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 -

  1. Are there more general existence and uniqueness theorems for ODEs?
  2. 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?


For Nagumo's theorem see




Foe Osgood's theorem see



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