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Sufficient conditions for exponential stability and dichotomy of evolution equations

We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u′(t) = A(t)u(t) (∗) on a Banach space X. Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform constants and A(·) has a sufficiently small Holder constant, then (∗) has exponential dichotomy. We further study robustness of exponential dichotomy under time dependent unbounded Miyadera-type perturbations. Our main tool is a characterization of exponential dichotomy of evolution families by means of the spectra of the so-called evolution semigroup on C0(R, X) or L(R, X).