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Almost periodic solutions of first- and second-order Cauchy problems. (English) Zbl 0879.34046
Let $$S(t)$$ be the shift group on the space $$B\cup C(\mathbb{R},X)$$ of all bounded uniformly continuous functions $$x: \mathbb{R}\to X$$, where $$X$$ is a complete $$B$$-space and let $$\overline S(t)$$ be the induced group on $$B\cup C(\mathbb{R}, X)/AP(\mathbb{R},X)$$ with generator $$\overline B$$. The authors use spectral properties of bounded groups to reformulate and prove the known Kadet’s result, namely it holds $$c_0 \not \subset X$$ iff $$\overline B$$ has no point spectrum, or, in other situations, if an ergodicity condition holds. Section 3 contains a spectral characterisation of almost periodic functions and the results are used in Section 4 to prove almost periodicity of solutions of some first- and second-order inhomogeneous Cauchy problems. The paper closes with an investigation of the case where the imaginary spectrum of the operator consists only of poles.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 47A10 Spectrum, resolvent 34G10 Linear differential equations in abstract spaces
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