Ergodicity and its applications in regularity and solutions of pseudo-almost periodic equations (Q5949361)

A relation between ergodicity and regularity in pseudo-almost periodic equations is established. For this purpose the following operator is considered NEWLINE\[NEWLINE L:C^\prime(\mathbb{R})^n\rightarrow C(\mathbb{R})^n,\quad y\rightarrow Ly\equiv y^\prime+A(t)y. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEIt is shown that the following three statements are equivalent:NEWLINENEWLINENEWLINE(1) The operator \(L\) is regular.NEWLINENEWLINENEWLINE(2) An solution to the homogeneous equation \(Ly=0\) exhibits an exponential dichotomy.NEWLINENEWLINENEWLINE(3) For every \(f\in PAP_0(\mathbb{R})^n\), the inhomogeneous equation \(Ly=f\) has a unique solution in \(C(\mathbb{R})^n\).NEWLINENEWLINENEWLINEHere, \(C(\mathbb{R})^n\) and \(C^\prime(\mathbb{R})^n\) are the \(n\)th powers of \(C(\mathbb{R})\), respectively, where \(C(\mathbb{R})\) is the space of the bounded continuous functions on the real line \(\mathbb{R}\) supplied with the \(\sup\) norm and \(C^\prime(\mathbb{R})\) is the space of differentiable functions \(\varphi\) with \(\varphi^\prime\in C(\mathbb{R})\) (with the norm \(\|\varphi\|_{C^\prime(\mathbb{R})}= \|\varphi\|_{C(\mathbb{R})}+\|\varphi^\prime\|_{C(\mathbb{R})}\)). \(PAP(\mathbb{R})\) is the space of pseudo-almost periodic functions and \(PAP_0(\mathbb{R})\subset PAP(\mathbb{R})\) consists of those \(\varphi\in PAP(\mathbb{R})\) for which \(M(\varphi)\equiv\lim_{T\to\infty}{1\over 2T} \int_{-T}^T |\varphi |dt=0\). NEWLINENEWLINENEWLINEThe main result states that, if the matrix \(A(t)\) is such that \(a_{ij}=0\) for all \(i>j\) and \(a_{ii}\), \(i=1,\dots n\), are ergodic, then the operator \(L\) is regular if and only if \(M(\text{Re } a_{ii})\neq 0\), \(i=1,\dots,n\). Furthermore, if \(A(t)\) and \(f\) are in \(PAP(\mathbb{R})^n\) then the unique solution \(y\) to \(Ly=f\) is again in \(PAP(\mathbb{R})^n\).