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Locally unique ODE solutions agree on overlaps and glue across a common endpoint
Statement
Locally unique solutions through the same data agree on every overlap. If two such solutions are defined on adjacent intervals, agree at their common endpoint, and solve the same continuous ODE there, their piecewise union is a solution on the union interval.
Facts & Assumptions
Given: Two solutions of the same locally state-Lipschitz ODE with a common value at some time in the overlap.
Coincident initial values give uniqueness on every common interval (The Grönwall estimate for two solutions of a Lipschitz ODE).
A subset of is connected if and only if it is order-convex (A subset of is connected if and only if it is order-convex, that is, an interval).
Proof
The agreement set in the overlap is nonempty, closed by continuity, and open because [L1] applies at each agreement time; the overlap is an interval and hence connected by [L2], so the agreement set is the whole overlap.
The piecewise union is therefore single-valued and continuous; away from the common endpoint it is a solution, while at the endpoint both one-sided derivatives equal the same value of the continuous vector field, so the union is differentiable there and solves the ODE.
Depends on
Used by
Dependency tree · two levels
24 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)