Almost Periodicity of Mild Solutions of Inhomogeneous Periodic Cauchy Problems

Abstract

We consider a mild solution u of a well-posed, inhomogeneous, Cauchy problem, u* (t)=A(t) u(t)+ f (t), on a Banach space X, where A( } ) is periodic. For a problem on R + , we 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 R, we show that a bounded, uniformly continuous solution u is almost periodic if f is almost periodic and various supplementary conditions are satisfied. We also show that there is a unique bounded solution subject to certain spectral assumptions on V, f and u. Academic Press

Boundedness and Almost Periodicity of Solutions of Partial Functional Differential Equations

“…Recently, evolution semigroups have been applied to study almost periodic solutions of evolution equations in [35]. In this direction see also [6,33,36], and especially [24] in which a systematic presentation has been made.…”

Asymptotic properties of mild solutions of nonautonomousevolution equations with applications to retarded differential equations

“…For a detailed account of the numerous other results in this direction we refer to [7,22]. Now assume that (1.2) is p-periodic, that is, A(t + p) = A(t), t ∈ R. It has been shown in [6,19,27,30] that under a certain spectral condition (nonresonance condition) on the monodromy operator U(p, 0) and the inhomogeneity f there is a p-periodic (respectively, almost periodic) mild solution of (1.1) provided that f has the same property. Moreover, u is unique subject to certain spectral assumptions.…”

Almost automorphic solutions of evolution equations

“…Actually, it goes back to the characterization of exponential dichotomy of linear ordinary differential equations by O. Perron. The reader can find many extensions of the classical result of Perron to the infinite-dimensional case in [4,11,15,17,27] and the references therein with results concerned with almost periodic solutions and bounded solutions. Recently, the interest in finding conditions for the existence of automorphic solutions has been regained (see, e.g., [21,26]).…”