Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Summary:
We give characterizations of the distributional derivatives $D^{1,1}$, $D^{1,0}$, $D^{0,1}$ of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.
References:
[1] P. Antosik, J. Mikusiński, R. Sikorski: Theory of Distributions—the Sequential Aproach. Elsevier Scientific Publishing Company, Amsterdam, Polish Scientific Publishers, Warsaw, 1973. MR 0365130
[2] J.A. Clarkson, C.R. Adams: On definitions of bounded variation for functions of two variables. Trans. Amer. Math. Soc. 35 (1933), 824–854. DOI 10.1090/S0002-9947-1933-1501718-2 | MR 1501718
[3] A. Halanay, D. Wexler: Qualitative Theory of Impulsive Systems. Moscow, 1971. (Russian)
[4] I. Halparin: Introduction to the Theory of Distributions. University of Toronto Press, Toronto, 1952. (Russian)
[5] T.H. Hildebrandt: Introduction to the Theory of Integration. Academic Press, New York-London, 1963. MR 0154957 | Zbl 0112.28302
[6] D. Idczak: Functions of several variables of finite variation and their differentiability. Annales Pol. Math. 60 (1994), no. 1, 47–56. DOI 10.4064/ap-60-1-47-56 | MR 1295107 | Zbl 0886.26007
[7] J. Kurzweil: Generalized ordinary differential equations. Czechoslovak Math. J. 8 (1958), no. 83, 360–388. MR 0111878 | Zbl 0102.07003
[8] J. Kurzweil: On generalized ordinary differential equations with discontinuous solutions. Prikl. Mat. Mech. 22 (1958), 27–45. MR 0111876
[9] J. Kurzweil: Linear differential equations with distributions as coefficients. Bull. Acad. Pol. Sci., ser. Math. Astr. Phys. 7 (1959), 557–560. MR 0111887 | Zbl 0117.34401
[10] V. Lakshmikantham: Trends in the theory of impulsive differential equations. Proceedings of the International Conference on Theory and Applications of Differential Equations, Ohio University, 1988. MR 1026202
[11] A. Lasota: Remarks on linear differential equations with distributional perturbations. Ordinary differential equations (Proc. NRL–MRC Conf. Math. Res. Center, Naval Res. Lab., Washington, D.C., 1971), New York, Academic Press, 1972, pp. 489–495. MR 0473282 | Zbl 0309.34007
[12] J. Ligeza: On distributional solutions of some systems of linear differential equations. Čas. pro Pěst. Mat. 102 (1977), 37–41. MR 0460757 | Zbl 0345.34003
[13] S. Łojasiewicz: An Introduction to the Theory of Real Functions. John Willey and Sons, Chichester, 1988.
[14] R. Pfaff: Generalized systems of linear differential equations. Proc. of the Royal Sci. of Edinburgh, sect. A 89 (1981), 1–14. MR 0628123 | Zbl 0475.34005
[15] W. W. Schmaedeke: Optimal control theory for nonlinear vector-differential equations containing measures. SIAM Control 3 (1965), no. 2, 231–280. MR 0189870 | Zbl 0161.29203
[16] L. Schwartz: Théorie des Distributions I. Paris, 1950.
[17] R. Sikorski: Funkcje Rzeczywiste I. PWN, Warszawa, 1958. MR 0105468
[18] S. Walczak: Absolutely continuous functions of several variables and their application to differential equations. Bull. Polish Acad. Sci. Math. 35 (1987), no. 11–12, 733–744. MR 0961712 | Zbl 0691.35029
[19] Z. Wyderka: Linear Differential Equations with Measures as Coefficients and Control Theory. Siles. Univ. Press, Katowice, 1994. MR 1292252 | Zbl 0813.34058
[20] Z. Wyderka: Linear differential equations with measures as coefficients and the control theory. Čas. pro Pěst. Mat. 114 (1989), no. 1, 13–27. MR 0990112 | Zbl 0664.34013
Search
Partner of
