| Title:
|
Some remarks on Nagumo's theorem (English) |
| Author:
|
Mejstrik, Thomas |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
235-242 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
ordinary differential equation |
| Keyword:
|
uniqueness |
| MSC:
|
34A12 |
| MSC:
|
34A34 |
| MSC:
|
34A45 |
| idZBL:
|
Zbl 1249.34021 |
| idMR:
|
MR2899747 |
| DOI:
|
10.1007/s10587-012-0008-7 |
| . |
| Date available:
|
2012-03-05T07:27:52Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142053 |
| . |
| Reference:
|
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| Reference:
|
[2] Athanassov, Z. S.: Uniqueness and convergence of successive approximations for ordinary differential equations.Math. Jap. 35 (1990), 351-367. Zbl 0709.34002, MR 1049101 |
| Reference:
|
[3] Bellman, R.: Stability Theory of Differential Equations.New York-London: McGraw-Hill Book Company (1953), (166). MR 0061235 |
| Reference:
|
[4] Constantin, A.: Solutions globales d'équations différentielles perturbées.C. R. Acad. Sci., Paris, Sér. I 320 (1995), 1319-1322. Zbl 0839.34002, MR 1338279 |
| Reference:
|
[5] Constantin, A.: On Nagumo's theorem.Proc. Japan Acad., Ser. A 86 (2010), 41-44. Zbl 1192.34014, MR 2590189 |
| Reference:
|
[6] Coppel, W. A.: Stability and Asymptotic Behavior of Differential Equations.Boston: D. C. Heath and Company (1965), 166. Zbl 0154.09301, MR 0190463 |
| Reference:
|
[7] Lipovan, O.: A retarded Gronwall-like inequality and its applications.J. Math. Anal. Appl. 252 (2000), 389-401. Zbl 0974.26007, MR 1797863, 10.1006/jmaa.2000.7085 |
| Reference:
|
[8] Nagumo, M.: Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung.Japanese Journ. of Math. 3 (1926), 107-112. 10.4099/jjm1924.3.0_107 |
| Reference:
|
[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. MR 2449681 |
| Reference:
|
[10] Sonoc, C.: On the pathwise uniqueness of solutions of stochastic differential equations.Port. Math. 55 (1998), 451-456. Zbl 0932.60067, MR 1672118 |
| . |
