Geodesic
The solvability of an infinite system of nonlinear algebraic equations with Toeplitz matrix
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 57 (2023) no. 3, pp. 69-78.

Voir la notice de l'article provenant de la source Math-Net.Ru

The work is focused on studying of the existence, uniqueness, and various qualitative properties of the constructive solution of an infinite system of algebraic equations with a concave nonlinearity property, which are generated by Toeplitz matrices. In addition to its independent mathematical interest, such systems have a significant application in several branches of mathematical physics and mathematical biology. Those particularly appear in discrete problems within radiative transfer theory, kinetic theory of gases, dynamic theory of $p$-adic strings, and the mathematical theory of epidemic propagation. We establish the existence of a positive solution for the system in the class of bounded sequences, as well as provide an iterative method to approximate to the solution. We also study the asymptotic behavior of the solution at infinity and the uniqueness of the nontrivial solution with non-negative elements in the class of bounded sequences. The last section of the paper provides examples of applications of the corresponding Toeplitz matrix and the function that describes the nonlinearity.
Keywords: nonlinearity, concavity, monotonicity, uniqueness of solution, asymptotics
Mots-clés : convergence 
@article{UZERU_2023_57_3_a0,
     author = {Kh. A. Khachatryan and V. G. Dilanyan},
     title = {The solvability of an infinite system of nonlinear algebraic equations with {Toeplitz} matrix},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
     pages = {69--78},
     publisher = {mathdoc},
     volume = {57},
     number = {3},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZERU_2023_57_3_a0/}
}
TY  - JOUR
AU  - Kh. A. Khachatryan
AU  - V. G. Dilanyan
TI  - The solvability of an infinite system of nonlinear algebraic equations with Toeplitz matrix
JO  - Proceedings of the Yerevan State University. Physical and mathematical sciences
PY  - 2023
SP  - 69
EP  - 78
VL  - 57
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/UZERU_2023_57_3_a0/
LA  - en
ID  - UZERU_2023_57_3_a0
ER  - 
%0 Journal Article
%A Kh. A. Khachatryan
%A V. G. Dilanyan
%T The solvability of an infinite system of nonlinear algebraic equations with Toeplitz matrix
%J Proceedings of the Yerevan State University. Physical and mathematical sciences
%D 2023
%P 69-78
%V 57
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/UZERU_2023_57_3_a0/
%G en
%F UZERU_2023_57_3_a0
Kh. A. Khachatryan; V. G. Dilanyan. The solvability of an infinite system of nonlinear algebraic equations with Toeplitz matrix. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 57 (2023) no. 3, pp. 69-78. http://geodesic.mathdoc.fr/item/UZERU_2023_57_3_a0/

[1] N. B. Engibaryan, “On a problem in nonlinear radiative transfer”, Astrophysics, 2:2 (1966), 12–14 | DOI | MR

[2] C. Cercignani, The Boltzman Equation and its Applications, Springer, New York, 1988 | MR | Zbl

[3] O. Diekmann, H. G. Kaper, “On the Bounded Solutions of a Nonlinear Convolution Equation”, Nonlinear Analysis, Theory Math., Appl., 2:6 (1978), 721–737 | DOI | MR | Zbl

[4] C. Atkinson, G. E. Reuter, “Deterministic Epidemic Waves”, Math. Proc. Cambridge Philos. Soc., 80:2 (1976), 315–330 | DOI | MR | Zbl

[5] V. S. Vladimirov, Ya. I. Volovich, “Nonlinear Dynamics Equation in p-Adic String Theory”, Theoret. and Math. Phys., 138:3 (2004), 297–309 | DOI | MR | Zbl

[6] L. G. Arabajian, M. T. Hakopian, “On One Infinite Algebraic System with a Toeplitz Matrix”, J. Integral Equations Math. Phys., 1:1 (1992), 32–43 | MR

[7] L. G. Arabadzhyan, “An Infite Algebraic System in the Irregular Case”, Math. Notes, 89 (2011), 3–10 | DOI | MR | Zbl

[8] Z. Presdorf, Some classes of singular integral equations, Mir, Moskva, 1979 | MR

[9] L. G. Arabadzhyan, N. B. Engibaryan, “Convolution Equations and Nonlinear Functional Equations”, Journal of Soviet Mathematics, 36 (1984), 745–791 | DOI | MR | Zbl

[10] Kh. A. Khachatryan, On solvability Some Classes of Nonlinear Integro-Differential Equations with Noncompact Operator, Doctorial Thesis, YSU Press, Yerevan, 2011, 235 pp. | MR

[11] Kh. A. Khachatryan, M. F. Broyan, “One-Parameter Family of Positive Solutions for a Class of Nonlinear Infinite Algebraic System with Teoplitz–Hankel Type Matrices”, Journal of Contemporary Mathematical Analysis, 48:5 (2013), 189–200 | DOI | MR