Given that C is a natural state space for (4.2), we now consider what sort of cones in C will yield useful order relations. The most natural such cones are those induced by cones in IRn. Let K be a cone in IRn with nonempty interior and K∗ denote the dual cone. All inequalities hereafter are assumed to be those induced on IRn by K. The cone K induces a cone CK in the Banach space C defined by
CK = {φ ∈ C : φ(θ) ≥ 0, −r ≤ θ ≤ 0}.
It has nonempty interior in C given by IntCK = {φ ∈ CK : φ(θ) 0, θ ∈ [−r, 0]}. The usual notation ≤, <, will be used for the various order relations on C generated by CK. In particular, φ ≤ ψ holds in C if and only if φ(s) ≤ ψ(s) holds in IRn for every s ∈ [−r, 0]. The same notation will also be used for the various order relations on IRn
but hopefully the context will alert the reader to the appropriate meaning. Cones in C that are not induced by a cone in IRn have also proved useful. See Smith and Thieme [200, 201, 193].
An immediate aim is to identify sufficient conditions on f for the semiflow Φ to be a monotone semiflow. The following condition should seem natural since it generalizes the condition (QM) for ordinary differential equations in the previous section. We refer to it here as the quasimonotone condition, (QMD) for short. “D” in the notation, standing for delay, is used so as not to confuse the reader with (QM) of the previous section. We follow this pattern in several definitions in this section.
(QMD) φ, ψ ∈ D, φ ≤ ψ and η(φ(0)) = η(ψ(0)) for some η ∈ K∗, implies η(f (φ)) ≤ η(f (ψ)).
For the special case K = IRn+, (QMD) becomes:
φ, ψ ∈ D, φ ≤ ψ and φi(0) = ψi(0), implies fi(φ)) ≤ fi(ψ).
As in section 3, it is convenient to consider the nonautonomous equation
x0(t) = f (t, xt) (4.4)
where f : Ω → IRn is continuous on Ω, an open subset of IR × C. Given (t0, φ) ∈ Ω, we write x(t, t0, φ, f ) and xt(t0, φ, f ) for the maximally defined solution and state of the system at time t satisfying xt0 = φ. We assume this solution is unique, which will be the case if f is Lipschitz in its second argument on each compact subset of Ω. We drop the last argument f from x(t, t0, φ, f ) when no confusion over which f is being considered will result.
f : Ω → IRn is said to satisfy (QMD) if f (t, •) satisfies (QMD) on Ωt ≡ {φ ∈ C : (t, φ) ∈ Ω} for each t.
The next Theorem not only establishes the desired monotonicity of the semiflow Φ but also allows comparisons of solutions between related functional differential equations.
It generalizes Theorem 3.2 of Chapter 3 to functional differential equations and is a generalization of Proposition 1.1 of [190] and Theorem 5.1.1 of [193] where K = IRn+ is considered. The quasimonotone condition for delay differential equations seems first to have appeared in the work of Kunisch and Schappacher [108] , Martin [125], and Ohta [148].
Theorem 4.1 Let f, g : Ω → IRn be continuous, Lipschitz on each compact subset of Ω, and assume that either f or g satisfies (QMD). Assume also that f (t, φ) ≤ g(t, φ) for all (t, φ) ∈ Ω. Then
φ, ψ ∈ Ωt0, φ ≤ ψ, t ≥ t0, =⇒ x(t, t0, φ, f ) ≤ x(t, t0, ψ, g) for all t for which both are defined.
Proof: Assume that f satisfies (QMD), a similar argument holds if g satisfies (QMD).
Let e ∈ IRn satisfy e 0, g(t, φ) := g(t, φ) + e and ψ := ψ + ˆe, for ≥ 0. If x(t, t0, ψ, g) is defined on [t0 − r, t1] for some t1 > t0, then x(t, t0, ψ, g) is also defined on this same interval for all sufficiently small positive and
x(t, t0, ψ, g) → x(t, t0, ψ, g), → 0,
for t ∈ [t0, t1] by Hale and Lunel [59], Theorem 2.2.2. We will show that x(t, t0, φ, f ) x(t, t0, ψ, g) on [t0 − r, t1] for small positive . The result will then follow by letting
→ 0 . If the assertion above were false for some , then applying the remark be-low Proposition 3.1, there exists s ∈ (t0, t1] such that x(t, t0, φ, f ) x(t, t0, ψ, g) for t0 ≤ t < s and η(x(s, t0, φ, f )) = η(x(s, t0, ψ, g)) for some nontrivial η ∈ K∗. As η(x(t, t0, φ, f )) < η(x(t, t0, ψ, g)) for t0 ≤ t < s, by Proposition 3.1, we conclude that
d
dt|t=sη(x(s, t0, φ, f )) ≥ dtd|t=sη(x(s, t0, ψ, g)). But d
dt|t=sη(x(s, t0, ψ, g)) = η(g(s, xs(t0, ψ, g))) + η(e)
> η(f (s, xs(t0, ψ, g)))
≥ η(f (s, xs(t0, φ, f )))
= d
dt|t=sη(x(s, t0, φ, f )),
where the last inequality follows from (QMD). This contradiction implies that no such s can exist, proving the assertion.
In the case of the autonomous system (4.2), taking f = g in Theorem 1.1 implies that xt(φ) ≤ xt(ψ) for t ≥ 0 such that both solutions are defined. In other words, the semiflow Φ defined by (4.3) is monotone. In contrast to Theorem 3.2 of the previous section, if φ < ψ we cannot conclude that x(t, φ) < x(t, ψ) or xt(φ) < xt(ψ) since Φt
is not generally one-to-one. A simple example is provided by the scalar equation (4.2) with r = 1 and f (φ) := max φ, which satisfies (QMD). Let φ < ψ be strictly increasing on [−1, −1/2], φ(−1) = ψ(−1) = 0, φ(−1/2) = ψ(−1/2) = 1, and φ(θ) = ψ(θ) = −2θ for −1/2 < θ ≤ 0. It is easy to see that x(t, φ) = x(t, ψ) for t ≥ 0.
It is useful to have sufficient conditions for the positive invariance of K. By this we mean that t0 ∈ J and φ ≥ 0 implies x(t, t0, φ) ≥ 0 for all t ≥ t0 for which it is defined.
The following result provides the expected necessary and sufficient condition. The proof is similar to that of Theorem 4.1; the result is the delay analog of Proposition 3.3.
Theorem 4.2 Assume that J ×K ⊂ Ω where J is an open interval. Then K is positively invariant for (4.4) if and only if for all t ∈ J
(PD) φ ≥ 0, λ ∈ K∗ and λ(φ(0)) = 0 implies λ(f (t, φ)) ≥ 0 holds.
Let L : J → L(C, IRn) be continuous, where L(C, IRn) denotes the space of bounded linear operators from C to IRn, and consider the initial value problem for the linear nonautonomous functional differential equation
x0 = L(t)xt, xt0 = φ. (4.5)
Observing that (PD) and (QMD) are equivalent for linear systems, we have the following corollary.
Corollary 4.3 Let x(t, t0, φ) be the solution of (4.5). Then x(t, t0, φ) ≥ 0 for all t ≥ t0
and all φ ≥ 0 if and only if for each t ∈ J, (PD) holds for L(t).
As in the case of ordinary differential equations, a stronger condition than (PD) for linear systems is that for every t ∈ J, there exists α ∈ IR such that L(t)φ + αφ(0) ≥ 0 whenever φ ≥ 0.
It is useful to invoke the Riesz Representation Theorem [167] in order to identify L(t) with a matrix of signed Borel measures η(t) = (η(t)ij):
L(t)φ = Z 0
−r
dη(t)φ (4.6)
The Radon-Nikodym decomposition of ηij with respect to the Dirac measure δ with unit mass at 0 gives ηij(t) = aij(t)δ + ˜ηij(t) where aij is a scalar and ˜ηij(t) is mutually singular with respect to δ. In particular, the latter assigns zero mass to {0}. Therefore,
L(t)φ = A(t)φ(0) + ˜L(t)φ, ˜L(t)φ :=
Z 0
−r
d˜η(t)φ. (4.7)
Continuity of the map t → A(t) follows from continuity of t → L(t). The decomposition (4.7) leads to sharp conditions for (PD) to hold for L(t).
Proposition 4.4 (PD) holds for L(t) if and only if (a) A(t) satisfies (P) of Proposition 3.3, and (b) ˜L(t)φ ≥ 0 whenever φ ≥ 0.
Proof: If (a) and (b) hold, φ ≥ 0, λ ∈ K∗and λ(φ(0)) = 0 then λ(L(t)φ) = λ(A(t)φ(0))+
λ( ˜L(t)φ) ≥ 0 because each summand on the right is nonnegative.
Conversely, if (PD) holds for L(t), v ∈ ∂K, λ ∈ K∗, and λ(v) = 0, define φn(θ) = enθv on [−r, 0]. Then φn ≥ 0 and φn converges point-wise to zero, almost everywhere with respect to ˜η(t). By (PD),
λ(L(t)φn) = λ(A(t)v + ˜L(t)φn) ≥ 0.
Letting n → ∞, we get λ(A(t)v) ≥ 0 implying that (P) holds for A(t). Let φ ≥ 0 be given and define φn(θ) = [1 − enθ]φ(θ) on [−r, 0], n ≥ 1. φn converges point-wise to φχ, where χ is the indicator function of the set [−r, 0), and φχ = φ almost everywhere with respect to ˜η(t). If λ ∈ K∗, then λ(φn(0)) = 0 so applying (PD) we get 0 ≤ λ(L(t)φn) = ˜L(t)φn. Letting n → ∞ we get (b).
For the remainder of this section, we suppose that Ω = J × D where J is a nonempty open interval and D ⊂ C is open. Suppose that ∂f∂φ(t, ψ) exists and is continuous on J ×D to L(C, IRn). In that case, x(t, t0, φ) is continuously differentiable in its last argument and y(t, t0, χ) = ∂x∂φ(t, t0, φ)χ satisfies the variational equation
y0(t) = ∂f
∂φ(t, xt(t, φ))yt, yt0 = χ. (4.8) See Theorem 2.4.1 of Hale and Lunel [59]. We say that f (or (4.4)) is K-cooperative if for all (t, χ) ∈ J × D the function ψ → ∂f∂φ(t, χ)ψ satisfies (PD). By Corollary 4.3 applied to the variational equation we have the following analog of Theorem 3.5 for functional differential equations. The proof is essentially the same.
Theorem 4.5 Let ∂f∂φ(t, ψ) exist and be continuous on J × D. If (QMD) holds for (4.4), then f is K-cooperative. Conversely, if D is p-convex and f is K-cooperative, then (QMD) holds for f .
Consider the nonlinear system
x0(t) = g(x(t), x(t − r1), x(t − r2), · · · , x(t − rm)) (4.9) where g(x, y1, y2, · · · , ym) is continuously differentiable on IR(m+1)n and rj+1 > rj > 0.
Then ∂f
∂φ(ψ) = ∂g
∂x(x, Y )δ +X
k
∂g
∂yk(x, Y ))δ−rk (4.10) where δ−rk is the Dirac measure with unit mass at {−rk} and x = ψ(0), yk = ψ(−rk) and (x, Y ) := (x, y1, y2, · · · , ym). By Theorem 4.5, Corollary 4.3, and Proposition 4.4, (QMD) holds if and only if for each (x, Y ), ∂g∂x(x, Y ) satisfies condition (P) and ∂y∂gk(x, Y ) is K-positive. If K = IRn+, the condition becomes ∂x∂gi
j(x, Y ) ≥ 0, for i 6= j and
∂gi
∂ykj(x, Y ) ≥ 0 for all i, j, k; if, in addition, n = 1 then ∂y∂gk(x, Y ) ≥ 0 for all k suf-fices.