A spectral countability condition for almost automorphy of solutions of differential equations

Abstract

We consider the almost automorphy of bounded mild solutions to equations of the form (*) dx/dt = A(t)x+f (t) with (generally unbounded) τ-periodic A(•) and almost automorphic f (•) in a Banach space X. Under the assumption that X does not contain c 0 , the part of the spectrum of the monodromy operator associated with the evolutionary process generated by A(•) on the unit circle is countable. We prove that every bounded mild solution of (*) on the real line is almost automorphic.

Almost automorphic delayed differential equations and Lasota-Wazewska model

“…Now, we note that AP (R, V ) and AA(R, V ) are vectorial space and AP (R, V ) is a proper subspace of AA(R, V ), since for instance ψ(t) = cos [2 + sin(t) + sin(t √ 2)] −1 is an almost periodic function but not almost authomorphic. Similarly, it is proven that the inclusion AP (R, V ) ⊂ BC(R, V ), for an extensive discussion consult [4,3,15,14,21,23,24,25,26,34,35,42,44,10,1,39,43,12].…”

“…Since there are plenty of results in literature, let us just quote, for their applications in engineering and life science, for example asymptotically almost periodic functions [17,18,19,20,27,29,30,31,32,33,40], and pseudo almost periodic functions [9,11,28]. Moreover, we recall that N'Guérékata has given a huge impulse to the study of almost automorphic solutions of differential equations [1,7,12,14,15,21,23,25,26]. For some recent results of almost automorphic differential equations consult also [5,6].…”

Equality of uniform and Carleman spectra for bounded measurable functions

“…The Carleman transform (see (1.3)), spectrum (see (1.7)) and the uniform spectrum are defined similarly. In [15], [24], the uniform Carleman spectrum sp Cu (φ) for φ ∈ BC(R, X) is introduced and it is shown that sp C (φ) ⊂ sp Cu (φ). Many properties of sp C (φ) are shown to hold true for sp Cu (φ) (see [15,Proposition 2.3]); however, equality is not established.…”