Evolution Semigroups and Spectral Criteria for Almost Periodic Solutions of Periodic Evolution Equations

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Abstract:

We investigate spectral criteria for the existence of (almost) periodic solutions to linear 1-periodic evolution equations of the form dx/dt=A(t) x+f(t) with (in general, unbounded) A(t) and (almost) periodic f. Using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set eisp(f), then the above equation has an almost periodic (mild) solution xf which is unique if one requires sp(xf)⊂{+2k, kZ, sp(f)}. We emphasize that our method allows us to treat the equations without assumption on the existence of Floquet representation. This improves recent results on the subject. In addition we discuss some particular cases, in which the spectrum of monodromy operator does not intersect the unit circle, and apply the obtained results to study the asymptotic behavior of solutions. Finally, an application to parabolic equations is considered. Copyright 1999 Academic Press.

Document Type: Research Article

Affiliations: Department of Mathematics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182, Japan

Publication date: March 1, 1999

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