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Peano’s 1886 existence theorem on first-order scalar differential equations: a review

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Abstract

In 1886 Giuseppe Peano presents the first proof of the existence of a solution of an initial value problem \(y^\prime =f(x,y)\), \(y(a)=b\), under the assumption of the continuity of the function f. The present paper gives a detailed description of Peano’s original statements and proofs, filling gaps, clarifying obscure points and avoiding ambiguous use of mathematical symbols. Peano’s 1886 work is compared with later papers of Peano himself as well as of Mie (Math Ann 43:553–568, 1893), Osgood (Monatsh Math Phys 9:331–345, 1898) and Perron (Math. Ann. 76:471–484, 1915).

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Notes

  1. Among the huge literature on differential equations, we have analyzed the work of a restricted number of authors. Our choice is based on the particular topic under study and cannot be exhaustive.

  2. We believe that our proposal to fill gaps in 1886 original proof is not the only possibility. The reader can try alternative ways.

  3. Here and in the sequel, we quote passages of Peano’s paper from the English translation in Kennedy [15] and, for a typographical convenience, we replace the symbols of derivative of the form \(\frac{d y}{d x}\), used by Peano, with \(y^\prime \).

  4. It is worth noticing that the family of the continuous piecewise linear functions \(y_{1}\) (resp. \(y_{2}\)), verifying the inequality \(y_{1}^\prime >f(x,y_{1})\) (resp. \(y_{2}^\prime <f(x,y_{2})\)) on each closed interval where are linear, is the minimum we need to support and to validate Peano’s 1886 arguments.

  5. In fact this convention relative to the definition of sub- and super-solution is rather natural and it is adopted also in the modern mathematical literature on differential equations [13].

  6. Observe that it is implicitly assumed that the inequality “\(D_{\pm }\gamma (x)>t\)” should be understood as “\(D_{+}\gamma (x)>t\)” (resp. “\(D_{-}\gamma (x)>t\)”), whenever x is the lower extremum (resp. upper extremum) of the domain of \(\gamma \). A similar convention holds for the other inequalities \(D_{\pm }\gamma (x)\ge t\), \(D_{\pm }\gamma (x)< t\), \(D_{\pm }\gamma (x)\le t\).

  7. We remark that the names super-and sub-solutions for the functions satisfying the differential inequalities (2.2) have been introduced for the first time by Perron [30], who used the terms Oberfunction and Unterfunktion and in [31] used these notions for solving partial differential equations.

  8. In proving Theorem 3 it is necessary only the existence of at least one strict sub-solution and one strict super-solution. In Peano’s original proof of (2.3) strict sub- and super-solutions are constructed as continuous piecewise linear functions. In this way the interval where the related differential inequalities (2.2) are valid can be enlarged. This technique allows Peano to assert (2.3), i.e., there are infinitely many strict super- and sub-solutions.

  9. Kennedy [14] has misunderstood this point of Peano’s proof and thus considered it wrong.

  10. Proof of the existence of \(\bar{\nu }_{n}\). Let \(\hat{b}\) and M be real numbers verifying the following four inequalities: (i) , (ii) \(\hat{b}>\nu _{n}(x_{1})\), (iii) and (iv) \(M(\hat{a}-x_{1})>\hat{b}-\nu _{n}(x_{1})\). Define the linear function \(\psi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) by \(\psi (x):=\nu _{n}(x_{1})+M(x-x_{1})\) and consider the real number \(\hat{x}:=x_{1}+ \frac{\hat{b}-\nu _{n}(x_{1})}{M}\). Clearly, \(x_{1}<\hat{x}<x_{1}+\frac{M(\hat{a} -x_{1})}{M}=\hat{a}\) and \(\nu _{n}(x_{1})=\psi (x_{1})\le \psi (x)\le \psi (\hat{x})=\hat{b}\) for every ; hence, by (iii) we have that \(\psi \) is a strict super-solution of (ODE) on . Thus, the required strict super-solution function \(\bar{\nu }_{n}\) is defined by \(\bar{\nu }_{n}(x):=\inf \{\nu _{n}(x),\gamma _{2}(x)\}\) for , \(\bar{\nu }_{n}(x):=\inf \{\psi (x),\gamma _{2}(x)\}\) for and \(\bar{\nu }_{n}(x):=\gamma _{2}(x)\) for .

  11. Proof of the left-continuity of \(\Phi \) at \(x_0\). By (3.18) one can deduce the inequality \(\limsup _{x\rightarrow x_{0}^{-}}\Phi (x)\le \Phi (x_{0})\). To prove the remaining inequality \(\Phi (x_{0})\le \liminf _{x\rightarrow x_{0}^{-}}\Phi (x)\), choose \(b_{1},b_{2},M,\delta \in {\mathbb {R}}\) such that \(0<\delta <x_{0}-a\) and

    Let , \(\mu \in {{\mathop {\mathrm {Sol}}}}_{>}(b;a,\hat{a})\) and let us define the linear function \(\psi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) by \(\psi (x):=\bar{\mu }(\tilde{x})+M(x-\tilde{x})\), where \(\bar{\mu }\) is a strict super-solution of (ODE) on , defined by \(\bar{\mu }:=\inf \{\mu ,\gamma _{2}\}\). From (i\(^*\)) and (iii\(^*\)) it follows that the solution \(\hat{x}:=\tilde{x}+\frac{2b_{2}-\bar{\mu }(\tilde{x})}{M}\) of the equation \(\psi (x)=2b_{2}\) satisfies:

    since \(\gamma _{1}\le \bar{\mu }\le \gamma _{2}\), \(\hat{x}> \tilde{x}+\frac{b_{2}}{M}> \tilde{x}+\frac{M\delta }{M}=x_{0}+\delta >x_{0}\), and . On the other hand, from (ii\(^*\)) and (iv\(^*\)) it follows that

    Now define a function by

    Footnote 11 continued

    in the case where \(\hat{x}<\hat{a}\); otherwise

    Consequently, since \(\bar{\mu }\) is a strict super-solution of (ODE) on , by (v\(^*\)) we have that \(\nu \) is a strict super-solution of (ODE) on . Then from the definition (3.8) of \(\Phi \) follows that \(\Phi (x_{0})\le \nu (x_{0})\le \psi (x_{0})=\bar{\mu }(\tilde{x})+M(x_{0}-\tilde{x})\le \mu (\tilde{x})+M(x_{0}-\tilde{x})\). Therefore, by the arbitrariness of and \(\mu \in {\mathop {\mathrm {Sol}}}(b;a,\hat{a})\), we have \(\Phi (x_{0})\le \Phi (\tilde{x})+M(x_{0}-\tilde{x})\) for every ; hence, \(\Phi (x_{0})\le \liminf _{\tilde{x}\rightarrow x_{0}^{-}}\Phi (\tilde{x})\). This concludes the proof of the left-continuity of \(\Phi \) at \(x_{0}\).

  12. I.e., the set \(\{y\in {\mathbb {R}}:(x,y)\in \mathrm {dom} (f)\}\) is a non-empty interval for every .

  13. I.e., argument based on the following proposition (see either Proposition (1.4.5) in Flett [6, p. 23] or Theorem in Sect. 1 of Fukuhara [8]): if f and \(\Phi \) are continuous and the equality \(D_{+}\Phi (x)=f(x,\Phi (x))\) holds, then \(\Phi \) is differentiable.

  14. Here, the above claimed “symmetry” is attained by using the characterization of \(C^{1}\)-function, first given by Peano in [25, p. 191] and [27].

References

  1. Arzelà, C.: Sull’integrabilità delle equazioni differenziali ordinarie. R. Accad. Sci. Istit. Bologna Mem. 5, 257–270 (1895–96)

  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)]

  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)]

  5. Dow, M.A., Výborný, R.: Elementary proofs of Peano’s existence theorem. J. Aust. Math. Soc. 15, 366–372 (1973)

    Article  MATH  Google Scholar 

  6. Flett, T.M.: Differential Analysis. Cambridge University Press, Cambridge (1980)

    Book  MATH  Google Scholar 

  7. Fukuhara, M.: Sur les systèmes des équations differentielles ordinaires. Jpn. J. Math. 5, 345–350 (1928)

  8. Fukuhara, M.: Sur le théorème d’existence des intégrales des équations différentielles ordinaires du premier ordre. Jpn. J. Math. 5, 239–251 (1928)

    MATH  Google Scholar 

  9. Gardner, C.: Another elementary proof of Peano’s existence theorem. Am. Math. Monthly 83, 556–560 (1976)

    Article  MATH  Google Scholar 

  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)

  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)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  15. Kennedy, H.C.: Selected works of Giuseppe Peano. Allen & Unwin Ltd, Sydney (1973)

  16. Kennedy, H.C.: Life and Works of Giuseppe Peano, Peremptory Edition (2nd edn.) (2006)

  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)

  18. López Pouso, R.: Peano’s Existence Theorem Revisited. arXiv:1202.1152 (2012)

  19. López Pouso, R.: Greatest solutions and differential inequalities: a journey in two directions. arXiv:1304.3576 (2013)

  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)

  21. Mie, G.: Beweis der Integribarkeit gewöhnlicher Differentialgleichungssysteme nach Peano. Math. Ann. 43, 553–568 (1893)

    Article  MathSciNet  MATH  Google Scholar 

  22. Osgood, W.F.: Beweis der Existenz einer Lösung der Differentialgleichung \(\frac{dy}{dx}=f(x, y)\) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung. Monatsh. Math. Phys. 9, 331–345 (1898)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  25. Peano, G.: Applicazioni geometriche del calcolo infinitesimale. Fratelli Bocca Editori, Torino (1887)

    MATH  Google Scholar 

  26. Peano, G.: Démonstration de l’intégrabilité des équations différentielles ordinaires. Math. Ann. 37, 182–228 (1890)

    Article  MathSciNet  MATH  Google Scholar 

  27. Peano, G.: Sur la définition de la dérivée. Mathesis 12, 12–14 (1892)

    MATH  Google Scholar 

  28. Peano, G.: Sulla definizione di integrale. Ann. Mat. Pura Appl. 23, 153–157 (1895)

    Article  MATH  Google Scholar 

  29. Peano, G.: Formulario Mathematico, 5th edn. Fratres Bocca, Torino (1908) (reprinted by Cremonese, Roma, 1960)

  30. Perron, O.: Ein neuer Existenzbeweis für die Integrale der Differentialgleichung \(y = f (x, y)\). Math. Ann. 76, 471–484 (1915)

    Article  MathSciNet  MATH  Google Scholar 

  31. Perron, O.: Eine neue behandlung der ersten randwertaufgabe für \(\Delta u=0\). Math. Z. 18, 42–54 (1923)

    Article  MathSciNet  MATH  Google Scholar 

  32. Volterra, V.: Sui principi del calcolo integrale. G. Mat. (Battaglini) 19, 333–372 (1881)

    Google Scholar 

  33. Walter, J.: On elementary proofs of Peano’s existence theorems. Am. Math. Monthly 80, 282–286 (1973)

    Article  MATH  Google Scholar 

  34. Walter, W.: There is an elementary proof of Peano’s existence theorem. Am. Math. Monthly 78, 170–173 (1971)

    Article  MATH  Google Scholar 

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  1. via Cesarini 27/2, 38121, Trento, Italy

    Gabriele H. Greco

  2. Department of Mathematics, University of Trento, 38123, Trento, Italy

    Sonia Mazzucchi

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Greco, G.H., Mazzucchi, S. Peano’s 1886 existence theorem on first-order scalar differential equations: a review. Boll Unione Mat Ital 9, 375–389 (2016). https://doi.org/10.1007/s40574-016-0052-6

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