CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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A globally state-Lipschitz vector field on $R \times R^{n}$ has global solutions

Statement

Let $F : R \times R^{n} \to R^{n}$ be continuous and suppose one $L \geq 0$ satisfies $∥ F (t, x) - F (t, y) ∥_{2} \leq L ∥ x - y ∥_{2}$ for all $t, x, y$ . Then every IVP for $x^{'} = F (t, x)$ has a unique global solution.

Facts & Assumptions

Given: The globally state-Lipschitz field and its maximal solution.

[L1]

Gronwall bounds a nonnegative function by its forcing and a linear integral term (Gronwall's integral inequality with variable and constant coefficients).

[L2]

At a finite maximal endpoint the solution leaves every compact subset of the ODE domain (At a finite maximal time an ODE solution leaves every compact subset of the domain).

[L3]

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).

[L4]

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

[L5]

Closed bounded subsets of $R^{n}$ are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[L6]

Every Picard–Lindelöf IVP has a unique maximal solution, and every other solution through the same data is its restriction (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval).

Proof

technique · contradiction

1.1

Let $x$ be the unique maximal solution supplied by [L6], and suppose, for contradiction, that one of its maximal endpoints is finite. On a finite time slab reaching toward it, [L3] bounds $∥ F (t, 0) ∥_{2}$ by $M$ and global Lipschitz continuity gives $∥ F (t, x) ∥_{2} \leq M + L ∥ x ∥_{2}$ ; [L4] and [L1] therefore bound $∥ x (t) ∥_{2}$ throughout the slab.

givenL1L3L4L6assume-contra

2.1

By [L5], step 1.1 places the graph near that endpoint in a compact time-state box, contradicting [L2]; hence both endpoints are infinite and the solution is global.

step 1.1L2L5discharge-contradiction∎

Depends on

Used by

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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)

A globally state-Lipschitz vector field on R×RnR×Rn has global solutions