On the Solvability of an Infinite System of Algebraic Equations with Monotone and Concave Nonlinearity

We study an infinite system of algebraic equations with monotone and concave nonlinearity. This system arises in the various discrete problems of the mathematical models of the natural sciences. In particular, these systems of such structure, with specific instances of nonlinearity and the corresponding infinite matrix, are encountered in the theory of radiative transfer, the kinetic theory of gases, and the mathematical theory of epidemic diseases. Under certain conditions on the entries of the corresponding infinite matrix and nonlinearity, we prove the existence and uniqueness theorems of a coordinatewise nonnegative nontrivial solution in the space of bounded sequences. Also, we obtain a uniform estimate of the corresponding successive approximations and prove that the so-constructed solution tends at infinity with speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ l_{1} $\end{document} to a positive fixed point of the function describing the nonlinearity of its system. The main tools are the Krasnoselskii method on constructing invariant cone segments for the corresponding nonlinear operator and the methods of the theory of discrete convolution operators, together with some geometric inequalities for concave and monotonic functions. Furthermore, we exhibit some particular examples of the corresponding infinite matrix and nonlinearity that satisfy all imposed hypotheses.