Picard's Uniqueness Theorem requires Lipschitz continuity on $f$ for uniqueness. But, I have also seen examples where $f$ is not Lipschitz, and yet the solution is unique. Similarly with Peano's Existence Theorem.
So, I have $2$ questions -
- Are there more general existence and uniqueness theorems for ODEs?
- If not, are there theorem which guarantee uniqueness even when $f$ is not Lipschitz, or existence even when $f$ is not continuous?
If the answers to any of the above are yes, could someone also recommend some resources from where I can learn about them?