Lecture Notes in Differential Equations - Bruce E. Shapiro

104 LESSON 12. EXISTENCE OF SOLUTIONS*
Proof. For k = 0, equation says
|φ 0 (t) − y 0 | ≤ M|t − t 0 | (12.24)
Since φ 0 (t) = y 0 , the left hand side is zero, so the right hands side, being
an absolute value, is ≥ 0.
For k > 0, prove the Lemma inductively. Assume that (12.23) is true and
use this prove that
|φ k+1 (t) − y 0 | ≤ M|t − t 0 | (12.25)
From the definition of φ k (see equation (12.10)),
which proves equation (12.25).
∫ t
φ k+1 (t) = y 0 + f(s, φ k (s))ds (12.26)
t 0
∫ t
|φ k+1 (t) − y 0 | =
∣ f(s, φ k (s))ds
∣ (12.27)
t 0
≤
∫ t
t 0
|f(s, φ k (s))|ds (12.28)
∫ t
≤ Mds (12.29)
t 0
= M|t − t 0 | (12.30)
Lemma 12.5. The sequence φ 0 , φ 1 , φ 2 , . . . converges if and only if the
series
∞∑
S(t) = s n (t) (12.31)
also converges, where
Proof.
k=1
s n (t) = [φ n (t) − φ n−1 (t)] (12.32)
φ n (t) = φ 0 + (φ 1 − φ 0 ) + (φ 2 − φ 1 ) + · · · + (φ n − φ n−1 ) (12.33)
n∑
= φ 0 + (φ k − φ k−1 ) (12.34)
= φ 0 +
k=1
n∑
s n (t) (12.35)
k=1