Remarks on generalized solutions of some ordinary nonlinear differential equations of second order in the Colombeau algebra
Abstract
In this article some equations of second order are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that the equations with additional conditions admit unique global solutions in the Colombeau algebra 𝓖(ℝ1).
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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. T. Dłotko, Aplication of the notation of rotation of a vector field in the theory of differential equations and their generalizations (in Polish), Prace Naukowe U.Śl. w Katowicach, 32 (1971).
5. V. Doležal, Dynamics of linear systems, Praha 1964.
6. T.H. Hildebrandt, On systems of linear differential Stieltjes integral equations, Illinois Jour. of Math. 3 (1959), 352-373.
7. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. Jour. 7 (1957), 418-447.
8. J. Kurzweil, Linear differential equations with distributions as coefficients, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 7 (1959), 557-560.
9. A. Lasota, J. Trapie, Nicoletti boundary value problem for system of linear differential equations with distributional perturbations, Prace Matematyczne UJ, Kraków 15 (1971), 103-108.
10. J. Ligęza, Weak solutions of ordinary differential equations, Prace Naukowe U. Śl. w Katowicach, 842 (1986).
11. J. Ligęza, Generalized solutions of ordinary linear differential equations in the Colombeau algebra, Math. Bohemica, 2 (1993), 123-146.
12. J. Ligęza, Generalized solutions of boundary value problems for ordinary linear differential equations of second order in the Colombeau algebra. Different aspect of differentiability, Dissertationes Mathematicae 340 (1995), 183-194.
13. R. Pfaff, Generalized systems of linear differential equations, Proc. of the Royal Soc. of Edingburgh, S. A. 89 (1981), 1-14.
14. M. Pelant, M. Tvrdý, Linear distributional differential equations in the space of regulated functions, Math. Bohemica, 4 (1993), 379-400.
15. J. Persson, The Cauchy system for linear distribution differential equations, Functial. Ekvac. 30 (1987), 162-168.
16. Š. Schwabik, M. Tvrdý, O. Vejvoda, Differential and integral equations, Praha 1979.
17. L. Schwartz, Sur l'impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris 239 (1954), 847-848.
18. Z. Wyderka, Some problems of optimal control for linear systems with measures as coefficient, Systems Science 5, 4 (1979), 425-431.
LigęzaJ. (1996). Remarks on generalized solutions of some ordinary nonlinear differential equations of second order in the Colombeau algebra. Annales Mathematicae Silesianae, 10, 87-101. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14196
Jan Ligęza
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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