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New criteria for exponential expansiveness of variational difference equations. (English) Zbl 1115.39005

The authors consider new criteria for exponential expansiveness of variational difference equations. They associate the linear system of variational difference equations to an input-output system, and characterize the uniform exponential expansiveness of the linear system in terms of the complete admissibility of pairs of sequence spaces related to its input-output system.
After introducing a general class of Banach sequence spaces \(\mathcal {Q}(\mathbb{N})\) and three sub-classes \(\mathcal {H}(\mathbb{N})\), \(\mathcal {L}(\mathbb{N})\) and \(\mathcal {R}(\mathbb{N})\), they define the complete admissibility of the pair \((\mathcal {B}(\Theta, V(\mathbb{N}, X)), U(\mathbb{N}, X))\) with \(U, V\in \mathcal {Q}(\mathbb{N})\), such that \(U(\mathbb{N}, X)\) is the input space and \(\mathcal {B}(\Theta, V(\mathbb{N}, X))\) is the output space. Then, they present a detailed study concerning the connections between the complete admissibility of the pair \((\mathcal {B}(\Theta, V(\mathbb{N}, X)), U(\mathbb{N}, X))\) and the uniform exponential expansiveness of the linear system. The main results are the following. If \(U\in \mathcal {L}(\mathbb{N})\) or \(V\in \mathcal {H}(\mathbb{N})\), the complete admissibility of the pair \((\mathcal {B}(\Theta, V(\mathbb{N}, X)), U(\mathbb{N}, X))\) implies the uniform exponential expansiveness, while the converse implication holds for \(V\in \mathcal {R}(\mathbb{N})\) and \(U\subset V\).

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
93D25 Input-output approaches in control theory
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