Geodesic
Special solutions of linear difference equations with infinite delay
Archivum mathematicum, Tome 30 (1994) no. 2, pp. 139-144

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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  $\epsilon \rightarrow 0$.
Classification : 34K30, 39A10, 39A70, 47B39
Keywords: difference equation; infinite delay; special solution 
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     author = {Medve\v{d}, Milan},
     title = {Special solutions of linear difference equations with infinite delay},
     journal = {Archivum mathematicum},
     pages = {139--144},
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     zbl = {0819.39001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1994__30_2_a5/}
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Medveď, Milan. Special solutions of linear difference equations with infinite delay. Archivum mathematicum, Tome 30 (1994) no. 2, pp. 139-144. http://geodesic.mathdoc.fr/item/ARM_1994__30_2_a5/