =
lim f(xn) =
lim xn+1 =
lim xn,
so the limit is a (the) fixed point of
f
Write d=d(x0,x1).
Then utilising the definition of a contracting function n times, d(xn+1,xn) <= d cn.
But this means that if m>n then
d(xm,xn) < d cn / (1-c), so our sequence is a Cauchy sequence. Thus it has a limit, which we saw is the fixed point of f.
Note that the above is a procedure for converging to the fixed point! Furthermore, we see that the rate of convergence is respectable -- a constant number of digits for each iteration.