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Discrete admissibility and exponential dichotomy for evolution families. (English) Zbl 1032.34048

Here, the authors study connections between admissibility and uniform exponential dichotomy for discrete and continuous evolution families. In the discrete case, they show that, for an evolution family with uniform exponential growth, an exponential dichotomy is equivalent to the admissibility of the pair \((c_0(X),c_{00}(X))\), where \(X\) is a Banach space and the sequence spaces \(c_0(X)=\{s:{\mathbb Z}_+\rightarrow X: \lim_{n\to\infty}s(n)=0\}\), \(c_{00}(X)=\{s\in c_0(X): s(0)=0\}\) are endowed with the sup-norm.
Having this result at hand, the authors are able to give a new proof of an admissibility theorem due to Nguyen Van Minh, F. Räbiger and R. Schnaubelt [Integral Equations Oper. Theory 32, 332–353 (1998; Zbl 0977.34056)]. This theorem states that, in a continuous setting, the uniform exponential dichotomy of a strongly continuous evolution family is equivalent to the admissibility of the pairs \((C_0(X),C_0(X))\) and \((C_0(X),C_{00}(X))\), respectively, with \(C_0(X)=\{s:{\mathbb R}_+\rightarrow X: \lim_{t\to\infty}s(t)=0\}\), \(C_{00}(X)=\{s\in C_0(X): s(0)=0\}\) equipped with the sup-norm.

MSC:

34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
39A12 Discrete version of topics in analysis
34G10 Linear differential equations in abstract spaces

Citations:

Zbl 0977.34056
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