Lecture Notes in Differential Equations - Bruce E. Shapiro

100 LESSON 12. EXISTENCE OF SOLUTIONS*
Results from Calculus and Analysis. We will need to use several results
from Calculus in this section. These are summarized here for review.
• Fundamental Theorem of Calculus.
1.
2.
d
dt
∫ b
a
∫ t
a
f(s)ds = f(t)
d
f(s)ds = f(b) − f(a)
ds
• Boundedness Theorem. Suppose |f| < M and a < b. Then
1.
2.
∫ b
a
∫ b
a
f(t)dt ≤ M(b − a)
f(t)dt ≤
∣
∫ b
a
∫ b
f(t)dt
∣ ≤ |f(t)|dt
a
• Mean Value Theorem. If f(t) is differentiable on [a, b] then there
is some number c ∈ [a, b] such that f(b) − f(a) = f ′ (c)(b − a).
• Pointwise Convergence. Let A, B ⊂ R, and suppose that f k (t) :
A → B for k = 0, 1, 2, ... Then the sequence of functions f k ,k =
0, 1, 2, ... is said to converge pointwise to f(t) if for every t ∈ A,
lim
k→∞ f k(t) = f(t), and we write this as f k (t) → f(t).
• Uniform Convergence. The sequence of functions f k ,k = 0, 1, 2, ...
is said to converge uniformly to f(t) if for every ε > 0 there exists an
integer N such that for every k > N, |f k (t) − f(t)| < ε for all t ∈ A.
Furthermore, If f k (t) is continuous andf k (t) → f(t) uniformly, then
1. f(t) is continuous.
∫ b
2. lim
k→∞
a f k(t)dt = ∫ b
a
lim f k(t)dt = ∫ b
k→∞
a f(t)dt
• Pointwise Convergence of a Series. The series ∑ ∞
k=0 f k(t) is
said to converge pointwise to s(t) if the sequence of partial sums
s n (t) = ∑ n
∑ k=0 f k(t) converges pointwise to s(t), and we write this as
∞
k=0 f k(t) = s(t).
• Uniform convergence of a Series. The series ∑ ∞
k=0 f k(t) is said
to converge uniformly to s(t) if the sequence of partial sums s n (t) =
∑ n
k=0 f k(t) converges uniformly to s(t).