Spectral Theory in Hilbert Space

130 C. LINEAR FIRST ORDER SYSTEMS
If we integrate the differential equation in (C.1) from c to x, using
the initial data, we get the integral equation
(C.2) u(x) = H(x) +
x
c
Au,
where H(x) = C+ x
B. Conversely, if u is continuous and solves (C.2),
c
then u has initial data H(c) = C and is locally absolutely continuous
(being an integral function). Differentiation gives u ′ = Au + B, so that
the initial value problem is equivalent to the integral equation (C.2).
In the case of Theorem 13.1, we put A = J −1 (Q − λW ) and B = 0
to get an equation of the form (C.1). We therefore need to show the
following theorems.
Theorem C.1. Suppose A has locally integrable, and H locally absolutely
continuous, elements. Then the integral equation (C.2) has a
unique, locally absolutely continuous solution.
Theorem C.2. Suppose that A depends analytically on a parameter
λ, in the sense that there is a matrix A ′ (x, λ) which is locally integrable
with respect to x, and such that
1 | J h (A(x, λ+h)−A(x, λ))−A′ (x, λ)| →
0 as h → 0, for all compact subintervals J of I, and all λ in some open
set Ω ⊂ C. Then the solution u(x, λ) of (C.2) is analytic for λ ∈ Ω,
locally uniformly in x.
Proof of Theorem C.1. We will find a series expansion for the
solution. To do this, we set u0 = H, and if uk is already defined, we set
uk+1(x) = x
c Auk. It is then clear that uk is defined for k = 0, 1, . . .
inductively, and all uk are (absolutely) continuous. I claim that
sup|uk|
≤ sup|H|
[c,x] [c,x]
1
k!
x c
k |A|
for k = 0, 1, . . . ,
for x > c, and a similar inequality with c and x interchanged for x < c.
Here |·| denotes a norm on n-vectors, and also the corresponding subordinate
matrix-norm (so that |Au| ≤ |A||u|). Indeed, the inequality
is trivial for k = 0, and supposing it valid for k, we obtain
|uk+1(x)| ≤
x
c
|A||uk| ≤ 1
k! sup
|H|
[c,x]
c
x
t k |A(t)| |A| dt
=
1
c
(k + 1)! sup
[c,x]
x |H|
c
k+1 |A| ,
for c < x, and a similar inequality for x < c. It follows that the series
u = ∞
k=0 uk is absolutely and uniformly convergent on any compact