Some remarks on Nagumo's theorem

Thomas Mejstrik

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 1, page 235-242
  • ISSN: 0011-4642

Abstract

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We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation x ' = f ( t , x ) , x ( 0 ) = 0 and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.

How to cite

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Mejstrik, Thomas. "Some remarks on Nagumo's theorem." Czechoslovak Mathematical Journal 62.1 (2012): 235-242. <http://eudml.org/doc/246686>.

@article{Mejstrik2012,
abstract = {We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation $x^\{\prime \}=f(t,x)$, $ x(0)=0$ and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.},
author = {Mejstrik, Thomas},
journal = {Czechoslovak Mathematical Journal},
keywords = {ordinary differential equation; uniqueness; ordinary differential equation; Picard successive approximations; Nagumo's uniqueness theorem},
language = {eng},
number = {1},
pages = {235-242},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some remarks on Nagumo's theorem},
url = {http://eudml.org/doc/246686},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Mejstrik, Thomas
TI - Some remarks on Nagumo's theorem
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 235
EP - 242
AB - We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation $x^{\prime }=f(t,x)$, $ x(0)=0$ and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.
LA - eng
KW - ordinary differential equation; uniqueness; ordinary differential equation; Picard successive approximations; Nagumo's uniqueness theorem
UR - http://eudml.org/doc/246686
ER -

References

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  1. Agarwal, R. P., Heikkilä, S., 10.1023/A:1015658621390, Acta Math. Hung. 94 (2002), 67-92. (2002) Zbl1006.34002MR1905788DOI10.1023/A:1015658621390
  2. Athanassov, Z. S., Uniqueness and convergence of successive approximations for ordinary differential equations, Math. Jap. 35 (1990), 351-367. (1990) Zbl0709.34002MR1049101
  3. Bellman, R., Stability Theory of Differential Equations, New York-London: McGraw-Hill Book Company (1953), (166). (166) MR0061235
  4. Constantin, A., Solutions globales d'équations différentielles perturbées, C. R. Acad. Sci., Paris, Sér. I 320 (1995), 1319-1322. (1995) Zbl0839.34002MR1338279
  5. Constantin, A., On Nagumo's theorem, Proc. Japan Acad., Ser. A 86 (2010), 41-44. (2010) Zbl1192.34014MR2590189
  6. Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations, Boston: D. C. Heath and Company (1965), 166. (1965) Zbl0154.09301MR0190463
  7. Lipovan, O., 10.1006/jmaa.2000.7085, J. Math. Anal. Appl. 252 (2000), 389-401. (2000) Zbl0974.26007MR1797863DOI10.1006/jmaa.2000.7085
  8. Nagumo, M., 10.4099/jjm1924.3.0_107, Japanese Journ. of Math. 3 (1926), 107-112. (1926) DOI10.4099/jjm1924.3.0_107
  9. Negrea, R., On a class of backward stochastic differential equations and applications to the stochastic resonance, "Recent advances in stochastic modeling and data analysis", pp. 26-33, World Sci. Publ., Hackensack, NJ, 2007. MR2449681
  10. Sonoc, C., On the pathwise uniqueness of solutions of stochastic differential equations, Port. Math. 55 (1998), 451-456. (1998) Zbl0932.60067MR1672118

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