Lecture Notes in Differential Equations - Bruce E. Shapiro

101
• Interchangeability of Limit and Summation. If ∑ ∞
∑ k=0 f k(t)
∞
converges uniformly to s(t): lim
t→a
k=0 f k(t) = ∑ ∞
k=0 lim f k(t) =
∑ t→a ∞
k=0
f(a) = s(a). In other words, the limit of the sum is the sum of
the limits. 5
Approach The method we will use to prove 12.1 is really a formalization
of the method we used in example 11.1:
1. Since f is continuous in the box R it is defined at every point in R,
hence it must be bounded by some number M. By theorem (12.3), f
is Lipshitz (Definition (12.2)). In Lemma (12.4) use this observation
to show that the Picard iterates exist in R and satisfy
|φ k (t) − y 0 | ≤ M|t − t 0 | (12.5)
2. Defining s n (t) = [φ n (t) − φ n−1 (t)], we will show in Lemma (12.5)
that the sequence φ 0 , φ 1 , φ 2 , . . . converges if and only if the series
also converges.
S(t) =
∞∑
s n (t) (12.6)
n=1
3. Use the Lipshitz condition to prove in Lemma (12.6) that
|s n (t)| = |φ n − φ n−1 | ≤ K n−1 M (t − t 0) n
n!
(12.7)
4. In Lemma (12.7), use equation (12.7) to show that S, defined by
(12.6), converges by comparing it with the Taylor series for an exponential,
and hence, in Lemma (12.8), that the sequence φ 0 , φ 1 , φ 2 , . . .
also converges to some function φ = lim
n→∞ φ n.
5. In Lemma (12.9), show that φ = lim
n→∞ φ n is defined and continuous
on the rectangle R.
6. Show that φ(t) satisfies the initial value problem (12.3),
Assumptions. We will make the following assumptions for the rest of this
section.
1. R is a rectangle of width 2a and height 2b, centered at (t 0 , y 0 ). Equations
(12.1) and (12.2) follow as a consequence.