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

We study an infinite system ofalgebraic equations with monotone and concave nonlinearity.This system arises in the various discrete problemsof the mathematical models of the natural sciences.In particular, these systems of such structure, withspecific instances of nonlinearity and the correspondinginfinite matrix, are encountered in the theoryof radiative transfer, the kinetic theory of gases,and the mathematical theory of epidemic diseases.Under certain conditions on the entries of the correspondinginfinite matrix and nonlinearity,we prove the existence and uniqueness theorems ofa coordinatewise nonnegative nontrivial solution in thespace of bounded sequences.Also, we obtain a uniformestimate of the corresponding successive approximations andprove that the so-constructed solutiontends at infinity with speed  $ l_{1} $ to a positive fixed point of the functiondescribing the nonlinearity of its system.The main tools are the Krasnoselskii method onconstructing invariant cone segments for the correspondingnonlinear operator and the methods of the theory ofdiscrete convolution operators, together with some geometricinequalities for concave and monotonic functions.Furthermore, we exhibit some particular examples ofthe corresponding infinite matrix and nonlinearitythat satisfy all imposed hypotheses.