On Favard's theorems (Q710520)

The paper is concerned with some generalizations of the classical Favard's theorems, that is Favard's theorem of the module containment and Favard's theorem of linear differential equations. The author first investigates the module containment of almost periodic/automorphic functions on \(\mathbb{R}\), and almost periodic/automorphic sequences on \(\mathbb{Z}\). Then he studies the Favard theory to the linear differential equation \[ \frac{d}{dt}x(t)=A(t)x(t)+B(t)x([t])+f(t) \] on \(\mathbb{R}^q\), which contains piecewise continuous delay \(t-[t]\), where \(A(t),\,B(t)\) are almost periodic functions with values in \(\mathbb{R}^{q\times q}\) and \(f(t)\) is an almost periodic function with values in \(\mathbb{R}^q\). The equivalence between almost automorphic functions and \(N\)-almost periodic ones is also discussed.