Palais’ Proof of Banach’s Fixed Point Theorem
The following short proof of Banach’s Fixed Point Theorem was given by Richard S. Palais
in 2007; see [1].
Theorem. Suppose that (X, d) is a complete metric space. Then every contraction F
on X has a uniquely determined fixed point.
Proof. Let α denote the contraction constant of F . Then, according to the triangle
inequality,
d(x1 , x2 ) ≤ d(x1 , F (x1 )) + d(F (x1 ), F (x2 )) + d(x2 , F (x2 ))
≤ d(x1 , F (x1 )) + αd(x1 , x2 ) + d(x2 , F (x2 )),
which means that
d(x1 , x2 ) ≤
d(x1 , F (x1 )) + d(x2 , F (x2 ))
1−α
(1)
for all points x1 , x2 ∈ X. This inequality immediately implies that F cannot have more
than one fixed point.
Let F n denote the composition of F with itself n times. It is easy to show that F n is a
contraction with contraction constant αn . If we now apply (1) to the points x1 = F m (x0 )
and x2 = F n (x0 ), where x0 ∈ X is arbitrary, we obtain that
d(F m (x0 ), F m (F (x0 ))) + d(F n (x0 ), F n (F (x0 )))
1−α
αm + αn
d(x0 , F (x0 )).
≤
1−α
d(F m (x0 ), F n (x0 )) ≤
Since 0 ≤ α < 1, this implies that the sequence (F n (x0 ))∞
n=1 is a Cauchy sequence and
therefore that F n (x0 ) → x for some x ∈ X. Finally, because F is continuous,
F (x) = F ( lim F n (x0 )) = lim F n+1 (x0 ) = x,
n→∞
n→∞
so x is a fixed point of F .
References
[1] Richard S. Palais, A simple proof of the Banach contraction principle, J. fixed point
theory appl. 2 (2007), 221–223.