PicardLindelöf

Picard-Lindelöf refers to a fundamental result in the theory of ordinary differential equations that guarantees both existence and uniqueness of solutions to initial value problems under a Lipschitz condition. The theorem is named after Charles-Émile Picard and, independently, Leopold Lindelöf, who contributed established proofs in the early 20th century.

Statement of the theorem: Consider an initial value problem y'(t) = f(t, y(t)), with y(t0) = y0. Suppose f is defined on a region containing a rectangle [t0 − a, t0 + a] × [y0 − b, y0 + b], is continuous in t and y there, and satisfies a Lipschitz condition in y: there exists a constant L such that for all t in the interval and all y1, y2 in the y-interval, ||f(t, y1) − f(t, y2)|| ≤ L ||y1 − y2||. Then there exists a T > 0 (depending on a, b, L) such that the initial value problem has a unique solution on [t0, t0 + T]. If f is globally Lipschitz in y on the whole domain and continuous in t, the solution may extend to all t in the domain.

Methods and consequences: The standard proof uses the contraction mapping principle via Picard iteration, defining successive approximations y_{n+1}(t) = y0 + ∫_{t0}^t f(s, y_n(s)) ds. The theorem provides a rigorous foundation for the local well-posedness of many differential equations and is a staple in analysis, dynamical systems, and applied mathematics. It contrasts with Peano’s existence theorem, which guarantees existence under continuity alone but not uniqueness.