On the Characterization of a Class of Difference Equations

Abstract

We focus on the behavior of solutions of the difference equation 𝑥𝑛=𝑏1𝑥𝑛−1+𝑏2𝑥𝑛−2+⋯+𝑏𝑛𝑥0+𝑦𝑛, 𝑛=1,2,…, where (𝑏𝑘) is a fixed sequence of complex numbers, and (𝑦𝑘) is a fixed sequence in a complex Banach space. We give the general solution of this difference equation. To examine the asymptotic behavior of solutions, we compute the spectra of operators which correspond to such type of difference equations. These operators are represented by upper triangular or lower triangular infinite banded Toeplitz matrices.