Implicit function theorem and PDE; do we get uniqueness?

Please see this page: The implicit function theorem: A PDE example.

In the implicit function theorem they quote, uniqueness is not mentioned. But the inverse function theorem (which is equivalent to the IMFT), we do get uniqueness. So am I right that the solution got by using the theorem is unique as well?

Answer

  1. Their statement of the Implicit Function theorem does contain a uniqueness part: “F(x,y)=0 if and only if”.

  2. However, this is a local statement, only for (x,y)U×V. It does not exclude the possibility of having other solutions outside of small neighborhood V.

Attribution
Source : Link , Question Author : hopo2 , Answer Author : Community

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