Geodesic
On the solvability of an infinite system of algebraic equations with monotone and concave nonlinearity
Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 2, pp. 82-94 
H. S. Petrosyan; Kh. A. Khachatryan. On the solvability of an infinite system of algebraic equations with monotone and concave nonlinearity. Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 2, pp. 82-94. http://geodesic.mathdoc.fr/item/VMJ_2024_26_2_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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