Spectral Theory in Hilbert Space

C. LINEAR FIRST ORDER SYSTEMS 131
subinterval of I. Therefore u is continuous, and
∞
∞
x
u(x) = uk(x) = H(x) + Auk
k=0
k=0
c
= H(x) +
x
c
A
∞
uk = H(x) +
k=0
x
c
Au .
Thus (C.2) has a solution. To prove the uniqueness, we need the following
lemma.
Lemma C.3 (Gronwall). Suppose f ∈ C(I) is real-valued, h is a
non-negative constant, and g is a locally integrable and non-negative
function. Suppose that 0 ≤ f(x) ≤ h + | x
gf| for x ∈ I. Then
c
f(x) ≤ h exp(| x
g|) for x ∈ I.
c
The uniqueness of the solution of (C.2) follows directly from this.
For suppose v is the difference of two solutions. Then v(x) = x
c Av,
so setting f = |v| and g = |A| we obtain 0 ≤ f(x) ≤ | x
gf|. Hence
c
f ≡ 0 by Lemma B.3, and thus v ≡ 0, so that (C.2) has at most one
solution.
It remains to prove the lemma.
Proof of Lemma C.3. We will prove the lemma for c < x, leaving
the other case as an exercise for the reader. Set F (x) = h + x
c gf.
Then f ≤ F and F ′ = gf so that F ′ ≤ gF . Multiplying by the integrating
factor exp(− x
d
g) we get c dx (F (x) exp(− x
g)) ≤ 0 so that
c
F (x) exp(− x
c g) is non-increasing. Thus F (x) exp(− x
g) ≤ F (c) = h
c
for x ≥ c. We obtain f(x) ≤ F (x) ≤ h exp( x
g), which was to be
c
proved.
Proof of Theorem C.2. It is clear by their definitions that the
functions uk in the proof of Theorem C.1 are analytic in Ω as functions
of λ, locally uniformly in x (this is a trivial induction). But the solution
u is the locally uniform limit, in x, λ, of the partial sums j k=1 uk. Since
uniform limits of analytic functions are analytic, we are done.