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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?

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For Nagumo's theorem see

https://projecteuclid.org/download/pdf_1/euclid.pja/1265033221

and

https://www.jstor.org/stable/pdf/2036312.pdf

Foe Osgood's theorem see

http://www.math.toronto.edu/mpugh/Teaching/MAT267_19/Osgood_Uniqueness_Theorem.pdf

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