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

1. P. Antosik, J. Mikusiński, R. Sikorski, Theory of distributions, The sequential approach, Amsterdam-Warsaw, Elsevier-PWN 1973.
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.
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Published : 1997-09-30


LigęzaJ. (1997). Generalized periodic solutions of ordinary linear differential equations in the Colombeau algebra. Annales Mathematicae Silesianae, 11, 67-87. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14185

Jan Ligęza 
Instytut Matematyki, Uniwersytet Śląski w Katowicach  Poland



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