Picard–Lindelöf theorem

The Picard–Lindelöf theorem guarantees a unique local solution to an ordinary differential equation’s initial-value problem when its defining function is continuous and locally Lipschitz in the dependent variable.

Contraction mapping theorem: A fixed-point theorem stating that a contraction on a complete metric space has exactly one fixed point. The theorem’s integral operator becomes a contraction on a sufficiently short time interval.
Ordinary differential equation: A differential equation involving an unknown function of one independent variable and its derivatives. The theorem concerns existence and uniqueness for this class of equations.
Peano existence theorem: A theorem guaranteeing local existence for an initial-value problem when the defining function is continuous. Unlike Picard–Lindelöf, it does not guarantee uniqueness.
Initial-value problem for a linear differential equation: An initial-value problem whose differential equation is linear in the unknown function and its derivatives. Continuous coefficients make the defining function locally Lipschitz in the state, so the theorem applies.
Banach fixed-point theorem: A theorem guaranteeing a unique fixed point for a contraction mapping on a complete metric space. Its fixed point is the function that satisfies the integral form of the initial-value problem.
Initial-value problem: A differential equation together with specified values of the unknown function at an initial point. The theorem guarantees a solution matching the prescribed initial value.
Osgood uniqueness theorem: A uniqueness criterion for differential equations that can replace a Lipschitz bound with an integral condition on a modulus of continuity. It establishes uniqueness under some weaker regularity conditions than a Lipschitz hypothesis.
Dynamical system: A mathematical model describing how a system’s state changes over time according to a rule. The theorem ensures a locally well-defined trajectory from each state under its hypotheses.
Picard iteration: An iterative method that approximates a solution by repeatedly substituting into an integral equation. These successive approximations converge to the local solution in the proof.
Local existence theorem: A result establishing that a mathematical problem has a solution near specified data or a point. Picard–Lindelöf is a specific local existence result with a uniqueness guarantee.

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