Picard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy.

Consider the initial value problem

Suppose  f  is uniformly Lipschitz continuous in y (meaning the Lipschitz constant can be taken independent of t) and continuous in t. Then, for some value ε > 0, there exists a unique solution y(t) to the initial value problem on the interval ${\displaystyle [t_{0}-\varepsilon ,t_{0}+\varepsilon ]}$.

The proof relies on transforming the differential equation, and applying fixed-point theory. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation

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