local existence and uniqueness theorem

E22820

mathematical theorem theorem in differential equations

The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.

Aliases (4)

Statements (47)

Predicate	Object
instanceOf	mathematical theorem → theorem in differential equations →
alsoKnownAs	Cauchy–Lipschitz theorem → Picard existence theorem → Picard–Lindelöf theorem →
appliesTo	equations of the form y' = f(t,y) → initial conditions of the form y(t0) = y0 → systems of ordinary differential equations →
assumes	f is continuous in t → f is locally Lipschitz in y →
category	existence and uniqueness theorem →
concerns	first-order ordinary differential equations → initial value problems →
concludes	existence of a unique solution on some neighborhood of t0 → existence of an interval around t0 where a solution exists →
ensures	solution is defined on some open interval containing t0 →
field	analysis → ordinary differential equations →
guarantees	existence of a solution through a given initial point → local existence of solutions → local uniqueness of solutions → uniqueness of a solution through a given initial point →
hasConsequence	deterministic behavior of solutions near initial data → well-posedness of local initial value problems →
hasScope	local in time →
historicallyAssociatedWith	Augustin-Louis Cauchy → Ernst Lindelöf → Rudolf Lipschitz → Émile Picard →
implies	solutions depend continuously on initial data (locally) →
isPartOf	theory of initial value problems →
isProvedBy	Banach fixed-point theorem → Picard iteration →
isTaughtIn	introductory analysis courses → undergraduate differential equations courses →
motivates	study of Lipschitz continuity → use of contraction mappings in analysis →
relatedTo	Peano existence theorem → global existence theorems →
requiresConditionOn	Lipschitz continuity of f in y → continuity of f in a neighborhood of (t0,y0) → local Lipschitz condition in y →
strongerThan	Peano existence theorem in terms of uniqueness →
usesConcept	Lipschitz condition → complete metric space → fixed-point theorem →
weakerThan	global existence and uniqueness theorems →

Referenced by (5)

Subject (surface form when different)	Predicate
local existence and uniqueness theorem ("Picard–Lindelöf theorem") → local existence and uniqueness theorem ("Cauchy–Lipschitz theorem") → local existence and uniqueness theorem ("Picard existence theorem") →	alsoKnownAs
local existence and uniqueness theorem ("existence and uniqueness theorem") →	category
implicit function theorem →	logicalForm