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

Dang Vu Giang

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arXiv · 1003.5091

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

Dang Vu Giang

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Abstract

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 $n\to\infty$ where $|\vartheta_1|=|\vartheta_2|=\cdots=|\vartheta_k|=1$.

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Dang Vu Giang. 2010-03-26. Spectrum of a bounded sequence and inhomogeneous delay linear difference equations in a Banach space. https://arxiv.org/abs/1003.5091

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