On the Qualitative Properties of a Solution to a System of Infinite Nonlinear Algebraic Equations

We study and solve some class of infinite systems of algebraic equations with monotone nonlinearity and Toeplitz-type matrices. Such systems for the specific representations of nonlinearities arise in the discrete problems of dynamic theory of clopen \( p \)-adic strings for a scalar field of tachyons, the mathematical theory of spatio-temporal spread of an epidemic, radiation transfer theory in inhomogeneous media, and the kinetic theory of gases in the framework of the modified Bhatnagar–Gross–Krook model. The noncompactness of the corresponding operator in the bounded sequence space and the criticality property (the presence of trivial nonphysical solutions) is a distinctive feature of these systems. For these reasons, the use of the well-known classical principles of existence of fixed points for such equations do not lead to the desired results. Constructing some invariant cone segments for the corresponding nonlinear operator, we prove the existence and uniqueness of a nontrivial nonnegative solution in the bounded sequence space. Also, we study the asymptotic behavior of the solution at \( \pm\infty \). In particular, we prove that the limit at \( \pm\infty \) of a solution is finite. Also, we show that the difference between this limit and a solution belongs to \( l_{1} \). By way of illustration, we provide some special applied examples.