Existence and uniqueness theorems for one infinite system of nonlinear algebraic equations

We study a infinite system of algebraic equations with monotone nonlinearities and with an infinite Toeplitz type matrix. This system has applications in discrete problems of the dynamical theory of $p$-adic open-closed strings, the kinetic theory of gases, and mathematical biology. Under certain restrictions on the nonlinearities and on the corresponding Toeplitz matrix, it is possible to prove existence and uniqueness theorems for a nontrivial solution in the class of bounded sequences. The main tool for proving the uniqueness theorem for a nontrivial solution is an auxiliary independent theorem on the asymptotic behavior of a nonnegative nontrivial and bounded solution on $\pm\infty.$ At the end of the paper, specific applied examples of nonlinearities and the corresponding matrix are given to illustrate the importance of the results obtained.