On the qualitative properties of a solution for one system of infinite nonlinear algebraic equations

The work is devoted to the study and solution of class of infinite systems of algebraic equations with monotone nonlinearity and Toeplitz type matrices. With specific representations of nonlinearities, this system arises in discrete problems of the dynamical theory of open-closed $p$-adic strings for a scalar field of tachyons, in the mathematical theory of the spatiotemporal propagation of an epidemic, in the theory of radiative transfer in inhomogeneous medium and in the kinetic theory of gases in the framework of a modified Bhatnagar–Gross–Krock models. A distinctive feature of these systems of nonlinear equations is the non-compactness of the corresponding operator in a space of bounded sequences and the criticality property (the presence of trivial non-physical solutions). For this reason, the use of well-known classical principles about the existence of fixed points for such equations does not give the desired results. In this paper, using methods for constructing invariant cone segments for the corresponding nonlinear operator, we prove the existence and uniqueness of a nontrivial nonnegative solution in the space of bounded sequences. The asymptotic behavior of the constructed solution on $\pm \infty$ is also studied. In particular, the finiteness of the limit of the solution on $\pm \infty,$ is proved, and it is established that the difference between the limit and the solution belongs to the space $l_1.$ At the end of the paper, special applied examples are given to illustrate the results obtained.