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Peano’s 1886 existence theorem on first-order scalar differential equations: a review. (English) Zbl 1354.01011

The authors focus on Peano’s 1886 theorem, stating the existence of a solution of an initial value problem \(y'=f(x,y)\), \(y(a)=b\), under the assumption of the continuity of the function \(f\). Someone lamented the lack of details in the proof, considering it “surprisingly unrigorous” (p. 387). Nevertheless, as the authors point out, these shortcomings are due to Peano’s choice to be “as simple and possible”. The lack of details must be interpreted in this perspective.
After a short introduction, the paper lays down the statement (Theorem 1, pp. 377–378) and the re-statement (Theorem 2, p. 379) of Peano’s existence theorem. Then, the authors state the more innovative part of Theorem 1, i.e., Theorem 3 (p. 380), called “the heart of Peano’s 1886 existence theorem”.
This last theorem will be proved in the following pages, filling in the gaps. As a matter of fact, the authors prove also Equations 3.14
\[ \frac{\Phi(x)-\Phi(x_0)}{x-x_0}\geq f(x_0,\Phi (x_0))-\varepsilon,\text{ for all }x\in ]x_0,x_1] \]
and 3.15
\[ \frac{\Phi(x)-\Phi(x_0)}{x-x_0}\leq f(x_0,\Phi (x_0))+\varepsilon,\text{ for all }x\in ]x_0,x_1], \]
which Peano left unproved (pp. 382–384). In other words, the authors re-state Peano’s original proof, emending it from its inacuracies.
Section 4 is devoted to the statement of analogous theorems by G. Peano himself [Math. Ann. 37, 182–228 (1890; JFM 22.0302.01)], G. Mie [ibid. 43, 553–568 (1893; JFM 25.0504.01)], W. F. Osgood [Monatsh. Math. Phys. 9, 331–345 (1898; JFM 29.0260.03)] and O. Perron [Math. Ann. 76, 471–484 (1915; JFM 45.0469.01)] (pp. 384–387), in order to make a comparison with Peano’s 1886 theorem and to show the originality of this last.
Section 5 (pp. 387–388) is a short conclusion which repeats the rationale of the author: the gaps in the proof of Theorem 1 are due to Peano’s choice to be as simple as possible and not to his floppiness.

MSC:

01A55 History of mathematics in the 19th century
34-03 History of ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable

Biographic References:

Peano, Giuseppe
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References:

[1] Arzelà, C.: Sull’integrabilità delle equazioni differenziali ordinarie. R. Accad. Sci. Istit. Bologna Mem. 5, 257-270 (1895-96) · JFM 26.0342.01
[2] Birkhoff, G., Merzbach, U.: A Source Book in Classical Analysis. Harvard University Press, Massachusetts (1973)
[3] de La Vallée Poussin, Ch.-J.: Mémoire sur l’intégration des équations différentielles. Mémoires. Acad. R. Belgique 47, 1-82 (1893) [de La Vallée Poussin, Ch.-J.: Collected Works, Oeuvres Scientifiques. In: Butzer, P., Mawhin, J., Vetro, P. (eds) Académie Royale de Belgique and Circolo Matematico di Palermo, vol. 2, pp. 405-486 (2001)] · JFM 25.0502.01
[4] de La Vallée Poussin, Ch.-J.: Sur l’intégration des équations différentielles. Ann. Soc. Sci. Bruxelles 17 (1e partie), 8-12 (1893) [de La Vallée Poussin, Ch.-J.: Collected Works, Oeuvres Scientifiques. In: Butzer, P., Mawhin, J., Vetro, P. (eds.) Académie Royale de Belgique and Circolo Matematico di Palermo, vol. 2, pp. 487-491 (2001)] · JFM 25.0503.01
[5] Dow, M.A., Výborný, R.: Elementary proofs of Peano’s existence theorem. J. Aust. Math. Soc. 15, 366-372 (1973) · Zbl 0272.34002 · doi:10.1017/S1446788700013276
[6] Flett, T.M.: Differential Analysis. Cambridge University Press, Cambridge (1980) · Zbl 0442.34002 · doi:10.1017/CBO9780511897191
[7] Fukuhara, M.: Sur les systèmes des équations differentielles ordinaires. Jpn. J. Math. 5, 345-350 (1928) · JFM 55.0843.01
[8] Fukuhara, M.: Sur le théorème d’existence des intégrales des équations différentielles ordinaires du premier ordre. Jpn. J. Math. 5, 239-251 (1928) · JFM 54.0451.04
[9] Gardner, C.: Another elementary proof of Peano’s existence theorem. Am. Math. Monthly 83, 556-560 (1976) · Zbl 0349.34002 · doi:10.2307/2319357
[10] Gilain, C.: Introduction to A. L. Cauchy: Équations différentielles ordinaires. Cours inédit. Fragment. Études Vivantes, Paris (1981)
[11] Greco, G.H., Mazzucchi, S.: The originality of Peano’s 1886 existence theorem for scalar differential equations. J. Convex Anal. 23 (2016) (to appear) · Zbl 1351.01012
[12] Greco, G.H., Mazzucchi, S.: Peano’s 1890 proof of existence theorem for systems of differential equations: a celebrated unknown (forthcoming, 2016)
[13] Hubbard, J.H., West, B.H.: Differential Equations: A Dynamical Systems Approach Ordinary Differential Equations. Springer, New York (1991) · Zbl 0987.34001
[14] Kennedy, H.C.: Is there an elementary proof of Peano’s existence theorem for first order differential equations? Am. Math. Monthly 76, 1043-1045 (1969) · doi:10.2307/2317137
[15] Kennedy, H.C.: Selected works of Giuseppe Peano. Allen & Unwin Ltd, Sydney (1973) · JFM 24.0248.05
[16] Kennedy, H.C.: Life and Works of Giuseppe Peano, Peremptory Edition (2nd edn.) (2006) · Zbl 0429.01015
[17] Kneser, H.: Uber die Lösungen eine system gewöhnlicher differential Gleichungen, das der lipschitzschen Bedingung nicht genügt S. B. Preuss Akad. Wiss. Phys. Math. Kl. 4, 171-174 (1923) · JFM 49.0302.03
[18] López Pouso, R.: Peano’s Existence Theorem Revisited. arXiv:1202.1152 (2012) · Zbl 0349.34002
[19] López Pouso, R.: Greatest solutions and differential inequalities: a journey in two directions. arXiv:1304.3576 (2013) · JFM 26.0313.02
[20] Mawhin, J.: Integration and the fundamental theory of ordinary differential equations : a historical sketch. In: Rassias, T.M. (ed.) Constantin Carathéodory, an International Tribute, vol. 2, pp. 828-849. World Scientific Publishing, Singapore (1991) · Zbl 0744.34003
[21] Mie, G.: Beweis der Integribarkeit gewöhnlicher Differentialgleichungssysteme nach Peano. Math. Ann. 43, 553-568 (1893) · JFM 25.0504.01 · doi:10.1007/BF01446453
[22] Osgood, W.F.: Beweis der Existenz einer Lösung der Differentialgleichung \[\frac{dy}{dx}=f(x, y)\] dydx=f(x,y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung. Monatsh. Math. Phys. 9, 331-345 (1898) · JFM 29.0260.03 · doi:10.1007/BF01707876
[23] Painlevé, P.: Existence de l’intégrale générale. In: Molk, J. (ed.) Encyclopédie des Sciences Mathématiques pures et appliquées, vol. 2.3, pp. 1-57. Gauthiers-Villars, Paris (1910)
[24] Peano, G.: Sull’integrabilità delle equazioni differenziali di primo ordine. Atti Reale Accad. Sci. Torino 21, 677-685 (1885-86) · Zbl 0275.34003
[25] Peano, G.: Applicazioni geometriche del calcolo infinitesimale. Fratelli Bocca Editori, Torino (1887) · JFM 19.0248.01
[26] Peano, G.: Démonstration de l’intégrabilité des équations différentielles ordinaires. Math. Ann. 37, 182-228 (1890) · JFM 22.0302.01 · doi:10.1007/BF01200235
[27] Peano, G.: Sur la définition de la dérivée. Mathesis 12, 12-14 (1892) · JFM 24.0248.05
[28] Peano, G.: Sulla definizione di integrale. Ann. Mat. Pura Appl. 23, 153-157 (1895) · JFM 26.0313.02 · doi:10.1007/BF02420514
[29] Peano, G.: Formulario Mathematico, 5th edn. Fratres Bocca, Torino (1908) (reprinted by Cremonese, Roma, 1960) · JFM 39.0084.01
[30] Perron, O.: Ein neuer Existenzbeweis für die Integrale der Differentialgleichung \[y = f (x, y)\] y=f(x,y). Math. Ann. 76, 471-484 (1915) · JFM 45.0469.01 · doi:10.1007/BF01458218
[31] Perron, O.: Eine neue behandlung der ersten randwertaufgabe für \[\Delta u=0\] Δu=0. Math. Z. 18, 42-54 (1923) · JFM 49.0340.01 · doi:10.1007/BF01192395
[32] Volterra, V.: Sui principi del calcolo integrale. G. Mat. (Battaglini) 19, 333-372 (1881) · JFM 13.0213.02
[33] Walter, J.: On elementary proofs of Peano’s existence theorems. Am. Math. Monthly 80, 282-286 (1973) · Zbl 0275.34003 · doi:10.2307/2318451
[34] Walter, W.: There is an elementary proof of Peano’s existence theorem. Am. Math. Monthly 78, 170-173 (1971) · Zbl 0207.08401 · doi:10.2307/2317624
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