In modern proofs of the Peano Existence Theorem for ordinary differential equations, Ascoli's theorem is used. Ascoli's theorem came after Peano's proof. Did Peano prove a form of Ascoli's theorem in his proof or did he do something else?


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Peano did do something else, and not quite right, apparently. In 1969 Kennedy published a note in the American Mathematical Monthly titled Is There an Elementary Proof of Peano's Existence Theorem, where he wrote:

"In 1886 Giuseppe Peano stated [8] that the initial value problem: $y' = f(x,y),\, y(a) = b$, has a solution on the sole condition that $f$ is continuous, and he gave an elementary proof of this. His theorem is correct, but a historical investigation into the work of Peano has led to the conclusion that his proof is invalid."

Kennedy's note stimulated a flurry of activity, some of which is reviewed in Pouso's Peano's Existence Theorem revisited. He divides proposed proofs into two groups: (A) constructing a sequence of approximate solutions, and (B) using fixed point arguments. There are also combinations of the two, like the one relying on the Arzela–Ascoli theorem:

"For instance, a well–known kind of mix of types (A) and (B) consists in approximating $f$ by polinomials $f_n$ (by virtue of the Stone–Weierstrass Theorem) so that the problems (1.1) with $f$ replaced by $f_n$ have unique solutions $y_n$ (by virtue of the Banach contraction principle) which form an approximating sequence."

Peano's attempted proof, on the other hand, is in (A). For those who read Italian, it can be found in his Sull'integrabilità delle equazioni differenziali del primo ordine, Atti Accad. Sci. Torino (1886) 21: 437–445 (1D) or Démonstration de l'intégrabilité des équations différentielles ordinaires, Mathematische Annalen, June (1890) 37, Issue 2, pp. 182–228 (multi D). There are also more accessible presentations using what came to be called Perron's method of superfunctions, apparently anticipated by Peano (Perron's proof appeared in 1915). Here is more from Pouso:

"More elementary proofs were devised in the seventies of the 20th century, partly as a reaction to a question posed in [10]. In [4, 6, 18, 20] we find proofs of type (A) in dimension $n = 1$ which avoid the Arzela–Ascoli Theorem. As already stated in those references, similar ideas do not work in higher dimension, at least without further assumptions. We also find in [4] a proof which uses Perron’s method, see [16], a refined version of Peano’s own proof in [14]. It is fair to acknowledge that probably the best application of Perron’s method to (1.1) in dimension $n=1$ is due to Goodman [7], who even allowed $f$ to be discontinuous with respect to the independent variable and whose approach has proven efficient in more general settings, see [1, 2, 8, 12]."

The [10] is Kennedy's note, and the [4] is Elementary proofs of Peano’s Existence Theorem by Dow and Vyborny in J. Aust. Math. Soc. 15 (1973), 366–372 (freely available), who characterize Perron's proof as non-elementary, and give an elementary version "in the spirit of Perron's original proof".


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