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.

我们研究并求解一类具有单调非线性和托普利茨型矩阵的无限代数方程组。这种非线性具体表示的系统出现在快子标量场的闭开\(p\) -adic弦动态理论、流行病时空传播的数学理论、非均匀辐射传输理论的离散问题中。介质，以及改进的 Bhatnagar-Gross-Krook 模型框架中的气体动力学理论。有界序列空间中相应算子的非紧性和临界性属性（平凡的非物理解的存在）是这些系统的显着特征。由于这些原因，对此类方程使用众所周知的不动点存在的经典原理不会产生预期的结果。为相应的非线性算子构造一些不变的锥段，证明了有界序列空间中非平凡非负解的存在性和唯一性。此外，我们还研究了\( \pm\infty \)处解的渐近行为 。特别是，我们证明解的\(\pm\infty \)处的极限 是有限的。此外，我们还表明该极限与解之间的差异属于 \( l_{1} \)。为了便于说明，我们提供了一些特殊的应用示例。