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

The paper discusses a difference equation of the formΣ︀𝑟𝑙=0 𝑎𝑘,𝑙𝑍𝑘+𝑙 = 𝑦𝑘 (𝑘 ∈ Z), where 𝑟 ∈ N, 𝑦 = {𝑦𝑘}𝑘∈Z is a given numerical sequence from the space 𝑙𝑝 (1 ⩽ 𝑝 < ∞), provided that the matrix 𝐴 = (𝑎𝑘,𝑙), 𝑎𝑘,𝑙 ∈ 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 thecoefficients 𝑎𝑘,𝑙, at which the equation has a unique solution 𝑍 = {𝑍𝑘}𝑘∈Z, belonging to the space 𝑙_𝑝. For the norm of this solution, a numerical estimate is given from above.