Peano existence theorem

The Peano existence theorem guarantees a local solution to an initial-value problem when the differential equation’s right-hand side is continuous near the initial point. It does not guarantee uniqueness.

Initial-value problem: A differential equation together with a specified value of its unknown function at an initial point. The theorem gives a local solution to this problem when its continuity hypothesis holds.
Picard–Lindelöf theorem: A theorem guaranteeing a unique local solution when the right-hand side is continuous and locally Lipschitz in the dependent variable. Its stronger hypothesis adds the uniqueness that Peano’s theorem does not provide.
Giuseppe Peano: An Italian mathematician and logician known for foundational work in mathematical logic and axiomatic arithmetic. The existence theorem bears his name and appeared in his work on differential equations.
Ordinary differential equation: An equation relating an unknown function of one independent variable to its derivatives. The theorem applies to local initial-value problems for ordinary differential equations.
Continuity: A property of a function whose values change arbitrarily little when its input changes sufficiently little. Continuity of the differential equation’s right-hand side is the theorem’s key hypothesis.
Local Lipschitz continuity: A condition requiring function values near a point to differ by at most a constant times the inputs’ distance. This stronger regularity condition supports local uniqueness alongside existence.
Cauchy–Lipschitz theorem: A classical local existence-and-uniqueness result for differential equations under suitable regularity conditions. Its history helps distinguish earlier uniqueness results from Peano’s continuity-only existence result.
Nonuniqueness in differential equations: The occurrence of multiple solutions to a differential equation with the same initial data. Peano’s hypotheses allow models with several trajectories from one initial state.
Euler method: A numerical method that approximates solutions of differential equations by successive tangent-line steps. Polygonal approximations from this method motivate a standard existence proof.
Uniqueness of solutions: The property that an initial-value problem has at most one solution on a specified domain. Peano guarantees existence but permits multiple solutions from the same initial data.

Linked from 8 pages