Sufficient conditions for the existence of the solution of an infinite-difference equation with variable coefficients

The paper discusses a difference equation of the form $\sum_{l=0}^{r}a_{k,l}Z_{k+l}=y_{k}\ (k\in \mathbb{Z})$, where $r\in \mathbb{N},\ y=\{y_k\}_{k\in \mathbb{Z}}$ is a given numerical sequence from the space ${{l}_{p}}\ (1\le p<\infty)$, provided that the matrix $A=(a_{k,l})$, $a_{k,l}\in \mathbb{R}$, satisfies some condition close to the presence of a dominant diagonal. With the help of the fixed point theorem, sufficient conditions are written for the coefficients $a_{k,l}$, at which the equation has a unique solution $Z=\{ Z_{k}\}_{k\in \mathbb{Z}}$, belonging to the space $l_p$. For the norm of this solution, a numerical estimate is given from above.