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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.

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