Existence and uniqueness theorem

An existence and uniqueness theorem gives conditions under which a differential equation has a solution and ensures that no other solution satisfies the same initial data.

Picard–Lindelöf theorem: A theorem guaranteeing a unique local solution to an initial-value problem when the differential equation’s right-hand side is continuous and locally Lipschitz in the dependent variable. It is the standard local result, with continuity securing existence and local Lipschitz continuity securing uniqueness.
Ordinary differential equation: A differential equation involving an unknown function of one independent variable and its derivatives. The theorem most commonly concerns whether an initial-value problem for this equation has exactly one solution.
Non-Lipschitz differential equation: A differential equation whose right-hand side fails to satisfy a Lipschitz condition in the dependent variable. Such equations can have multiple solutions despite continuous right-hand sides.
Deterministic dynamical system: A system whose state evolves according to fixed rules, so a given state determines its future trajectory when evolution is unique. Uniqueness makes the trajectory from a specified state unambiguous.
Cauchy–Lipschitz theorem: A name for the local existence and uniqueness theorem for ordinary differential equations under a Lipschitz condition. This common alternate name highlights the regularity condition that prevents distinct solutions from sharing initial data.
Initial value problem: A differential equation together with specified values of the unknown function at an initial point. Existence and uniqueness are claims about solutions matching these prescribed data.
Failure of uniqueness: A situation in which distinct solutions satisfy the same differential equation and initial data. It demonstrates that existence alone does not determine a unique evolution.
Numerical analysis of differential equations: The study of algorithms for approximating solutions to differential equations and controlling their errors. A well-defined exact solution gives numerical approximations a clear target.
Local Lipschitz continuity: A property in which a function satisfies a Lipschitz bound within a neighborhood of each point in its domain. This condition on the equation’s right-hand side is a key sufficient hypothesis for uniqueness.
Local existence theorem: A theorem ensuring that a solution exists on some neighborhood of an initial point. The standard result guarantees a nearby solution, not necessarily one defined for all time.

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