Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 51. no. 2. P. 57 — 72. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-51-2-57-72
Research Article
Full text in Russian
MSC 35L35,35L25,35S15

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On the Correct Solvability of a Nonlocal Boundary Value Problem for Loaded Fourth-Order Hyperbolic Equations with Impulse Effects

Sh. Sh. Yusubov¹, M. N. Huseynova²^{\ast}

¹Baku State University, AZ1148, Baku, Z. Khalilov, 23, Azerbaijan
²National Aviation Academy, AZ1045, Mardakan ave, 30, Azerbaijan

Abstract. This article discusses the problem with nonlocal boundary and integral conditions for a loaded fourth-order hyperbolic equation with impulsive effects. The investigated equation can be interpreted as a generalized Bussinesque-Lev equation, which arises when modeling transverse vibrations of short thick rods, as well as when studying wave processes in periodically layered media. The considered equation generally has nonsmooth coefficients belonging to the Lp space. The solution to the posed problem is sought in a functional space of the Sobolev type. The representation of elements of this space is used significantly. The elements of this functional space allow discontinuities of the first kind along lines parallel to the characteristic curves. Using this representation, the problem is reduced to an equivalent integral equation. It is proven that for establishing a homeomorphism of the operator corresponding to the posed problem between certain pairs of Banach spaces, it is necessary and sufficient for the corresponding integral equation
to have a unique solution. Furthermore, the existence and uniqueness of the solution to the posed problem are established. Next, the corresponding conjugate integral equation is constructed, and using an a priori estimate, the existence and uniqueness of its solution are proven. The fundamental solution to the posed problem is defined as a particular case of the conjugate integral equation. Based on the fundamental solution, an integral representation of the solution to the posed problem is obtained.

Key words: hyperbolic equation, nonlocal problem, problems with impulse effects, loaded equations.

Received: 11.06.2025; Revised: 13.08.2025; Accepted: 29.08.2025; First online: 17.09.2025

For citation. Yusubov Sh. Sh., Huseynova M. N. On the correct solvability of a nonlocal boundary value problem for loaded fourth-order hyperbolic equations with impulse effects. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 51: 2, 57-72. EDN: FZBWYU. https://doi.org/10.26117/2079-6641-2025-51-2-57-72.

Funding. The work was carried out without the support of funds.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: minayye.huseynova@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Yusubov Sh. Sh., Huseynova M. N., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

References

1. Lowden D. F. Optimal trajectories for space navigation. Moscow, Mir, 1966. 152 p. (In Russian).
2. Kalanay A., Veksler D., Quality Theory of Impulse systems. Moscow, Mir, 1971, 209 p. (In Russian).
3. Pandit S. G., Deo S. G. Differential systems involving impulses Lecture Notes. Notes Math, 1982, vol. 954, no.8, 102 p.
4. Lakshmikantham V., Baionov D. D., Simeonov P. S. Theory of impulsive differential equation. Singapure, World Sc, 1989, 434 p.
5. Il’in V. A., Moiseev E. I. An a priori estimate for the solution of a problem associated with a nonlocal boundary value problem of the first kind, Differ. Uravn., 1988, Vol 24, no. 5, pp. 795–804.(In Russian).
6. Nakhushev A. M. Loaded Equations and Their Application. Moscow, Nauka, 2012, 232 p. (In Russian).
7. Dzhenaliev M. T., Ramazanov M. I. Loaded Equations as Perturbations of Differential Equations. Almaty, Gylym 2010, 336 p. (In Russian).
8. Attaev A. Kh. On a nonlocal boundary value problem for a model hyperbolic nonlocal equations, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2022, vol. 42, no. 3, pp. 7–15. DOI: 10.26117/2079-6641-2022-40-3-7-15 (In Russian).
9. Zikirov O. S., Kholikov D. K., “On some problem for a loaded pseudoparabolic equation of the third order, Mathematical notes of NEFU, 2016, vol. 23, no. 2, pp. 19–30. (In Russian).
10. Yusubov Sh. Sh. Non-local problem with integral conditions for the three-dimensional high order hyperbolic equations, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2020, vol. 33, no. 4, pp. 51–62. DOI: 10.26117/2079-6641-2020-33-4-51-62 (In Russian).
11. Yusubov Sh. Sh. Nonlocal problem with integral conditiond for a high order hyperbolic equation, Ukrains’kyi Matematychnyi Zhurnal, 2017, vol. 69, no. 1, pp. 121–131. DOI: 10.1007/s11253-017-1353-3
12. Asanova A. T. On the correct solvability of a nonlocal boundary value problem for systems of hyperbolic equations with impulsive actions, Ukrains’kyi Matematychnyi Journal, 2015, vol. 67, no. 3, pp. 291–303.
13. Bainov D. D., Minchev E., Myshkis A. On the correct solvability of a nonlocal boundary value problem for systems of hyperbolic equations with impulsive actions, Commun. Appl. Anal., 1997, vol. 1, no. 4, pp. 1–14.
14. Belarbi A., Benchohra M. Existence theory for perturbed impulsive hyperbolic differential inclusions with variable times, J. Math. Anal.Appl., 2007, vol. 327, pp. 1116–1129.
15. Perestyuk N. A., Tkach A. B. Periodic solutions for weakly nonlinear partial system with pulse influense, Ukrains’kyi Matematychnyi Journal, 1997, vol. 49, no. 4, pp. 601–605. DOI: 10.1007/BF02487331
16. Yusubov Sh. Sh. On correct solvability and representation of solutions to third-order hyperbolic equations with impulsive action, Bulletin of the Baku University, 2000, no. 1, pp. 124–130.
17. Yusubov Sh. Sh. On the Fredholm property and representation of the solution of an impulsive integro-differential equation, Proceedings of IMM AN Azerb, 1996, vol. V, no. XIII, pp. 187–192.
18. Grigorieva A. I. Conjugation problems for some analogs of the longitudinal wave equation with a discontinuous coefficient, Chelyabinsk Journal of Physics and Mathematics, 2018, vol. 3, no. 3, pp. 276–294. DOI: 10.24411/2500-0101-2018-13302 (In Russian).
19. Kozhanov A. I., Shadrina N. N. Investigation of the influence of parameters on the correctness of the conjugation problem for the Bussinesque-Lyav differential equation, Chelyabinsk Journal of Physics and Mathematics, 2022, vol. 7, no. 1, pp. 30–42. (In Russian).
20. Yuldashev T. K. On a nonlocal problem for impulsive differential equations with mixed maxima, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2022, vol. 38, no. 1, pp. 40–53. DOI: 10.26117/2079-6641-2022-38-1-40-53
21. Yusubov Sh. SH., Huseynova M. N. On the Unique Solvability of a Nonlocal Boundary Value Problem for Fourth-Order Hyperbolic Equation with Impulsive Actions, In: 2023 5th International Conference on Problems of Cybernetics and Informatics (PCI) (28-30 August 2023). IEEE, Institute of Electrical and Electronics Engineers., 2023, pp. 1–4 DOI: 10.1109/PCI60110.2023.10325919
22. Whitham G. B. Linear and Nonlinear Waves. London, John Wiley-Sons, 1974, 660 p.
23. Rao S. Advanced theory of biration. New York, Wiley, 1992, 431 p. (In Russian).
24. Bakhvalov N. S., Eglit M. E. Effective equations with dispersion for wave propagation in periodic media, Reports of the Academy of Sciences, 2000, vol. 370, no. 1, pp. 7–10. (In Russian).
25. Pulkina L. S., Beylin A. B. Nonlocal approach to problems on longitudinal vibration in a short bar, Electronic Journal of Differential equations, 2019, no. 29, pp. 42–54. DOI: 10.26117/2079-6641-2020-32-3-42-54 (In Russian).

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