CauchyLipschitz

CauchyLipschitz refers to a fundamental existence and uniqueness result in the theory of ordinary differential equations. The standard formulation concerns the initial value problem dy/dt = f(t, y), with y(t0) = y0. The theorem is associated with Augustin-Louis Cauchy and Rudolf Lipschitz, and is often presented together with Picard’s method, leading to the alternative name Picard–Lindelöf.

A key condition is that f is continuous in t and Lipschitz in y on a rectangle R = [t0 - a, t0 + a] × [y0 - b, y0 + b]. The Lipschitz condition means there exists a constant L such that |f(t, y1) - f(t, y2)| ≤ L|y1 - y2| for all (t, y) in R. Under these hypotheses, there exists a T > 0 and a unique function y(t) defined on [t0, t0 + T] that solves the IVP. If f is globally Lipschitz, the solution exists and is unique on the whole interval where the problem is posed.

Idea and methods: The problem can be reformulated as the integral equation y(t) = y0 + ∫_{t0}^{t} f(s, y(s)) ds. On the space of continuous functions with the sup norm, the right-hand side is a contraction on a suitably small interval, and the Banach fixed point theorem yields a unique fixed point, i.e., a unique solution. Picard iterations provide successive approximations to this solution.

Remarks: The Lipschitz condition is sufficient but not necessary; Peano’s existence theorem guarantees at least one solution under mere continuity of f, but uniqueness may fail without Lipschitz continuity. The CauchyLipschitz theorem applies to systems of first-order ODEs and has various generalizations, including Carathéodory conditions.