Article
| Title: | A roller coaster approach to integration and Peano's existence theorem (English) |
| Author: | López Pouso, Rodrigo |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 157-177 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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. (English) |
| Keyword: | primitive |
| Keyword: | Newton integral |
| Keyword: | Peano's existence theorem |
| MSC: | 26A27 |
| MSC: | 26A36 |
| MSC: | 26A39 |
| DOI: | 10.21136/CMJ.2023.0514-22 |
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| Date available: | 2025-03-11T15:59:51Z |
| Last updated: | 2025-03-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152902 |
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| Reference: | [1] Bendová, H., Malý, J.: An elementary way to introduce a Perron-like integral.Ann. Acad. Sci. Fenn., Math. 36 (2011), 153-164. Zbl 1225.26016, MR 2797688, 10.5186/aasfm.2011.3609 |
| Reference: | [2] Bongiorno, B.: A new integral for the problem of antiderivatives.Matematiche 51 (1996), 299-313 Italian. Zbl 0929.26007, MR 1488074 |
| Reference: | [3] Bongiorno, B., Piazza, L. Di, Preiss, D.: A constructive minimal integral which includes Lebesgue integrable functions and derivatives.J. Lond. Math. Soc., II. Ser. 62 (2000), 117-126. Zbl 0980.26006, MR 1771855, 10.1112/S0024610700008905 |
| Reference: | [4] Bruckner, A. M., Fleissner, R. J., Foran, J.: The minimal integral which includes Lebesgue integrable functions and derivatives.Colloq. Math. 50 (1986), 289-293. Zbl 0604.26006, MR 0857865, 10.4064/cm-50-2-289-293 |
| Reference: | [5] Černý, I., Rokyta, M.: Differential and Integral Calculus of One Real Variable.Karolinum, Prague (1998). |
| Reference: | [6] Piazza, L. Di: A Riemann-type minimal integral for the classical problem of primitives.Rend. Ist. Mat. Univ. Trieste 34 (2002), 143-153. Zbl 1047.26005, MR 2013947 |
| Reference: | [7] Dow, M. A., Výborný, R.: Elementary proofs of Peano's existence theorem.J. Aust. Math. Soc. 15 (1973), 366-372. Zbl 0272.34002, MR 0335905, 10.1017/S1446788700013276 |
| Reference: | [8] Gardner, C.: Another elementary proof of Peano's existence theorem.Am. Math. Mon. 83 (1976), 556-560. Zbl 0349.34002, MR 0425221, 10.2307/2319357 |
| Reference: | [9] Goodman, G. S.: Subfunctions and the initial-value problem for differential equations satisfying Carathéodory's hypotheses.J. Differ. Equations 7 (1970), 232-242. Zbl 0276.34003, MR 0255880, 10.1016/0022-0396(70)90108-7 |
| Reference: | [10] Henstock, R.: Lectures on the Theory of Integration.Series in Real Analysis 1. World Scientific, Singapore (1988). Zbl 0668.28001, MR 0963249, 10.1142/0510 |
| Reference: | [11] Kawasaki, T.: On Newton integration in vector spaces.Math. Jap. 46 (1997), 85-90. Zbl 0913.46004, MR 1466120 |
| Reference: | [12] Kennedy, H. C.: Is there an elementary proof of Peano's existence theorem for first order differential equations?.Am. Math. Mon. 76 (1969), 1043-1045. MR 1535642, 10.2307/2317137 |
| Reference: | [13] Koliha, J. J.: Metrics, Norms and Integrals: An Introduction to Contemporary Analysis.World Scientific, Hackensack (2008). Zbl 1169.26001, MR 2484181, 10.1142/7090 |
| Reference: | [14] Koliha, J. J.: Mean, meaner, and the meanest mean value theorem.Am. Math. Mon. 116 (2009), 356-361. Zbl 1229.26009, MR 2503322, 10.1080/00029890.2009.11920948 |
| Reference: | [15] Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter.Czech. Math. J. 7 (1957), 418-449. Zbl 0090.30002, MR 0111875, 10.21136/CMJ.1957.100258 |
| Reference: | [16] Leng, N. W., Yee, L. P.: An alternative definition of the Henstock-Kurzweil integral using primitives.N. Z. J. Math. 48 (2018), 121-128. Zbl 1405.26008, MR 3884908, 10.53733/31 |
| Reference: | [17] Mikusińksi, P., Ostaszewski, K.: Embedding Henstock integrable functions into the space of Schwartz distributions.Real Anal. Exchange 14 (1988), 24-29. MR 1051926, 10.2307/44153614 |
| Reference: | [18] Mikusiński, J., Sikorski, R.: The elementary theory of distributions. I.Rozprawy Mat. 12 (1957), 52 pages. Zbl 0078.11101, MR 0094702 |
| Reference: | [19] Peano, G.: Sull' integrabilità delle equazioni differenziali di primo ordine.Atti. Accad. Sci. Torino 21 (1886), 677-685 Italian \99999JFM99999 18.0284.02. |
| Reference: | [20] Peano, G.: Démonstration de l'integrabilité des équations différentielles ordinaires.Math. Ann. 37 (1890), 182-228 French \99999JFM99999 22.0302.01. MR 1510645, 10.1007/BF01200235 |
| Reference: | [21] Perron, O.: Ein neuer Existenzbeweis für die Integrale der Differentialgleichung $y'=f(x,y)$.Math. Ann. 76 (1915), 471-484 German \99999JFM99999 45.0469.01. MR 1511836, 10.1007/BF01458218 |
| Reference: | [22] Stromberg, K. R.: An Introduction to Classical Real Analysis.Wadsworth International Mathematics Series. Wadsworth, Belmont (1981). Zbl 0454.26001, MR 0604364, 10.1090/chel/376 |
| Reference: | [23] Talvila, E.: The distributional Denjoy integral.Real Anal. Exch. 33 (2008), 51-82. Zbl 1154.26011, MR 2402863, 10.14321/realanalexch.33.1.0051 |
| Reference: | [24] Tevy, I.: Une définition de l'intégrale de Lebesgue à l'aide des fonctions primitives.Rev. Roum. Math. Pures Appl. 19 (1974), 1159-1163 French. Zbl 0326.28010, MR 0364575 |
| Reference: | [25] Walter, J.: On elementary proofs of Peano's existence theorem.Am. Math. Mon. 80 (1973), 282-286. Zbl 0275.34003, MR 0316785, 10.2307/2318451 |
| Reference: | [26] Walter, W.: There is an elementary proof of Peano's existence theorem.Am. Math. Mon. 78 (1971), 170-173. Zbl 0207.08401, MR 0276524, 10.2307/2317624 |
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