Banach Contraction Principle
Let (X,d) be a complete metric
space, then each contraction map f : X → X has a unique fixed point.
Let h be a contraction constant of the mapping f . We will explicitly construct
a sequence converging to the fixed point. Let x0 be an arbitrary but fixed element
in X. Define a sequence of iterates {xn} in X by
xn = f(xn−1) (= f n(x0)), for all n ≥ 1.
Since f is a contraction, we have
d(xn, xn+1) = d(f(xn−1), f(xn)) ≤ hd(xn−1, xn), for any n ≥ 1.
Thus, we obtain
d(xn, xn+1) ≤ hnd(x0, x1), for all n ≥ 1.
Hence, for any m > n, we have
d(xn, xm) ≤
hn +hn+1 +···+hm−1
d(x0, x1) ≤
hn
1−h
d(x0, x1).
We deduce that {xn} is Cauchy sequence in a complete space X. Let xn → p ∈ X.
Now using the
continuity of the map f , we get
p = lim
n→∞ xn = lim
n→∞ f(xn−1) = f(p).
Finally, to show f has at most one fixed point in X, let p and q be fixed points of f .
Then,
d(p,q) = d(f(p), f(q)) ≤ hd(p,q).
Since h < 1, we must have p = q
The proof of the BCP yields the following useful information about the rate of
convergence towards the fixed point.
Comments