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Almost periodicity of mild solutions of inhomogeneous periodic Cauchy problems. (English) Zbl 0934.34049
The authors consider a mild solution \(u\) to the well-posed inhomogeneous Cauchy problem \[ u'(t)= A(t)u(t)+ f(t), \] on a Banach space \(X\), where \(A(\cdot)\) is periodic. For a problem on \(\mathbb{R}_+\) the authors show that \(u\) is asymptotically almost-periodic if \(f\) is asymptotically almost-periodic, \(u\) is bounded, uniformly continuous and totally ergodic, and the spectrum of the monodromy operator \(V\) contains only countably many points of the unit circle. For a problem on \(\mathbb{R}\), the authors prove that a bounded uniformly continuous solution \(u\) is almost-periodic if \(f\) is almost-periodic and various supplementary conditions are satisfied. Moreover, the authors prove that there is a unique bounded solution subject to certain spectral assumptions on \(V\), \(f\) and \(u\).

34G10 Linear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34L05 General spectral theory of ordinary differential operators
Full Text: DOI
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