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

Chapter 7
Delay Differential Equations
7.1 Basic Theory
Definition 7.1 (Delay Differential Equation). Let
0 ≤ τ 1 < τ 2 < · · · < τ m (7.1)
be a set of m positive numbers representing the delays, the largest delay being denoted
by τ m . If y appears explicitly without any delay then τ 1 = 0. Then we may define
the scalar delay differential equation (DDE) as
y ′ (t) = f(t, y(t − τ 1 ), y(t − τ 2 ), . . . , y(t − τ m )) (7.2)
y(t) = g(t), t 0 − τ m ≤ t ≤ t 0 (7.3)
Theorem 7.1. Let D ∈ [t 0 , t 0 + β) × R m be convex; suppose that f : D ↦→ R is
continuous on D; and that g(t) : [t 0 − τ m ] ↦→ D. Then if r 1 ≠ 0 the DDE (equations
7.2, 7.3) has a unique solution on [t 0 , t 0 + α), where 0 < α ≤ β. Furthermore, if
α < β then y(t) approaches the boundary of D as t → α. If τ 1 = 0 then the DDE
has at most one solution.
Proof. The proof is constructive. We consider two cases:
Case 1: τ 1 ≥ 0: For t 0 ≤ t ≤ τ 1 + t 0 , each y(t − τ i ) = g(t − τ i ), hence we have an
explicit expression for f(t), viz.,
y ′ (t) = f(t, g(t − τ 1 ), g(t − τ 2 ), . . . , g(t − τ m )) (7.4)
y(0) = g(0) (7.5)
Hence we can integrate on (0, t) where t < τ 1 ,
y(t) =
∫ t
t 0
y ′ (t)dt = g(0) +
∫ t
t 0
f(x, g(x − τ 1 ), . . . , g(x − τ m ))dx (7.6)
This uniquely determines y(t), t 0 ≤ t ≤ τ 1 . So we can repeat the process on
[t 0 + τ 1 , t 0 + min(β, 2τ 1 )]. We can repeat this process until we reach β. This
process is called the method of steps.Thus by construction, a solution exists.
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