For the difference equation $(\epsilon )\,\, x_{n+1} = Ax_n + \epsilon \sum _{k = -\infty }^n R_{n-k}x_k$,where $x_n \in Y,\, Y$ is a Banach space, $\epsilon $ is a parameter and $A$ is a linear, bounded operator. A sufficient condition for the existence of a unique special solution $y = \lbrace y_n\rbrace _{n=-\infty }^{\infty }$ passing through the point $x_0 \in Y$ is proved. This special solution converges to the solution of the equation (0) as $\epsilon \rightarrow 0$.
@article{107502,
author = {Milan Medve\v d},
title = {Special solutions of linear difference equations with infinite delay},
journal = {Archivum Mathematicum},
volume = {030},
year = {1994},
pages = {139-144},
zbl = {0819.39001},
mrnumber = {1292565},
language = {en},
url = {http://dml.mathdoc.fr/item/107502}
}
Medveď, Milan. Special solutions of linear difference equations with infinite delay. Archivum Mathematicum, Tome 030 (1994) pp. 139-144. http://gdmltest.u-ga.fr/item/107502/
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