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Method of steps gives a unique retarded-delay solution

Statement

Let $d \geq 1$ , $T, τ > 0$ , and let $ϕ \in C_{τ}$ . Suppose $f : U \times R^{d} \times R^{d} \to R^{d}$ is continuous on an open interval $U \supseteq [0, T]$ and satisfies the global state bound in Retarded delay equations and their history space with constant $L \geq 0$ . Then the retarded-delay initial-value problem has exactly one solution on $[- τ, T]$ . The construction is the method of steps: on each interval $[k τ, (k + 1) τ] \cap [0, T]$ , the delayed path is already known, so the current equation is an ordinary differential equation.

Facts & Assumptions

Given: The delay, continuous history, vector field, and global Lipschitz constant in the Statement.

[F1]

A retarded-delay solution equals $ϕ$ on $[- τ, 0]$ , satisfies $y (0) = ϕ (0)$ , and obeys $y^{'} (t) = f (t, y (t), y (t - τ))$ on $[0, T]$ (Retarded delay equations and their history space).

[F2]

A continuous vector field locally Lipschitz in its state variable has a unique local IVP solution; on a compact cylinder, a bound $h M \leq r$ and $L h < 1$ give a unique solution staying in that cylinder (Picard-Lindelöf local existence and uniqueness for first-order systems).

[F3]

For an integrable vector-valued function, $∥ \int_{a}^{b} g ∥_{2} \leq \int_{a}^{b} ∥ g ∥_{2}$ when $a \leq b$ (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

[F4]

A continuous real-valued function on a nonempty compact metric space is bounded (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F5]

The Euclidean norm satisfies the triangle inequality (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F6]

A differentiable vector path with integrable derivative has increments equal to integrals of that derivative (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[L1]

If $u$ obeys a scalar retarded integral inequality with history bound $H$ and coefficient $L$ , then its running maximum is bounded by $(H + \int_{0}^{t} q) e^{2 L t}$ (A Gronwall bound for retarded-delay inequalities).

Proof

1.1givenF1F3F5F6algebra

Put $q (t) = ∥ f (t, 0, 0) ∥_{2}$ and $H = ∥ ϕ ∥_{\infty}$ . For any partial method-of-steps solution, [F6] supplies its integral equation; the norm integral bound [F3], the triangle inequality [F5], and the Lipschitz bound then give $∥ y (t) ∥_{2} \leq H + \int_{0}^{t} (q (s) + L (∥ y (s) ∥_{2} + ∥ y (s - τ) ∥_{2})) d s$ .

2.1step 1.1L1

By [L1], every partial solution constructed on $[0, t]$ remains in the fixed ball of radius $R = (H + \int_{0}^{T} q (s) d s) e^{2 L T}$ . The delayed values used on the next step are therefore bounded by $R_{*} : = \max {H, R}$ .

3.1F1F2F4F5step 2.1construct

On $I_{k} = [k τ, \min {(k + 1) τ, T}]$ , define the known delayed input $z_{k} (t) = y (t - τ)$ from the history and already constructed pieces. Extend $z_{k}$ constantly a short distance past each endpoint. Then $F_{k} (t, x) = f (t, x, z_{k} (t))$ is continuous and $L$ -Lipschitz in $x$ . Choose a compact time neighborhood $U_{0} \subset U$ of $[0, T]$ and put $Q_{*} : = \sup_{t \in U_{0}} ∥ f (t, 0, 0) ∥_{2}$ ; this is finite by [F4] applied to the continuous norm of $f (t, 0, 0)$ . The extended delayed values remain bounded by $R_{*} : = \max {H, R}$ . The triangle inequality [F5] and state Lipschitz bound show that on $∥ x ∥_{2} \leq R + 1$ the vector field is bounded by $M_{*} : = Q_{*} + L (R + 1 + R_{*})$ . Choose a common $δ > 0$ inside the time-domain margin with $L δ < 1$ and $δ M_{*} \leq 1$ . [F2] gives a unique local solution from each current endpoint, and finitely many subintervals of length at most $δ$ cover $I_{k}$ ; the bound in step 2.1 keeps every endpoint in the required ball.

4.1F2step 3.1induction∎

Starting with $k = 0$ , the delayed input is $ϕ (t - τ)$ and step 3.1 constructs a unique solution on $I_{0}$ . If a unique solution has been constructed through $k τ$ , its delayed path on $I_{k}$ is fixed by that past; step 3.1 constructs exactly one continuation there. Induction over the finite number $⌈ T / τ ⌉$ of intervals gives existence on $[0, T]$ . Any two solutions coincide on $I_{0}$ by [F2], and the same induction makes them coincide on each later $I_{k}$ , proving uniqueness.

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Sources

M. Zennaro, The Numerical Solution of Delay Differential Equations, Dobbiaco 2006 summer-school lecture notes (standard reference, not scraped)
Nicola Guglielmi and Ernst Hairer, Solving Delay Differential Equations, Chapter 1 (standard reference, not scraped)