THE SOLVABILITY OF AN INFINITE SYSTEM OF NONLINEAR ALGEBRAIC EQUATIONS WITH TOEPLITZ MATRIX

Abstract

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.