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Uniform dichotomy and exponential dichotomy of evolution families on the half-line. (English) Zbl 1113.34038

The author characterizes uniform and exponential dichotomies of evolution equations on half-lines. With a discrete evolution family \(\Phi = \{\Phi(m,n)\}_{(m,n) \in \Delta}\) associate the subspace \(X_1 = \{x \in X : \Phi(\cdot,0)x \in \ell^\infty(\mathbb{N},X)\}\). Supposing that \(X_1\) is closed and complemented, he proves that the admissibility of the pair \((\ell^\infty(\mathbb{N},X), \ell_0^1(\mathbb{N},X))\) implies the uniform dichotomy of \(\Phi\). Under the same hypothesis on \(X_1\), he obtains that the admissibility of the pair \((\ell^\infty(\mathbb{N},X), \ell_0^p(\mathbb{N},X))\) with \(p \in (1,\infty]\) is a sufficient condition for an exponential dichotomy of \(\Phi\), which becomes necessary when \(\Phi\) has exponential growth. The author gives characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.

MSC:

34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
39A12 Discrete version of topics in analysis
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