Peano’s Existence Theorem

Existence Theorem

The existence theorem states that if a function is continuous, then the Cauchy problem for an ordinary differential equation admits at least one solution.

This result applies to Cauchy problems, which is why it’s also referred to as the Cauchy - Peano theorem.

However, the solution is not necessarily unique.

Existence and Uniqueness Theorem

If the function is continuous and also Lipschitz continuous, then the Cauchy problem admits a unique solution.

In this case, existence and uniqueness are both guaranteed.

And so on.