Geodesic
Generalized solutions of ordinary linear differential equations in the Colombeau algebra
Mathematica Bohemica, Tome 118 (1993) no. 2, pp. 123-146
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\Cal L(\bold R^1)$.
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 $\Cal L(\bold R^1)$.
DOI : 10.21136/MB.1993.126054
Classification : 34A10, 34A12, 34A30, 34G10, 46F05, 46F99, 46N20
Keywords: linear Cauchy problem; Colombeau algebra of generalized distributions; existence; uniqueness; generalized ordinary differential equation; Cauchy problem generalized function; distribution 
@article{10_21136_MB_1993_126054,
     author = {Lig\k{e}za, Jan},
     title = {Generalized solutions of ordinary linear differential equations in the {Colombeau} algebra},
     journal = {Mathematica Bohemica},
     pages = {123--146},
     year = {1993},
     volume = {118},
     number = {2},
     doi = {10.21136/MB.1993.126054},
     mrnumber = {1223478},
     zbl = {0776.34046},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1993.126054/}
}
TY  - JOUR
AU  - Ligęza, Jan
TI  - Generalized solutions of ordinary linear differential equations in the Colombeau algebra
JO  - Mathematica Bohemica
PY  - 1993
SP  - 123
EP  - 146
VL  - 118
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1993.126054/
DO  - 10.21136/MB.1993.126054
LA  - en
ID  - 10_21136_MB_1993_126054
ER  - 
%0 Journal Article
%A Ligęza, Jan
%T Generalized solutions of ordinary linear differential equations in the Colombeau algebra
%J Mathematica Bohemica
%D 1993
%P 123-146
%V 118
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1993.126054/
%R 10.21136/MB.1993.126054
%G en
%F 10_21136_MB_1993_126054
Ligęza, Jan. Generalized solutions of ordinary linear differential equations in the Colombeau algebra. Mathematica Bohemica, Tome 118 (1993) no. 2, pp. 123-146. doi: 10.21136/MB.1993.126054

[1] P. Antosik: On the modulus of a distribution. Bull. Acad. Polon. Sci. Ser. Sci. Mat. Astronom. Phys. 15 (1967), 717-722. | MR | Zbl

[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

[3] P. Antosik J. Mikusinski R. Sikorski: Theory of Distributions. The sequential approach. Elsevier-PWN, Amsterodam-Warsaw. | MR

[4] J. F. Colombeau: Elementary Introduction to New Generalized Functions. North Holland, Amsterodam, New York, Oxford, 1985. | MR | Zbl

[5] J. F. Colombeau: A multiplication of distribution. Jour. of Math. Anal. and Appl. 94 (1983), 96-115. | DOI | MR

[6] S. G. Deo S. G. Pandit: Differential Systems Involving Impulses. vol. 954, Lecture Notes, 1982. | MR

[7] V. Doležal: Dynamics of Linear Systems. Praha, 1964. | MR

[8] T. H. Hildebrandt: On systems of linear differential Stieltjes integral equations. Illinois Jour. of Math. 3 (1959), 352-373. | DOI | MR

[9] J. Kurzweil: Generalized ordinary differential equations. Czechoslov. Math. Jour. 8 (1958), 360-389. | MR | Zbl

[10] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslov. Math. Jour. 7(1957), 418-449. | MR | Zbl

[11] J. Kurzweil: Linear differential equations with distributions as coefficients. Bull. Acad. Polon. Sci. Ser. Sci. Mat. Astronom. Phys. 7 (1959), 557-580. | MR | Zbl

[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

[13] J. Ligęza: On distributional solutions of some systems of linear differential equations. Časopis pro pěstování mat. 102 (1977), 37-41. | MR

[14] J. Ligęza: Weak solutions of ordinary differential equations. Prace Naukowe Uniwersytetu Slaskiego, Katowice 842 (1986). | MR

[15] J. Persson: The Cauchy problem for linear distribution differential equations. Funkcial. Ekvac. 30 (1987), 163-168. | MR | Zbl

[16] R. Pfaff: Generalized systems of linear differential equations. Proc. of the Royal Soc. of Edingburh, S. A. 89(1981), 1-14. | MR | Zbl

[17] Š. Schwabik M. Tvrdý O. Vejvoda: Differential and Integral Equations. Praha, 1979. | MR

[18] Š. Schwabik: Generalized differential equations. Rozp. Česk. Akad. Věd. 956(1985). | Zbl

[19] L. Schwartz: Sur l'impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239 (1954), 847-848. | MR | Zbl

Cité par Sources :