An Infinite Algebraic System in the Irregular Case

We obtain sufficient conditions for the nontrivial solvability of systems of the form
$$ \varphi_i=b_i+\lambda_i\sum^\infty_{j=0} a_{i-j}\varphi_j,\qquad i\in\mathbb Z_+\overset{\text{def}}{=}\{0,1,2,\dots,n,\ldots\}, $$
and of the corresponding homogeneous systems. It is assumed that the sequences $b=(b_0,b_1,b_2,\ldots)$ and $\lambda=(\lambda_0,\lambda_1,\lambda_2,\ldots)$ and the Toeplitz matrix $A=(a_{i-j})$ satisfy the conditions
\begin{gather*} a_j\ge 0,\quad j\in {\mathbb Z},\qquad \sum^\infty_{j=-\infty}a_j=1,\qquad \sum^\infty_{j=-\infty}|j|a_j<\infty,\qquad \sum^\infty_{j=-\infty}ja_j<0, \\ b_j\ge 0,\quad j\in {\mathbb Z}_+,\qquad \sum^\infty_{j=0}b_j<\infty,\qquad 1\le\lambda_i\le \biggl(\,\sum^i_{j=-\infty}a_j\biggr)^{-1},\quad i\in {\mathbb Z_+}. \end{gather*}
Under these conditions, we construct bounded solutions of homogeneous and inhomogeneous systems of the form indicated above.