Special solutions of linear difference equations with infinite delay

Medveď, Milan

Archivum mathematicum, Tome 30 (1994) no. 2, pp. 139-144.

Résumé

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  $\epsilon \rightarrow 0$.

Détail

Classification : 34K30, 39A10, 39A70, 47B39
Keywords: difference equation; infinite delay; special solution