Lecture Notes in Differential Equations - Bruce E. Shapiro

106 LESSON 12. EXISTENCE OF SOLUTIONS*
which is what we wanted to prove (equation (12.39)).
Lemma 12.7. The series S(t) converges.
Proof. By Lemma (12.6)
∞∑
∞∑
|φ n (t) − φ(t)| ≤ K n−1 M |t − t 0| n
n!
n=1
n=1
= M ∞∑ |K(t − t 0 )| n
K n!
n=1
(
= M ∞
)
∑ |K(t − t 0 )| n
− 1
K
n!
n=0
= M (
)
e K|t−t0| − 1
K
(12.47)
(12.48)
(12.49)
(12.50)
≤ M K eK|t−t0| (12.51)
Since each term in the series for S is absolutely bounded by the corresponding
term in the power series for the exponential, the series S converges
absolutely, hence it converges.
Lemma 12.8. The sequence φ 0 , φ 1 , φ 2 , . . . converges to some limit φ(t).
Proof. Since the series for S(t) converges, then by Lemma (12.5), the sequence
φ 0 , φ 1 , . . . converges to some function φ(t).
Lemma 12.9. φ(t) is defined and continous on R.
Proof. For any s, t,
|φ n (s) − φ n (t)| =
∣
=
∣
≤
∣
Hence taking the limit,
∫ s
t
∫
0
s
t
∫ s
t
∫ t
f(x, φ n (x))dx − f(x, φ n (x))dx
∣ (12.52)
t 0 f(x, φ n (s))dx
∣ (12.53)
Mdx
∣ (12.54)
= M|s − t| (12.55)
lim |φ n(s) − φ n (t)| ≤ M|s − t| (12.56)
n→∞