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

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