The Computable Differential Equation Lecture ... - Bruce E. Shapiro

136 CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS
Case 2: τ 1 = 0: For t 0 ≤ t ≤ t 0 + τ 2 ,
y ′ (t) = f(t, g(t), g(t − τ 2 ), . . . , g(t − τ m )) (7.7)
= f(t, y(t), g(t − τ 2 ), . . . , g(t − τ m )) (7.8)
y(0) = g(0) (7.9)
By the fundamental theorem, If f is Lipshitz in y then at most one solution
exists.
Lemma 7.1 (Gronwall Inequality). Suppose that f, g : [a, b] ↦→ R are continuous,
non-negative functions such that
f(t) ≤ K +
∫ t
for some non-negative constant K. Then
(∫ t
f(t) ≤ Kexp
Proof. Define the functions
Then
The assumption 7.10 becomes
F (t) = K +
G(t) =
∫ t
hence by differentiating 7.12 we obtain
a
∫ t
Multiplying both sides through by exp (−G(t)),
a
a
f(x)g(x)dx (7.10)
a
)
g(x)dx
(7.11)
f(x)g(x)dx (7.12)
g(x)dx (7.13)
F (a) = K (7.14)
G(a) = 0 (7.15)
G ′ (t) = g(t) (7.16)
f(t) ≤ F (t) (7.17)
F ′ (t) = f(t)g(t) ≤ F (t)g(t) (7.18)
F ′ (t)e −G(t) ≤ F (t)g(t)e −G(t) (7.19)
Differentiating F (t)e −G(t) and applying equations 7.16 and 7.18 gives
d
dt F (t)e−G(t) = F ′ (t)e −G(t) − F (t)G ′ (t)e −G(t) (7.20)
= ( F ′ (t) − F (t)G ′ (t) ) e −G(t) (7.21)
= ( F ′ (t) − F (t)g(t)) ) e −G(t) ≤ 0 (7.22)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007