Abstract: In this paper, we study an infinite system of algebraic equations with monotone and concave nonlinearity. This system arises in various discrete problems when studying mathematical models in the natural sciences. In particular, systems of such a structure, with specific representations of nonlinearity and the corresponding infinite matrix, are encountered in the theory of radiative transfer, in the kinetic theory of gases, and in the mathematical theory of epidemic diseases. Under certain conditions on the elements of the corresponding infinite matrix and on the non-linearity, existence and uniqueness theorems with respect to the coordinatewise non-negative non-trivial solution in the space of bounded sequences are proved. In the course of proving the existence theorem, we also obtain a uniform estimate for the corresponding successive approximations. It is also proved that the constructed solution tends at infinity to a positive fixed point of the function describing the nonlinearity of the given system with the speed \(l_1.\) The main tools for proving the above facts are the method of M. A. Krasnoselsky on the construction of invariant cone segments for the corresponding nonlinear operator, methods of the theory of discrete convolution operators, as well as some geometric inequalities for concave and monotonic functions. At the end of the paper, concrete particular examples of the corresponding infinite matrix and non-linearity are given that satisfy all the conditions of the proven statements.
1. Engibaryan, N. B. On a Problem in Nonlinear Radiative Transfer, Astrophysics, 1966, vol. 2, pp. 12-14. DOI: 10.1007/BF01014505.
2. Khachatryan, A. Kh. and Khachatryan, Kh. A. Qualitative Difference Between Solutions for a Model of the Boltzmann Equation in the Linear and Nonlinear Cases, Theoretical and Mathematical Physics, 2012, vol. 172, no. 3, pp. 1315-1320. DOI: 10.1007/s11232-012-0116-4.
3. Kogan, M. N. Rarefied Gas Dynamics, New York, Plenum Press, 1969.
4. Diekmann O. Threshold and Travelling Waves for the Geographical Spread of Infection,
Journal of Mathematical Biology, 1978, vol. 6, pp. 109-130. DOI: 10.1007/BF02450783.
5. Khachatryan, A. Kh. and Khachatryan, Kh. A. On Solvability of one Infinite System of Nonlinear Functional Equations in the Theory of Epidemics, Eurasian Mathematical Journal, 2020, vol. 11, no. 2, pp. 52-64. DOI: 10.32523/2077-9879-2020-11-2-52-64.
6. Vladimirov, V. S. and Volovich, Y. I. Nonlinear Dynamics Equation in \(p\)-Adic String Theory, Theoretical and Mathematical Physics, 2004, vol. 138, no. 3, pp. 297-309. DOI: 10.1023/B:TAMP.0000018447.02723.29.
7. Aref'eva, I. Ya. Rolling Tachyon on Non-BPS Branes and \(p\)-Adic Strings, Proceedings of the Steklov Institute of Mathematics, 2004, vol. 245, pp. 40-47.
8. Vladimirov, V. S. Nonlinear Equations for \(p\)-adic Open, Closed, and Open-Closed Strings,
Theoretical and Mathematical Physics, 2006, vol. 149, no. 3, pp. 1604-1616. DOI: 10.1007/s11232-006-0144-z.
9. Khachatryan, Kh. A. Voprosy razreshimosti nekotoryh nelinejnyh integral'nyh i integro-differencial'nyh uravnenij s nekompaktnymi operatorami v kriticheskom sluchae [Questions of solvability of some nonlinear integral and integro-differential equations with non-compact operators in the critical case], Doctoral Dissertation, Yerevan State University, 2011, 231 p. (in Russian).
10. Khachatryan, Kh. A. and Broyan, M. F. On Some Nonlinear Infinite Systems of Algebraic Equations with Toeplitz-Hankel Type Matrices, Matematicheskie voprosy kibernetiki i vychislitel'noj mekhaniki [Mathematical Issues of Cybernetics and Computational Mechanics],
2012, vol. 38, no. 2, pp. 32-33 (in Russian).
11. Khachatryan, Kh. A. and Andriyan, S. M. On the Solvability of a Class of Discrete Matrix Equations with Cubic Nonlinearity, Ukrainian Mathematical Journal, 2020, vol. 71, no. 12, pp. 1910-1928. DOI: 10.1007/s11253-020-01755-4.
12. Khachatryan, Kh. A. and Broyan, M. F. One-Parameter Family of Positive Solutions for a Class of Nonlinear Infinite Algebraic Systems with Teoplitz-Hankel Type Matrices, Journal of Contemporary Mathematical Analysis, 2013, vol. 48, no. 5, pp. 209-220. DOI: 10.3103/S1068362313050026.
13. Engibaryan, N. B. Renewal Equations on the Semi-Axis, Izvestiya: Mathematics, 1999, vol. 63, no. 1, pp. 57-71. DOI: 10.1070/im1999v063n01ABEH000228.
