Fixed-Point Theorem for an Infinite Toeplitz Matrix and Its Extension to General Infinite Matrices

In [V. M. Abramov, Bull. Austral. Math. Soc., 104, 108 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present work, we provide an alternative proof of the existence of a positive solution in the general case. This proof is based on the application of a version of the M. A. Krasnosel’skii fixed-point theorem. The results are then extended to the equations with infinite matrices of the general type.