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I was recently reading a book about the implicit function theorem (IFT): The implicit function theorem: history, theory, and applications, and before that I learned that Peano's existence theorem can be used to prove the IFT (the existence of implicit functions); in this book, an idea of using the IFT to prove Peano's existence theorem (PET) is given: idea of using the implicit function theorem to prove Peano's existence theorem

Theorem 3.4.10 mentioned in it is as follows (with its proof): Theorem 3.4.10 with proof

where $d_{2}G(x,y)$ is defined as d_2

Apart from this book, I haven't found other papers or books describing IFT to prove PET (maybe this view is of little value?), so I have to consult you here.

My questions are as follows: Is this proof of PET using IFT correct? If correct does this mean that IFT is equivalent to PET?

YCor
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anyon
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    This is perhaps a bit pedantic and certainly unhelpful as far as the actual question goes, but I don't like this use of the word "equivalent". What exactly does it mean to claim that two true statements are (or are not) equivalent to each other? (For example, is the fundamental theorem of calculus "equivalent" to Fermat's last theorem? In any formal sense, the answer would seem to have to be yes since one holds if and only if the other does.) – Christian Remling Jul 21 '22 at 14:19
  • Perhaps a naive question, but I have to ask: what's the other view on the existence theorem for ODEs? The only time I saw a proof of this was when I took a course on ODEs as an undergrad, and my vague recollection is that it was proven using the implicit function theorem. – Anurag Sahay Jul 21 '22 at 15:44
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    By the way, @ChristianRemling: perhaps the discussion in the following answer might help alleviate your irritation? https://mathoverflow.net/a/284231/37327

    Certainly this is not what people "mean" when they say two theorems are equivalent to one another, but I think of this as being morally what they are saying [the same way as a theorem in a non-set-theory paper "is" morally its formalization in, say, ZFC].

     – Anurag Sahay Jul 21 '22 at 15:52
  • Rudin was rather fond of listing equivalent statements in his books. I can think of four equivalent (and very useful) ways to characterize any Appell polynomial sequence. Using the umbral notation with $(a.)^k = a_k$, given $A_0=1,$ 1} $D_x A_n(x) = nA_{n-1}(x)$, 2) e.g.f. is $e^{a.t}e^{xt}$, 3) $A_n(x) = (a.+x)^n$, and 4) the raising op, defined by $RA_n(x) = A_{n+1}(x)$, is $R = D_{t = D_x} \ln[e^{a.t}e^{xt}]$. Take any one as a definition, the others can be treated as theorems. 1) doesn't allow for direct construction, 2) to 4) do given the moments $a_k$. Each has its use. – Tom Copeland Jul 21 '22 at 16:21
  • When I think of an inverse pair of formal e.g.f.s $f(x)$ and $g(x)$, I keep in mind equivalent perspectives: 1) an autonomous o.d.e., 2) integral curves constructed from a vector field of tangents, 3} reflection through $y=x$, 4) iterated Graves-Lie infinitesimal generators, 5] refined Euler characteristic polynomials of associahedra, 6) marked phylogenetic trees, 6} punctured Riemann surfaces, 7} moduli spaces of certain marked curves, 8) weighted noncrossing partitions, 9) $f(x) =x h(f(x))$, 10) . . . , and, of course, the IFT, but never Fermat's LT. – Tom Copeland Jul 21 '22 at 16:55
  • Nice ref. Thanks. Ferraro also discusses Newton and the implicit function theorem in the "The Rise and Development of the Theory of Series up to the Early 1820s". (Wouldn't look for any help from Whitehead and Russell.) – Tom Copeland Jul 21 '22 at 19:07
  • @AnuragSahay :Thanks for the experience you mentioned.But The proofs of the PET in the textbooks I have found so far do not use IFT, which textbook did you use or maybe your teacher did not use the method in the textbook? Please tell me more about it, it's really important to me. – anyon Jul 24 '22 at 05:29

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