Abstract
The relation between the following two properties of linear difference equations with infinite delay is investigated: (i) exponential stability and (ii) -input -state stability (Perron's property) which means that solutions of the non-homogeneous equation with zero initial data belong to when non-homogeneous terms are in . It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted -space with an exponentially fading weight (the phase space). Our main result states that whenever and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and -input -state stabilities do not depend on the choice of a phase space and parameters p and q. -input -state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.
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