Multiperiodic solutions of quasilinear systems of integro-differential equations with $D_c$-operator and $\epsilon$-period of hereditarity

Zh. A. Sartabanov ; G. M. Aitenova ; G. A. Abdikalikova

Eurasian mathematical journal, Tome 13 (2022) no. 1, pp. 86-100.

Résumé

We investigate a quasilinear system of partial integro-differential equations with the operator of differentiation in the direction of a vector field, which describes the process of hereditary propagation with an $\epsilon$-period of heredity. Under some conditions on the input data, conditions for the solvability of the initial problem for a quasilinear system of integro-differential equations are obtained. On this basis, sufficient conditions for the existence of multiperiodic solutions of integro-differential systems are found under the exponential dichotomy additional assumption on the corresponding homogeneous integro-differential system. The unique solvability of an operator equation in the space of smooth multiperiodic functions is proved, to which the main question under consideration reduces. Thus, sufficient conditions are established for the existence of a unique multiperiodical in all time variables solution of a quasilinear system of integro-differential equations with the differentiation operator in the directions of a vector field and a finite period of hereditarity.

Bibliographie

[1] G. A. Abdikalikova, “Construction of an almost periodic solution of one quasilinear parabolic system”, Izv. MON NAN RK. Ser. fiz. mat., 2001, no. 3, 3–8

[2] A. T. Assanova, Z. S. Tokmurzin, “A nonlocal multipoint problem for a system of fourth-order partial differential equations”, Eurasian Mathematical Journal, 11:3 (2020), 3–20

[3] E. A. Barbashin, Introduction to sustainability theory, Nauka, M., 1967 (in Russian)

[4] A. B. Berzhanov, E. K. Kurmangaliyev, “Solution of a countable system of quasilinear partial differential equations multiperiodic in a part of variables”, Ukrainian Mathematical Journal, 61:2 (2009), 336–345 | DOI

[5] R. N. Butris, “Periodic solution of nonlinear system of integro-differential equation depending on the Gamma distribution”, Gen. Math. Notes, 15:1 (2013), 56–71

[6] Y. V. Bykov, On some problems of the theory of integro-differential equations, Kirgiz. Gos. un-t, Frunze, 1957 (in Russian)

[7] K.-S. Chiu, “Periodic solutions for nonlinear integro-differential systems with piecewise constant argument”, The Scientific World Journal, 2014

[8] D. S. Dzhumabaev, S. T. Mynbayeva, “New general solution to a nonlinear Fredholm integro-differential equation”, Eurasian Mathematical Journal, 10:4 (2019), 24–33

[9] A. A. Ilyushin, B. E. Pobedria, Fundamentals of the mathematical theory of thermovisco-elasticity, Nauka, M., 1970 (in Russian)

[10] M. I. Imanaliev, Nonlinear partial integro-differential equations, Ilim, Bishkek, 1992 (in Russian)

[11] V. Kh. Kharasakhal, Almost periodic solutions of ordinary differential equations, Nauka, Alma-Ata, 1970 (in Russian)

[12] A. A. Kulzhumiyeva, Zh. A. Sartabanov, “On multiperiodic integrals of a linear system with the differentiation operator in the direction of the main diagonal in the space of independent variables”, Eurasian Mathematical Journal, 8:1 (2017), 67–75

[13] A. A. Kulzhumiyeva, Zh. A. Sartabanov, “Integration of a linear equation with differential operator, corresponding to the main diagonal in the space of independent variables, and coefficients, constant on the diagonal”, Russian Mathematics, 63:6 (2019), 29–41

[14] V. Lakshmikantham, R. Mohana Rao, Theory of integro-differential equations, Gordon and Breach Science Publishers, Amsterdam, 1995

[15] A. M. Nakhushev, Equations of mathematical biology, Vysshaia shkola, M., 1995 (in Russian)

[16] B. Zh. Omarova, Zh. A. Sartabanov, “On multiperiodic solutions of perturbed nonlinear autonomous systems with the differentiation operator on a vector field”, Eurasian Mathematical Journal, 12:1 (2021), 68–81

[17] Yu. N. Rabotnov, Elements of hereditary mechanics of solids, Nauka, M., 1977 (in Russian)

[18] X. Ros Oton, Integro-differential equations: Regularity theory and Pohozaev identities, Universitat Politecnica de Catalunya Programa de Doctorat de Matematica Aplicada, Barcelona, 2014

[19] A. M. Samoilenko, N. I. Ronto, Numerical and analytical methods for studying solutions of boundary value problems, Naukova Dumka, Kiev, 1995 (in Russian)

[20] Zh. Sartabanov, “Pseudoperiodic solutions of a system of integrodifferential equations”, Ukrainian Mathematical Journal, 41:1 (1989), 116–120

[21] Zh. A. Sartabanov, “The multiperiodic solutions of a linear system of equations with the operator of differentiation along the main diagonal of the space of independent variables and delayed arguments”, AIP Conference Proceedings, 1880 (2017), 040020, 5 pp.

[22] V. S. Sergeev, “Forced extremely periodic motions in systems with aftereffect described by integro-differential equations of Volterra type”, Trudy MFTI, 9:3 (2017), 57–65

[23] D. U. Umbetzhanov, Almost periodic solutions of evolutionary equations, Nauka, Alma-Ata, 1990 (in Russian)

[24] V. Volterra, Lecons sur les equations integrals et les equations integro-differentielles, Paris, 1913

[25] V. Volterra, “Vibrazione elastiche nel caso della eredita”, Rend. Accad. Lincei, 5:21 (1912), 3–12

[26] V. Volterra, “Sur les equations integro differentielles et leurs applications”, Acta Math., 35 (1912), 295–356

[27] V. Volterra, Theory of functionals, integral and integro-differential equations, Nauka, M., 1982 (in Russian)