TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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Continuous dependence of ODE solutions on initial data and parameters

Statement

Let $F (t, x, λ)$ be continuous on an open time-state-parameter domain and locally state-Lipschitz with one constant on compact cylinders. Near fixed data $(t_{0}, x_{0}, λ_{0})$ , the solutions supplied by Picard-Lindelof exist on one common compact time interval and depend jointly and uniformly continuously there on the initial time, initial state, and parameter. Quantitatively, if the common time interval has length at most $H$ , $∥ F ∥_{2} \leq M$ , and $F$ has state-Lipschitz constant $L$ , then solutions through $(t_{0}, x_{0})$ and $(s, y_{0})$ satisfy

$∥ x (t) - y (t) ∥_{2} \leq e^{L H} (∥ x_{0} - y_{0} ∥_{2} + M ∣ s - t_{0} ∣ + H ω (∥ λ - μ ∥_{2})),$

where $ω (r) \to 0$ is a uniform modulus for the parameter dependence of $F$ on that cylinder.

Facts & Assumptions

Given: Two nearby parameterized IVPs and a compact time-state-parameter cylinder around the fixed data on which the common state-Lipschitz constant exists.

[L1]

A continuous map on a compact metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[L2]

Gronwall's integral inequality converts an additive forcing error into an exponential stability bound (Gronwall's integral inequality with variable and constant coefficients).

[L3]

On a time-state cylinder where $∥ F ∥_{2} \leq M$ , the state-Lipschitz constant is $L$ , $h M \leq r$ , and $L h < 1$ , Picard–Lindelöf gives exactly one solution on the full interval of half-length $h$ whose graph lies in that cylinder (Picard-Lindelöf local existence and uniqueness for first-order systems).

[L4]

A solution satisfies its associated Volterra integral equation (A first-order initial value problem is equivalent to its Volterra integral equation).

[L5]

The norm of a vector integral is at most the integral of the Euclidean norm (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).

[L6]

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).

Proof

technique · direct

1.1

Choose a smaller compact time-state-parameter cylinder around $(t_{0}, x_{0}, λ_{0})$ . By [L6] the field norm has one bound $M$ there, while the stated compact-cylinder hypothesis supplies one state-Lipschitz constant $L$ . Choose positive spatial and temporal margins and one $h, r$ with $h M \leq r$ and $L h < 1$ . Applying [L3] separately to every parameter slice and nearby initial datum gives a solution on $[s - h, s + h]$ inside the same state cylinder; after restricting to $∣ s - t_{0} ∣ < h / 2$ , all these intervals contain the fixed common interval $[t_{0} - h / 2, t_{0} + h / 2]$ . On the full compact parameter cylinder, [L1] bounds $∥ F (t, z, λ) - F (t, z, μ) ∥_{2}$ by a modulus $ω (∥ λ - μ ∥_{2})$ tending to zero with the parameter distance.

givenL1L3L6algebra

2.1

Use [L4] to rebase the second Volterra equation from $s$ to $t_{0}$ , which costs at most $M ∣ s - t_{0} ∣$ , then split the remaining integrand into the state difference and the discrepancy of step 1.1; [L5] gives the stated errors plus $L$ times the accumulated state error, so [L2] yields the displayed bound and joint continuous dependence.

step 1.1L2L4L5algebra∎

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Sources

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Ch. 2 (standard reference, not scraped)
Jiri Lebl, Basic Analysis I, Section 6.3 (standard reference, not scraped)