Linear Algebra, Theory And Applications, 2012a

436 APPLICATIONS TO DIFFERENTIAL EQUATIONS
It follows from Lemma C.7.1 that if |x (t, a)| is bounded by δ as asserted, then it must be
thecasethata + + ∫ ∞
Φ
0 + (−s) g + (x (s, a)) ds = 0. Consequently, it must be the case that
(
a−
x (t) =Φ(t)
0
)
+
( ∫ t
0 Φ − (t − s) g − (x (s, a)) ds
− ∫ ∞
Φ
t + (t − s) g + (x (s, a)) ds
)
(3.39)
Letting t → 0, this requires that for a solution to the initial value problem to exist and also
satisfy |x (t)| 0itmustbethecasethat
(
)
a
x (0) =
−
− ∫ ∞
Φ
0 + (−s) g + (x (s, a)) ds
where x (t, a) is the solution of
(
)
x ′ a
= Ax + g (x) , x (0) =
−
− ∫ ∞
Φ
0 + (−s) g + (x (s, a)) ds
This is because in (3.39), if x is bounded by δ then the reverse steps show x is a solution of
the above differential equation and initial condition.
It follows if I can show that for all a − sufficiently small and a =(a − , 0) T , there exists a
solution to (3.39) x (s, a) on(0, ∞) for which |x (s, a)| 0 such that if |x| , |y| ≤δ,
then
|g (x) − g (y)|