Asymptotic behavior of solutions to abstract functional differential equations (Q1025016)

The authors consider ergodicity and spectra of functions taking values in a Banach space \(X\) and defined on \(\mathbb{R}\) or \(\mathbb{R}^+\). Let \(G\) be a closed translation-invariant subspace of \(BUC(\mathbb{R}; X)\) and \(E\subset BUC(\mathbb{R}^+; X)\) a translation bi-invariant subspace. For \(\nu\in BUC(\mathbb{R}, X)\), define the spectrum \(sp^{\mathbb{R}}_G\) of \(\nu\) with respect to \(G\) as the set of \(\eta\in\mathbb{R}\) such that, for any \(\varepsilon> 0\), there exists \(f\in L^1(\mathbb{R})\) such that the support of the Fourier transform of \(f\) is contained in \((\eta- \varepsilon, \eta+ \varepsilon)\) and \(f\) convolved with \(\nu\) is not in \(G\). For \(u\in BUC(\mathbb{R}^+, X)\), define the the spectrum \(sp^{\mathbb{R}^+}_E\) of \(u\) with respect to \(E\) as the set of \(\eta\in\mathbb{R}\) such that, for any \(\varepsilon> 0\), there exists \(f\in L^1(\mathbb{R})\) such that the support of the Fourier transform of \(f\) is contained in \((\eta-\varepsilon,\eta+\varepsilon)\) and the restriction to \(\mathbb{R}^+\) of \(f\) convolved with \(u\) is not in \(E\). The authors prove that, if \(C^+_0(\mathbb{R};X)\subset G\) and \(E= G\), where \(G\) restricted to \(\mathbb{R}^+\), and if \(u=\nu\) restricted to \(\mathbb{R}^+\) with \(\nu\in BUC(\mathbb{R}; X)\), then \(\text{sp}^{\mathbb{R}^+}_E(u)= sp^{\mathbb{R}}_G(\nu)\). This result is applied to prove that, if \(u\in BUC(\mathbb{R}; X)\) is such that \(sp^{\mathbb{R}}_G(u)\) is countable and if for some real \(s\) the translation \(u_s\) (restricted to \(\mathbb{R}^+\)) is totally ergodic on \(isp^{\mathbb{R}}_G(u)\) and for any \(\eta\in sp^{\mathbb{R}}_G(u)\) one has \(M_\eta(u_s)\in E\), then \(u\in G\). Here \(M_\eta\) denotes the Cesàro limit and \(u_s\) is restricted to \(\mathbb{R}^+\). These results are used to describe the asymptotic behavior of mild solutions of the equation \[ u'(t)= Au(t)+ Bu(t)+ f(t),\qquad t\in \mathbb{R}. \] The work is related to and extends results in [\textit{W. Arendt} and \textit{C. J. K. Batty}, J. Differ. Equations 137, No.~2, 363--383 (1997; Zbl 0879.34046); \textit{B. Basit}, Some problems concerning different types of vector-valued almost periodic functions. Diss. Math. 338 (1995; Zbl 0828.43004)].