On associated solutions of the system of non-autonomous differential equations in the Lebesgue spaces

  • Anastasia I. Zhuk Brest State Technical University, 267 Maskoŭskaja Street, Brest 224023, Belarus https://orcid.org/0000-0002-6612-9582
  • Helena N. Zashchuk Brest State Technical University, 267 Maskoŭskaja Street, Brest 224023, Belarus
DOI: https://doi.org/10.33581/2520-6508-2022-1-6-13

Abstract

Herein, we investigate systems of non-autonomous differential equations with generalised coefficients using the algebra of new generalised functions. We consider a system of non-autonomous differential equations with generalised coefficients as a system of equations in differentials in the algebra of new generalised functions. The solution of such a system is a new generalised function. It is shown that the different interpretations of the solutions of the given systems can be described by a unique approach of the algebra of new generalised functions. In this paper, for the first time in the literature, we describe associated solutions of the system of non-autonomous differential equations with generalised coefficients in the Lebesgue spaces Lp(T).

Author Biographies

Anastasia I. Zhuk, Brest State Technical University, 267 Maskoŭskaja Street, Brest 224023, Belarus

PhD (physics and mathematics), docent; associate professor at the department of higher mathematics, faculty of electronic informational systems

Helena N. Zashchuk, Brest State Technical University, 267 Maskoŭskaja Street, Brest 224023, Belarus

PhD (physics and mathematics), docent; associate professor at the department of higher mathematics, faculty of electronic informational systems

References

  1. Antosik P, Legeza J. Products of measures and functions of finite variations. In: Proceedings of the International conference on generalized functions and operational calculus; 29 September – 6 October 1975; Varna, Bulgaria. Sophia: Bulgarian Academy of Sciences; 1979. p. 20–26.
  2. Das PC, Sharma RR. Existence and stability of measure differential equations. Czechoslovak Mathematical Journal. 1972;22(1):145–158. DOI: 10.21136/cmj.1972.101082.
  3. Ligeza J. On generalized solutions of some differential non-linear equations of order n. Annales Polonici Mathematici. 1975;31(2):115–120. DOI: 10.4064/ap-31-2-115-120.
  4. Zavalishchin ST, Sesekin AN. Dynamic impulse systems: theory and applications. Dordrecht: Kluwer Academic Publishers; 1997. 271 p. (Mathematics and its applications; volume 394).
  5. Pandit SG, Deo SG. Differential systems involving impulses. Berlin: Springer; 1982. 106 p. (Lecture notes in mathematics; volume 954). DOI: 10.1007/BFb0067476.
  6. Lazakovich NV. [Stochastic differentials in the algebra of generalised random processes]. Doklady of the National Academy of Sciences of Belarus. 1994;38(5):23–27. Russian.
  7. Karimova TI, Yablonskii OL. About associated solutions of system of non-homogeneous equations in differentials in the algebra of generalized stochastic processes. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2009;2:81–86. Russian.
  8. Zhuk AI, Yablonskiy OL. Spaskov SA. Associated solutions of the system of non-autonomous differential equations with generalized coefficients. Mixed cas. Vesci BDPU. Seryja 3. Fizika. Matjematyka. Infarmatyka. Bijalogija. Geagrafija. 2019;4:16–22. Russian.
  9. Zhuk AI, Yablonskiy OL. [Systems of differential equations in the algebra of generalised functions]. Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series. 2011;1:12–16. Russian.
  10. Zhuk AI, Yablonskii OL. Non-autonomous systems of differential equations in the algebra of generalized functions. Trudy Instituta matematiki. 2011;19(1):43–51. Russian.
  11. Zhuk AI, Yablonski OL. Estimation of a convergence rate to associated solutions of differential equations with generalized coefficients in the algebra of mnemofunctions. Doklady of the National Academy of Sciences of Belarus. 2015;59(2):17–22. Russian.
  12. Zhuk AI, Khmyzov AK. Systems of quasidifferential equations with generalized right part is investigated. Associated solution of the initial problem is considered in the direct product of algebras of mnemonic functions. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2010;2:87–93. Russian.
Published
2022-04-14
Keywords: algebra of new generalised functions, differential equations with generalised coefficients, functions of finite variation
How to Cite
Zhuk, A. I., & Zashchuk, H. N. (2022). On associated solutions of the system of non-autonomous differential equations in the Lebesgue spaces. Journal of the Belarusian State University. Mathematics and Informatics, 1, 6-13. https://doi.org/10.33581/2520-6508-2022-1-6-13
Issue
No 1 (2022): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

Copyright (c) 2022 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)