We study and solve some class of infinite systems ofalgebraic equations with monotone nonlinearity and Toeplitz-type matrices.Such systemsfor the specific representations of nonlinearities arise in the discrete problems ofdynamic theory of clopen $ p $ -adic strings for a scalar field of tachyons,the mathematical theory of spatio-temporal spread of an epidemic, radiation transfer theoryin inhomogeneous media, and the kinetic theory of gases in the framework of the modified Bhatnagar–Gross–Krookmodel. The noncompactness of the corresponding operator in the bounded sequence spaceand the criticality property (the presence of trivial nonphysicalsolutions) is a distinctive feature of these systems.For these reasons, the use of the well-known classical principles of existenceof fixed points for such equations do not lead to the desired results.Constructing some invariant cone segments for the correspondingnonlinear operator, we prove the existence and uniqueness of a nontrivialnonnegative 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 betweenthis limit and a solution belongs to $ l_{1} $ .By way of illustration, we provide some special applied examples.
