Spectrum of a bounded sequence and inhomogeneous delay linear difference equations in a Banach space

We study the asymptotic behavior of a bounded solution of an inhomogeneous delay linear difference equation in a Banach space by using the spectrum of bounded sequences. We get a significant extension of excellent results in [1]. A new simple proof is also found for the famous Gelfand spectral radius theorem. Moreover, among other things we prove that if the spectrum of a bounded sequence $\{x_n{\}}_n$ is finite then $x_n=c_1{\vartheta }_1^n+c_2{\vartheta }_2^n+\cdots +c_k{\vartheta }_k^n+o(1)$ as $n\to \mathrm{\infty }$ where $\mathrm{\mid }{\vartheta }_1\mathrm{\mid }=\mathrm{\mid }{\vartheta }_2\mathrm{\mid }=\cdots =\mathrm{\mid }{\vartheta }_k\mathrm{\mid }=1$.