CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-21

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Nearby initial values share one Picard–Lindelöf time interval and one state cylinder

Statement

Under the hypotheses of Picard-Lindelof at $(t_{0}, x_{0})$ , there are $h, r, η > 0$ such that every initial value $(s, y)$ with $∣ s - t_{0} ∣ < η$ and $∥ y - x_{0} ∥_{2} < η$ has a unique solution on $[s - h, s + h]$ , and all these solution graphs lie in one compact time-state cylinder.

Facts & Assumptions

Given: A compact cylinder contained in the open ODE domain and a smaller cylinder with positive distance from its boundary.

[L1]

On a cylinder with bound $M$ , state-Lipschitz constant $L$ , $h M \leq r$ , 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).

[L2]

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

Proof

technique · direct

1.1givenL2

On the larger compact cylinder, [L2] bounds $∥ F ∥_{2}$ by a common $M$ , while local state-Lipschitz continuity and a finite compact cover give a common $L$ ; the smaller cylinder has a positive spatial and temporal boundary margin.

2.1step 1.1L1algebra∎

Choose one radius $r > 0$ below the spatial margin and one $h > 0$ below the temporal margin so that $h M \leq r$ and $L h < 1$ ; then [L1] applies with these same data to every initial point in the smaller cylinder, giving the asserted common time and graph cylinder.

Depends on

Used by

Pointwise local existence does not force one global uniform time interval False statement
Smooth dependence of ODE solutions on parameters Theorem
The fundamental theorem for autonomous smooth ODEs Theorem

Dependency tree · two levels

47 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)