Previous |  Up |  Next

Article

Keywords:
linear Cauchy problem; Colombeau algebra of generalized distributions; existence; uniqueness; generalized ordinary differential equation; Cauchy problem generalized function; distribution
Summary:
In this paper first order systems of linear of ODEs are considered. It is shown that these systems admit unique solutions in the Colombeau algebra $\Cal L(\bold R^1)$.
References:
[1] P. Antosik: On the modulus of a distribution. Bull. Acad. Polon. Sci. Ser. Sci. Mat. Astronom. Phys. 15 (1967), 717-722. MR 0512036 | Zbl 0173.41703
[2] P. Antosik J. Ligęza: Products of measures and functions of finite variation. In: Proc. of the conference on generalized functions and operational calculus, Varna 1975, Sofia 1979, 20-26. MR 0547326
[3] P. Antosik J. Mikusinski R. Sikorski: Theory of Distributions. The sequential approach. Elsevier-PWN, Amsterodam-Warsaw. MR 0365130
[4] J. F. Colombeau: Elementary Introduction to New Generalized Functions. North Holland, Amsterodam, New York, Oxford, 1985. MR 0808961 | Zbl 0584.46024
[5] J. F. Colombeau: A multiplication of distribution. Jour. of Math. Anal. and Appl. 94 (1983), 96-115. DOI 10.1016/0022-247X(83)90007-0 | MR 0701451
[6] S. G. Deo S. G. Pandit: Differential Systems Involving Impulses. vol. 954, Lecture Notes, 1982. MR 0674119
[7] V. Doležal: Dynamics of Linear Systems. Praha, 1964. MR 0169744
[8] T. H. Hildebrandt: On systems of linear differential Stieltjes integral equations. Illinois Jour. of Math. 3 (1959), 352-373. MR 0105600
[9] J. Kurzweil: Generalized ordinary differential equations. Czechoslov. Math. Jour. 8 (1958), 360-389. MR 0111878 | Zbl 0102.07003
[10] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslov. Math. Jour. 7(1957), 418-449. MR 0111875 | Zbl 0090.30002
[11] J. Kurzweil: Linear differential equations with distributions as coefficients. Bull. Acad. Polon. Sci. Ser. Sci. Mat. Astronom. Phys. 7 (1959), 557-580. MR 0111887 | Zbl 0117.34401
[12] A. Lasota J. Traple: Nicoletti boundary value problem for system of linear differential equations with distributional perturbations. Prace Mathematyczne Uniwersytetu Jagiellonskiego, Kraków 15(1971), 103-108. MR 0369785
[13] J. Ligęza: On distributional solutions of some systems of linear differential equations. Časopis pro pěstování mat. 102 (1977), 37-41. MR 0460757
[14] J. Ligęza: Weak solutions of ordinary differential equations. Prace Naukowe Uniwersytetu Slaskiego, Katowice 842 (1986). MR 0868863
[15] J. Persson: The Cauchy problem for linear distribution differential equations. Funkcial. Ekvac. 30 (1987), 163-168. MR 0915270 | Zbl 0643.34004
[16] R. Pfaff: Generalized systems of linear differential equations. Proc. of the Royal Soc. of Edingburh, S. A. 89(1981), 1-14. DOI 10.1017/S0308210500032303 | MR 0628123 | Zbl 0475.34005
[17] Š. Schwabik M. Tvrdý O. Vejvoda: Differential and Integral Equations. Praha, 1979. MR 0542283
[18] Š. Schwabik: Generalized differential equations. Rozp. Česk. Akad. Věd. 956(1985). Zbl 0594.34002
[19] L. Schwartz: Sur l'impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239 (1954), 847-848. MR 0064324 | Zbl 0056.10602
Partner of
EuDML logo