Abstract
This is a didactic proposal on how to introduce the Newton integral in just three or four sessions in elementary courses. Our motivation for this paper were Talvila’s work on the continuous primitive integral and Koliha’s general approach to the Newton integral. We introduce it independently of any other integration theory, so some basic results require somewhat nonstandard proofs. As an instance, showing that continuous functions on compact intervals are Newton integrable (or, equivalently, that they have primitives) cannot lean on indefinite Riemann integrals. Remarkably, there is a very old proof (without integrals) of a more general result, and it is precisely that of Peano’s existence theorem for continuous nonlinear ODEs, published in 1886. Some elements in Peano’s original proof lack rigor, and that is why his proof has been criticized and revised several times. However, modern proofs are based on integration and do not use Peano’s original ideas. In this note we provide an updated correct version of Peano’s original proof, which obviously contains the proof that continuous functions have primitives, and it is also worthy of remark because it does not use the Ascoli-Arzelà theorem, uniform continuity, or any integration theory.
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References
H. Bendová, J. Malý: An elementary way to introduce a Perron-like integral. Ann. Acad. Sci. Fenn., Math. 36 (2011), 153–164.
B. Bongiorno: A new integral for the problem of antiderivatives. Matematiche 51 (1996), 299–313. (In Italian.)
B. Bongiorno, L. Di Piazza, D. Preiss: A constructive minimal integral which includes Lebesgue integrable functions and derivatives. J. Lond. Math. Soc., II. Ser. 62 (2000), 117–126.
A. M. Bruckner, R. J. Fleissner, J. Foran: The minimal integral which includes Lebesgue integrable functions and derivatives. Colloq. Math. 50 (1986), 289–293.
I. Černý, M. Rokyta: Differential and Integral Calculus of One Real Variable. Karolinum, Prague, 1998.
L. Di Piazza: A Riemann-type minimal integral for the classical problem of primitives. Rend. Ist. Mat. Univ. Trieste 34 (2002), 143–153.
M. A. Dow, R. Výborný: Elementary proofs of Peano’s existence theorem. J. Aust. Math. Soc. 15 (1973), 366–372.
C. Gardner: Another elementary proof of Peano’s existence theorem. Am. Math. Mon. 83 (1976), 556–560.
G. S. Goodman: Subfunctions and the initial-value problem for differential equations satisfying Carathéodory’s hypotheses. J. Differ. Equations 7 (1970), 232–242.
R. Henstock: Lectures on the Theory of Integration. Series in Real Analysis 1. World Scientific, Singapore, 1988.
T. Kawasaki: On Newton integration in vector spaces. Math. Jap. 46 (1997), 85–90.
H. C. Kennedy: Is there an elementary proof of Peano’s existence theorem for first order differential equations? Am. Math. Mon. 76 (1969), 1043–1045.
J. J. Koliha: Metrics, Norms and Integrals: An Introduction to Contemporary Analysis. World Scientific, Hackensack, 2008.
J. J. Koliha: Mean, meaner, and the meanest mean value theorem. Am. Math. Mon. 116 (2009), 356–361.
J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (1957), 418–449.
N. W. Leng, L. P. Yee: An alternative definition of the Henstock-Kurzweil integral using primitives. N. Z. J. Math. 48 (2018), 121–128.
P. Mikusińksi, K. Ostaszewski: Embedding Henstock integrable functions into the space of Schwartz distributions. Real Anal. Exchange 14 (1988), 24–29.
J. Mikusiński, R. Sikorski: The elementary theory of distributions. I. Rozprawy Mat. 12 (1957), 52 pages.
G. Peano: Sull’ integrabilità delle equazioni differenziali di primo ordine. Atti. Accad. Sci. Torino 21 (1886), 677–685. (In Italian.)
G. Peano: Démonstration de l’integrabilité des équations différentielles ordinaires. Math. Ann. 37 (1890), 182–228. (In French.)
O. Perron: Ein neuer Existenzbeweis für die Integrale der Differentialgleichung y′ = f (x, y). Math. Ann. 76 (1915), 471–484. (In German.)
K. R. Stromberg: An Introduction to Classical Real Analysis. Wadsworth International Mathematics Series. Wadsworth, Belmont, 1981.
E. Talvila: The distributional Denjoy integral. Real Anal. Exch. 33 (2008), 51–82.
I. Tevy: Une définition de l’intégrale de Lebesgue à l’aide des fonctions primitives. Rev. Roum. Math. Pures Appl. 19 (1974), 1159–1163. (In French.)
J. Walter: On elementary proofs of Peano’s existence theorem. Am. Math. Mon. 80 (1973), 282–286.
W. Walter: There is an elementary proof of Peano’s existence theorem. Am. Math. Mon. 78 (1971), 170–173.
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Dedicated to the memory of Professor Jaroslav Kurzweil
Open access funding provided by Universidade de Santiago de Compostela.
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López Pouso, R. A roller coaster approach to integration and Peano’s existence theorem. Czech Math J (2023). https://doi.org/10.21136/CMJ.2023.0514-22
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DOI: https://doi.org/10.21136/CMJ.2023.0514-22