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Keywords:
generalized ordinary differential equations; Cauchy problem; distributions; Colombeau algebra
Summary:
In this paper first order linear ordinary differential equations are considered. It is shown that the Cauchy problem for these systems has a unique solution in $ {\Cal G}^n (\Bbb R) $, where $ {\Cal G} (\Bbb R) $ denotes the Colombeau algebra.
References:
[1] J. F. Colombeau: Elementary Introduction to New Generalized Functions. North Holland, Amsterdam, 1985. MR 0808961 | Zbl 0584.46024
[2] J. F. Colombeau: Multiplication of Distributions. Lecture Notes in Math. 1532, Springer, Berlin, 1992. MR 1222643 | Zbl 0815.35002
[3] S. G. Deo S. G. Pandit: DifferentiaІ Systems Involving Impulses. Lecture Notes in Math. 954, Springer, Berlin, 1982. MR 0674119
[4] V. Doležal: Dynamics of Linear Systems. Academia, Praha, 1967. MR 0220530
[5] Y. Egorov: A theory of generalized functions. Uspehi Math. Nauk 455 (1990), 3-40. (In Russian.) MR 1084986
[6] A. F. Filippov: Differential Equations with Discontinuous Right Part. Nauka, Moscow, 1985. (In Russian.) MR 0790682
[7] I. M. Geľfand G. E. Shilov: Generalized Functions I. Academic Press, NeW York, 1964. MR 0166596
[8] T. H. Hildebrandt: On systems of linєar differential Stieltjes integral equations. Illinois J. Math. 3 (1959), 352-373. MR 0105600
[9] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7 (1957), 418-449. MR 0111875 | Zbl 0090.30002
[10] J. Kurzweil: Linear differential equations with distributions coefficients. Bull. Acad. Polon. Sci. Ser. Math. Phys. 7 (1959), 557-560. MR 0111887
[11] J. Ligęza: On distributional solutions of some systems of linear differential equations. Časopis Pěst. Mat. 102 (1977). 37-41. MR 0460757
[12] J. Ligęza: Weak Solutions of Ordinary Differential Equations. Prace Nauk. Uniw. Śląsk. Katowic. 842, 1986. MR 0868863
[13] J. Ligęza: Generalized solutions of ordinary linear dilferential equations in the Colombeau algebra. Math. Bohem. 118 (1993), 123-146. MR 1223478
[14] M. Oberguggenberger: Hyperbolic systems with discontinuous coefficients: Generalized solutions and a transmission problem in acoustics. J. Math. Anal. Appl. 142 (1989), 452-467. DOI 10.1016/0022-247X(89)90014-0 | MR 1014590 | Zbl 0705.35146
[15] M. Pelant M. Tvrdý: Linear distributional differential equations in the space of regulated functions. Math. Bohem. 118 (1993), 379-400. MR 1251883
[16] J. Persson: The Cauchy system for linear distribution differential equations. Functial Ekvac. 30 (1987), 162-168.
[17] R. Pfaff: Gewöhnliche lineare Differentialgleichungen n-ter Order mit Distributionskoefizienten. Proc. Roy. Soc. Edingburgh, Sect. A. 85 (1980), 291-298. DOI 10.1017/S0308210500011860 | MR 0574022
[18] Š. Schwabik M. Tvrdý O. Vejvoda: Differential and Integral Equations. Academia, Praha, 1979. MR 0542283
[19] E. E. Rosinger: Nonlinear Partial Differential Equations, Sequential and Weak Solutions. Math. Studies 44, North-Holland, 1980. MR 0590891 | Zbl 0447.35001
[20] L. Schwartz: Sur l'impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris Sér. I Math. 239 (1954), 847-848. MR 0064324 | Zbl 0056.10602
[21] Z. Wyderka: Some problems of optimal control for linear systems with measures as coefficients. Systems Sci. 5 (1979), 425-431. MR 0572042 | Zbl 0442.49002
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