Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation

Dariusz Idczak

Czechoslovak Mathematical Journal (1998)

  • Volume: 48, Issue: 1, page 145-171
  • ISSN: 0011-4642

Abstract

top
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.

How to cite

top

Idczak, Dariusz. "Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation." Czechoslovak Mathematical Journal 48.1 (1998): 145-171. <http://eudml.org/doc/30409>.

@article{Idczak1998,
abstract = {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.},
author = {Idczak, Dariusz},
journal = {Czechoslovak Mathematical Journal},
keywords = {distributional derivatives; locally finite variation; impulsive hyperbolic equation},
language = {eng},
number = {1},
pages = {145-171},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation},
url = {http://eudml.org/doc/30409},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Idczak, Dariusz
TI - Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 1
SP - 145
EP - 171
AB - 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.
LA - eng
KW - distributional derivatives; locally finite variation; impulsive hyperbolic equation
UR - http://eudml.org/doc/30409
ER -

References

top
  1. Theory of Distributions—the Sequential Aproach, Elsevier Scientific Publishing Company, Amsterdam, Polish Scientific Publishers, Warsaw, 1973. (1973) MR0365130
  2. 10.1090/S0002-9947-1933-1501718-2, Trans. Amer. Math. Soc. 35 (1933), 824–854. (1933) MR1501718DOI10.1090/S0002-9947-1933-1501718-2
  3. Qualitative Theory of Impulsive Systems, Moscow, 1971. (Russian) (1971) 
  4. Introduction to the Theory of Distributions, University of Toronto Press, Toronto, 1952. (Russian) (1952) 
  5. Introduction to the Theory of Integration, Academic Press, New York-London, 1963. (1963) Zbl0112.28302MR0154957
  6. 10.4064/ap-60-1-47-56, Annales Pol. Math. 60 (1994), no. 1, 47–56. (1994) Zbl0886.26007MR1295107DOI10.4064/ap-60-1-47-56
  7. Generalized ordinary differential equations, Czechoslovak Math. J. 8 (1958), no. 83, 360–388. (1958) Zbl0102.07003MR0111878
  8. On generalized ordinary differential equations with discontinuous solutions, Prikl. Mat. Mech. 22 (1958), 27–45. (1958) MR0111876
  9. Linear differential equations with distributions as coefficients, Bull. Acad. Pol. Sci., ser. Math. Astr. Phys. 7 (1959), 557–560. (1959) Zbl0117.34401MR0111887
  10. Trends in the theory of impulsive differential equations, Proceedings of the International Conference on Theory and Applications of Differential Equations, Ohio University, 1988. (1988) MR1026202
  11. 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. (1972) Zbl0309.34007MR0473282
  12. On distributional solutions of some systems of linear differential equations, Čas. pro Pěst. Mat. 102 (1977), 37–41. (1977) Zbl0345.34003MR0460757
  13. An Introduction to the Theory of Real Functions, John Willey and Sons, Chichester, 1988. (1988) 
  14. Generalized systems of linear differential equations, Proc. of the Royal Sci. of Edinburgh, sect. A 89 (1981), 1–14. (1981) Zbl0475.34005MR0628123
  15. Optimal control theory for nonlinear vector-differential equations containing measures, SIAM Control 3 (1965), no. 2, 231–280. (1965) Zbl0161.29203MR0189870
  16. Théorie des Distributions I, Paris, 1950. (1950) 
  17. Funkcje Rzeczywiste I, PWN, Warszawa, 1958. (1958) MR0105468
  18. Absolutely continuous functions of several variables and their application to differential equations, Bull. Polish Acad. Sci. Math. 35 (1987), no. 11–12, 733–744. (1987) Zbl0691.35029MR0961712
  19. Linear Differential Equations with Measures as Coefficients and Control Theory, Siles. Univ. Press, Katowice, 1994. (1994) Zbl0813.34058MR1292252
  20. Linear differential equations with measures as coefficients and the control theory, Čas. pro Pěst. Mat. 114 (1989), no. 1, 13–27. (1989) Zbl0664.34013MR0990112

NotesEmbed ?

top

You must be logged in to post comments.