Lecture Notes in Differential Equations - Bruce E. Shapiro

222 LESSON 26. GENERAL EXISTENCE THEORY*
Proof. Use induction. For n = 1, the formula gives
which is true.
‖T y − y‖ ≤ 1 − K ‖T y − y‖ = ‖T y − y‖ (26.24)
1 − K
For n > 1 suppose that equation 26.23 holds. Then
‖T n+1 y − y‖ = ‖T n+1 y − T n y + T n y − y‖ (26.25)
≤ ‖T n+1 y − T n y‖ + ‖T n y − y‖ (triangle ineqality) (26.26)
≤ ‖T n+1 y − T n y‖ + 1 − Kn
‖T y − y‖ (by (26.23)) (26.27)
1 − K
= ‖T n T y − T n y‖ + 1 − Kn
‖T y − y‖ (26.28)
1 − K
≤ K n ‖T y − y‖ + 1 − Kn
‖T y − y‖ (because T is a contraction)
1 − K
(26.29)
= (1 − K)Kn + (1 − K n )
‖T y − y‖ (26.30)
1 − K
= 1 − Kn+1
‖T y − y‖ (26.31)
1 − K
which proves the conjecture for n + 1.
Definition 26.9. Let V be a vector space let T be an operator on V. Then
we say y is a fixed point of T if T y = y.
Note that in the vector space of functions, since the vectors are functions,
the fixed point is a function.
Theorem 26.10. Contraction Mapping Theorem 1 Let T be a contraction
on a normed vector space V . Then T has a unique fixed point
u ∈ V such that T u = u. Furthermore, any sequence of vectors v 1 , v 2 , . . .
defined by v k = T v k−1 converges to the unique fixed point T u = u. We
denote this by v k → u.
Proof. 2 Let ɛ > 0 be given and let v ∈ V.
1 The contraction mapping theorem is sometimes called the Banach Fixed Point Theorem.
2 The proof follows “Proof of Banach Fixed Point Theorem,” Encyclopedia of Mathematics
(Volume 2, 54A20:2034), PlanetMath.org.