When working with ordinary differential equations, I understand that there are two parts to the theorem on existence and uniqueness. While looking in several textbooks and online notes, I have found examples where existence of a solution is guaranteed, but uniqueness is not.
I am wondering if it is possible to be able to prove the uniqueness part of the theorem, but not have the existence part be met?
Or can we only look into uniqueness if we have already proven that a solution exists?
Thank You!
Generally speaking, if a solution does not exist, it makes no sense to prove it is unique. It's like saying the devil does not exist, but he must be blue :-).
It is possible, if you don't know the solution exists or not, to prove that if a solution exists, then the solution must be unique. However, such result will ultimately still have to rely on existence to classify solutions.