Averaging for ordinary differential equations perturbed by a small parameter

Mustapha Lakrib; Tahar Kherraz; Amel Bourada

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 2, page 143-151
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Lakrib, Mustapha, Kherraz, Tahar, and Bourada, Amel. "Averaging for ordinary differential equations perturbed by a small parameter." Mathematica Bohemica 141.2 (2016): 143-151. <http://eudml.org/doc/276986>.

@article{Lakrib2016,
abstract = {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.},
author = {Lakrib, Mustapha, Kherraz, Tahar, Bourada, Amel},
journal = {Mathematica Bohemica},
keywords = {ordinary differential equation; method of averaging},
language = {eng},
number = {2},
pages = {143-151},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Averaging for ordinary differential equations perturbed by a small parameter},
url = {http://eudml.org/doc/276986},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Lakrib, Mustapha
AU - Kherraz, Tahar
AU - Bourada, Amel
TI - Averaging for ordinary differential equations perturbed by a small parameter
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 2
SP - 143
EP - 151
AB - 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.
LA - eng
KW - ordinary differential equation; method of averaging
UR - http://eudml.org/doc/276986
ER -

References

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  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. (1955) MR0071596
  5. Lochak, P., Meunier, C., 10.1007/978-1-4612-1044-3, Applied Mathematical Sciences 72 Springer, New York (1988). (1988) Zbl0668.34044MR0959890DOI10.1007/978-1-4612-1044-3
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  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. (2008) Zbl1277.49034MR2515085
  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. (2008) Zbl1277.34053MR2512749
  9. Plotnikova, N. V., 10.1007/s10625-005-0248-5, Differ. Equ. 41 (2005), 1049-1053 Differ. Uravn. 41 997-1000 (2005), Russian. (2005) Zbl1109.34306MR2201995DOI10.1007/s10625-005-0248-5
  10. Sanders, J. A., Verhulst, F., Murdock, J., Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences 59 Springer, New York (2007). (2007) Zbl1128.34001MR2316999
  11. Verhulst, F., 10.1007/0-387-28313-7, Texts in Applied Math. 50 Springer, New York (2005). (2005) Zbl1148.35006MR2148856DOI10.1007/0-387-28313-7
  12. Verhulst, F., Nonlinear Differential Equations and Dynamical Systems, Universitext Springer, Berlin (1996). (1996) Zbl0854.34002MR1422255

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