Fixed point theorem for an infinite Toeplitz matrix and its extension to general infinite matrices

TL;DR

This paper extends fixed point theorems for infinite matrices, particularly Toeplitz matrices, providing a more general proof of positive solutions using Krasnoselskii's theorem and applying it to broader classes of infinite matrices.

Contribution

It offers a new, more general proof for the existence of positive solutions for infinite matrix equations, extending previous results to more general matrix types.

Findings

Established conditions for positive solutions of infinite Toeplitz matrix equations.

Extended fixed point results to general infinite matrices.

Provided an alternative proof using Krasnoselskii's fixed point theorem.

Abstract

In [V. M. Abramov, \emph{Bull. Aust. Math. Soc.} \textbf{104} (2021), 108--117] the fixed point equation for an infinite nonnegative Toeplitz matrix has been studied. It was found the conditions for existence of a positive solution and bounded positive solution. However, the proof of the existence of a positive solution was entirely straightforward, not admitting extensions for more general types of matrices. In the present note, we provide an alternative proof for the existence of a positive solution in more general case. The presented proof is based on an application of a variant of the fixed point theorem of M. A. Krasnoselskii. The results are then extended for the equations with infinite matrices of a general type.