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Exponential dichotomy roughness on Banach spaces. (English) Zbl 1093.34022

The author proves roughness results for exponential dichotomies of infinite-dimensional linear equations \[ \dot x = A(t)x . \] An exponential dichotomy on \(\mathbb R\) is rough if there exists \(\delta>0\) such that for \(\sup_{t \in \mathbb R} \| B(t)\| < \delta\) the perturbed equation \[ \dot x = [A(t) + B(t)]x \] also admits an exponential dichotomy. This problem is the topic of many research papers. The author gives an overview in the Introduction. The main contribution is to compute an explicit lower bound for \(\delta\) in infinite-dimensional Banach spaces by extending existing results on exponential dichotomy roughness, more precisely, the main problem of the existence of a bounded projection for the subspace of initial values of bounded solutions is tackled by extending an idea from W. A. Coppel [Dichotomies in stability theory. Lecture Notes in Mathematics. 629. Berlin-Heidelberg-New York: Springer-Verlag (1978; Zbl 0376.34001)].

MSC:

34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
47N20 Applications of operator theory to differential and integral equations

Citations:

Zbl 0376.34001
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Full Text: DOI

References:

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