The distributional solutions of some systems of linear differential equations
Abstract
In this paper we prove a theorem on the existence and uniqueness of the solution of the Cauchy problem for linear differential equations with distributional coefficients.
References
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12. T.H. Hildebrandt, On systems of linear differential Stieltjes - inteqral equations, Illinois J. Math. 3 (1959), 352-373.
13. И.T. Kигypaдзe [I.T. Kiguradze], Нeкoтopыe cингyляpныe кpaeвыe зaдaчи для oбыкнoвeнных диффepeнциaльных ypaвнeний, Изд. Tбилиccкoгo Унивepcитeтa 1975.
14. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Čechoslovak Math. J. 7 (1957), 418-449.
15. J. Kurzweil, Generalized ordinary differential equations, Čechoslovak Math. J. 8 (1958), 360-389.
16. J. Kurzweil, Linear differential equations with distributions as coefficients, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 7 (1959), 557-560.
17. A. Lasota, Applications of generalized functions to contingent equations and control theory, In: Seminar Lectures of the University of Maryland 1971 - 1972, Maryland, 41-53.
18. A. Lasota, F.H. Szafraniec, Application of differential equations with distributional coefficients to optimal control theory, In: Prace Matematyczne 12. Kraków UJ 1968, 31-37.
19. A. Lasota, J. Traple, Nicoletti boundary value problem for systems of linear differential equations with distributional perturbations, In: Prace Matematyczne 15. Kraków UJ 1971, 103-108.
20. J. Ligęza, On generalized periodic solutions of some linear differential equations of order n, Ann. Polon. Math. 33 (1977), 210-218.
21. J. Ligęza, On distributional solutions of some systems of linear differential equations, Časopis Pěst. Mat. 102 (1977), 30-36.
22. J. Ligęza, On an integral inequality, Prace Nauk. Uniw. Śląsk., Prace Matematyczne 7 (1977), 22-25.
23. A.Д. Mышкиc, A.M. Caмoйлeнкo [A.D. Myshkis, A.M. Samolenko], Cиcтeмa c тoлчкaми в зaдaныe мoмeнты вpeмeнии, Maт. cбopник 6: 2 1967, 202-208.
24. R. Pfaff, Gewöhnliche lineare Differentialgleichungen n-ter Ordnung mit Distributionskoefizienten, Proc. Edingburgh Math. Soc. 85 (1980), 291-298.
25. R. Pfaff, Generalized systems of linear differential equations, Proc. Edingburgh Math. Soc. 89 (1981), 1-14.
26. Š. Schwabik, M. Tvrdy, O. Vejvoda, Differential and integral equations, Praha, 1979.
27. J. Traple, Boundary value problem for linear differential equations with distributional perturbations, In: Prace Matematyczne 17. Kraków UJ 1975, 119-126.
28. Z. Wyderka, On the linear concave convex differential open - loop game with measures as coefficients, Prace Nauk. Uniw. Śląsk., Prace Matematyczne 8 (1978), 88-92.
29. Z. Wyderka, Some optimal control problem for linear systems, Systems Sci. 4 (1978), 53-59.
30. Z. Wyderka, Some problems of optimal control for linear systems with measures as coefficients, Systems Sci. 5 (1979), 425-431.
31. Z. Wyderka, On the attainable set for linear systems with measures as coefficients, Systems Sci. 6 (1980), 385-390.
2. P. Antosik, On the modulus of a distribution, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 717-722.
3. P. Antosik, The symmetric value of a distribution at a point, In: Proceedings of the Scientific Research Centre in Zabrze, Polish Academy of Sciences, Institute of Mathematics, Warsaw, Polish Academy of Sciences, 1973.
4. P. Antosik, J. Ligęza, Products of measures and functions of finite variation, In: Proceedings of the conference on generalized functions and operational calculus, Varna 1975. Sofia 1979, 20-26.
5. P. Antosik, J. Mikusiński, R. Sikorski, Theory of distributions. The sequential approach, Amsterdam - Warsaw 1973.
6. G. Buttazzo, D. Percivale, On the approximation of the elastic bounce problem on Riemannian manifolds, J. Differential Equations 47 (1983), 227-245.
7. M. Carie, E. Pascali, Il problemma del rimbalzo unidemensionale e sul approssimazioni con penalizzazioni non convesse, Rend. Mat. 13 (1980), 541-553.
8. T. Dłotko, J. Klamka, J. Ligęza, Controllability and minimum energy control of linear systems, Podstawy sterowania 6 (1976), 211-217.
9. V. Doležal, Dynamics of linear systems, Praha 1964.
10. И.M. Гeльфaнд, Г. Шилoв [I.M. Gelfand, G.J. Shilov], Oбoбщённыe фyнкции и дeйcтвия нaд ними, T.I., Mocквa 1958.
11. A. Хaльaнaй, Д. Вeкcлep [A. Halanay, D. Wexler], Кaчecтвeннaя тeopия импyльcных cиcтeм, Mocквa 1971.
12. T.H. Hildebrandt, On systems of linear differential Stieltjes - inteqral equations, Illinois J. Math. 3 (1959), 352-373.
13. И.T. Kигypaдзe [I.T. Kiguradze], Нeкoтopыe cингyляpныe кpaeвыe зaдaчи для oбыкнoвeнных диффepeнциaльных ypaвнeний, Изд. Tбилиccкoгo Унивepcитeтa 1975.
14. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Čechoslovak Math. J. 7 (1957), 418-449.
15. J. Kurzweil, Generalized ordinary differential equations, Čechoslovak Math. J. 8 (1958), 360-389.
16. J. Kurzweil, Linear differential equations with distributions as coefficients, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 7 (1959), 557-560.
17. A. Lasota, Applications of generalized functions to contingent equations and control theory, In: Seminar Lectures of the University of Maryland 1971 - 1972, Maryland, 41-53.
18. A. Lasota, F.H. Szafraniec, Application of differential equations with distributional coefficients to optimal control theory, In: Prace Matematyczne 12. Kraków UJ 1968, 31-37.
19. A. Lasota, J. Traple, Nicoletti boundary value problem for systems of linear differential equations with distributional perturbations, In: Prace Matematyczne 15. Kraków UJ 1971, 103-108.
20. J. Ligęza, On generalized periodic solutions of some linear differential equations of order n, Ann. Polon. Math. 33 (1977), 210-218.
21. J. Ligęza, On distributional solutions of some systems of linear differential equations, Časopis Pěst. Mat. 102 (1977), 30-36.
22. J. Ligęza, On an integral inequality, Prace Nauk. Uniw. Śląsk., Prace Matematyczne 7 (1977), 22-25.
23. A.Д. Mышкиc, A.M. Caмoйлeнкo [A.D. Myshkis, A.M. Samolenko], Cиcтeмa c тoлчкaми в зaдaныe мoмeнты вpeмeнии, Maт. cбopник 6: 2 1967, 202-208.
24. R. Pfaff, Gewöhnliche lineare Differentialgleichungen n-ter Ordnung mit Distributionskoefizienten, Proc. Edingburgh Math. Soc. 85 (1980), 291-298.
25. R. Pfaff, Generalized systems of linear differential equations, Proc. Edingburgh Math. Soc. 89 (1981), 1-14.
26. Š. Schwabik, M. Tvrdy, O. Vejvoda, Differential and integral equations, Praha, 1979.
27. J. Traple, Boundary value problem for linear differential equations with distributional perturbations, In: Prace Matematyczne 17. Kraków UJ 1975, 119-126.
28. Z. Wyderka, On the linear concave convex differential open - loop game with measures as coefficients, Prace Nauk. Uniw. Śląsk., Prace Matematyczne 8 (1978), 88-92.
29. Z. Wyderka, Some optimal control problem for linear systems, Systems Sci. 4 (1978), 53-59.
30. Z. Wyderka, Some problems of optimal control for linear systems with measures as coefficients, Systems Sci. 5 (1979), 425-431.
31. Z. Wyderka, On the attainable set for linear systems with measures as coefficients, Systems Sci. 6 (1980), 385-390.
LigęzaJ. (1990). The distributional solutions of some systems of linear differential equations. Annales Mathematicae Silesianae, 3, 82-90. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14306
Jan Ligęza
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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