The Peano Existence Theorem guarantees that an ordinary differential equation with a continuous right-hand side has at least one solution through any given initial point, even though continuity alone does not guarantee that solution is unique. Named for Giuseppe Peano, it is weaker than the Picard-Lindelof Theorem, which adds a stronger smoothness condition on the right-hand side in exchange for also guaranteeing that the solution found is the only one.
Facts
Partially Attested
Proof YearPeano first published the theorem in 1886 with an incorrect proof and republished a corrected proof in 1890; 1890 is recorded as Value, the corrected proof year StatementLet D be an open subset of R x R with f: D to R a continuous function and y'(t) = f(t, y(t)) a first-order differential equation defined on D, then every initial value problem y(t0) = y0 for f with (t0, y0) in D has a local solution. 1 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Peano existence theorem, Wikipedia
Sources
1. Peano existence theorem, Wikipedia
Statement section
Let D be an open subset of R times R with f: D to R a continuous function and y'(t) = f(t, y(t)) a first-order differential equation defined on D, then every initial value problem y(t0) = y0 for f with (t0, y0) in D has a local solution.
History section
Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations.
- In Branch: Differential Equations, Lead sentence
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