| Title:
|
Averaging for ordinary differential equations perturbed by a small parameter (English) |
| Author:
|
Lakrib, Mustapha |
| Author:
|
Kherraz, Tahar |
| Author:
|
Bourada, Amel |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
143-151 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we prove and discuss averaging results for ordinary differential equations perturbed by a small parameter. The conditions we assume on the right-hand sides of the equations under which our averaging results are stated are more general than those considered in the literature. Indeed, often it is assumed that the right-hand sides of the equations are uniformly bounded and a Lipschitz condition is imposed on them. Sometimes this last condition is relaxed to the uniform continuity in the second variable uniformly with respect to the first one. In our results, we assume only that the right-hand sides of the equations are bounded by some locally Lebesgue integrable functions with the property that their indefinite integrals satisfy a Lipschitz-type condition. Also, we consider that they are only continuous in the second variable uniformly with respect to the first one. (English) |
| Keyword:
|
ordinary differential equation |
| Keyword:
|
method of averaging |
| MSC:
|
34C15 |
| MSC:
|
34C29 |
| MSC:
|
34K25 |
| idZBL:
|
Zbl 06587859 |
| idMR:
|
MR3499781 |
| DOI:
|
10.21136/MB.2016.12 |
| . |
| Date available:
|
2016-05-19T09:03:13Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145709 |
| . |
| Reference:
|
[1] Arnold, V. I.: Geometrical Methods in the Theory of Ordinary Differential Equations.Fundamental Principles of Mathematical Sciences 250 Springer, New York (1988). MR 0947141 |
| Reference:
|
[2] Artstein, Z.: Averaging of time-varying differential equations revisited.J. Differ. Equations 243 (2007), 146-167. Zbl 1135.34031, MR 2371783, 10.1016/j.jde.2007.01.022 |
| Reference:
|
[3] Gama, R., Smirnov, G.: Stability and optimality of solutions to differential inclusions via averaging method.Set-Valued Var. Anal. 22 (2014), 349-374. Zbl 1307.34001, MR 3207744 |
| Reference:
|
[4] Krasnosel'skiĭ, M. A., Kreĭn, S. G.: On the principle of averaging in nonlinear mechanics.Usp. Mat. Nauk, N.S. 10 Russian (1955), 147-152. MR 0071596 |
| Reference:
|
[5] Lochak, P., Meunier, C.: Multiphase Averaging for Classical Systems. With Applications to Adiabatic Theorems.Applied Mathematical Sciences 72 Springer, New York (1988). Zbl 0668.34044, MR 0959890, 10.1007/978-1-4612-1044-3 |
| Reference:
|
[6] Mesquita, J. G., Slavík, A.: Periodic averaging theorems for various types of equations.J. Math. Anal. Appl. 387 (2012), 862-877. Zbl 1243.34060, MR 2853180, 10.1016/j.jmaa.2011.09.038 |
| Reference:
|
[7] Nosenko, T. V., Stanzhyts'kyi, O. M.: Averaging method in some problems of optimal control.Nelinijni Kolyvannya 11 (2008), 512-519 Ukrainian translated in Nonlinear Oscil., N.Y. (electronic only) 11 (2008), 539-547. Zbl 1277.49034, MR 2515085 |
| Reference:
|
[8] Plotnikov, V. A., Romanyuk, A. V.: Averaging of differential equations with Perron-integrable right-hand side.Nelinijni Kolyvannya 11 (2008), 387-395 Russian translated in Nonlinear Oscil., N.Y. (electronic only) 11 (2008), 407-415. Zbl 1277.34053, MR 2512749 |
| Reference:
|
[9] Plotnikova, N. V.: The Krasnosel'skii-Krein theorem for differential inclusions.Differ. Equ. 41 (2005), 1049-1053 Differ. Uravn. 41 997-1000 (2005), Russian. Zbl 1109.34306, MR 2201995, 10.1007/s10625-005-0248-5 |
| Reference:
|
[10] Sanders, J. A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems.Applied Mathematical Sciences 59 Springer, New York (2007). Zbl 1128.34001, MR 2316999 |
| Reference:
|
[11] Verhulst, F.: Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics.Texts in Applied Math. 50 Springer, New York (2005). Zbl 1148.35006, MR 2148856, 10.1007/0-387-28313-7 |
| Reference:
|
[12] Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems.Universitext Springer, Berlin (1996). Zbl 0854.34002, MR 1422255 |
| . |
