Generalized periodic solutions of ordinary linear differential equations in the Colombeau algebra
Abstract
It is shown that from the fact that the unique periodic solution of homogeneous system of equations is the trivial one it follows the existence of periodic solutions of nonhomogeneous systems of equations in the Colombeau algebra.
Keywords
generalized ordinary differential equations; periodic solutions; Colombeau algebra
References
2. J.F. Colombeau, Elementary introduction to new generalized functions, Amsterdam, New York, Oxford, North Holland 1985.
3. S.G. Deo, S.G. Pandit, Differential systems involving impulses, Lecture Notes 954 (1982).
4. V. Doležal, Dynamics of linear systems, Praha 1964.
5. T.H. Hildebrandt, On systems of linear differential Stieltjes integral equations, Illinois Jour. of Math. 3 (1959), 352-373.
6. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 17 (1957), 418-449.
7. J. Kurzweil, Linear differential equations with distributions coefficients, Bull. Acad. Polon. Sci. Ser. Math. Phys. 7 (1959), 557-560.
8. A. Lasota, Z. Opial, Sur les solutions periodiqnes des equations differentielles ordinaires, Ann. Polon. Math. 16 (1964), 69-94.
9. J. Ligęza, On generalized periodic solutions of linear differential equations of order n, Ann. Polon. Math. 33 (1977), 209-218.
10. J. Ligęza, Weak solutions of ordinary differential equations, Prace Nauk. Uniwersytetu Śląskiego w Katowicach 842 (1986).
11. J. Ligęza, Generalized solutions of ordinary linear differential equations in the Colombeau algebra, Mathematica Bohemica 2 (1993), 123-146.
12. J. Ligęza, Periodic solutions of ordinary linear differential equations of second order in the Colombeau algebra, Different aspect of differentiability, Integral transforms and special functions, V.4, N. 1-2, (1996), 121-140.
13. J. Ligęza, M. Tvrdy, On linear algebraic equations in the Colombeau algebra, Math. Bohemica (to appear).
14. R. Pfaff, Generalized systems of linear differential equations, Proc. of the Royal Soc. of Edingburgh, S.A. 89 (1981), 1-14.
15. M. Pelant, M. Tvrdy, Linear distributional differential equations in the space of regulated functions, Math. Bohemica 4 (1993), 379-400.
16. J. Person, The Cauchy system for linear distribution differential equations, Functional Ekvac. 30 (1987), 162-168.
17. Š. Schwabik, M. Tvrdy, O. Vejvoda, Differential and integral equations, Praha 1979.
18. L. Schwartz, Sur L'impossibilite' de la multiplication des distributions, C. R. Acad. Sci. Paris 239 (1954), 847-848.
19. K. Skórnik, Hereditarily periodic distributions, Studia Math. 43 (1972), 245-272.
20. Z. Wyderka, Some problems of optimal control for linear systems with measures as coefficients, Systems Science 5, 4 (1979), 425-431.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
