Separation conditions and stability properties in almost periodic systems

Almost Periodic Solutions of Nonlinear Volterra Difference Equations with Unbounded Delay

“…Moreover, by (14), we have ρ(w 0 , 0) ≤ δ 0 . However, this implies that ρ(w n , 0) → 0 as n → ∞; this contradicts (15). This proves the theorem.…”

Almost periodic linear differential equations with non-separated solutions

“…By summarizing the previous discussion, we showed that those almost periodic solutions to (12) which are representable over (Ω, Ψ ), are in one-to-one correspondence with the continuous solutions of (11). The next proposition is concerned with the discontinuous solutions to the same equation.…”

“…The proof is the same as for the quasi-periodic case; we give it hereafter just for the convenience of the reader. It is clear that, for every τ ∈ R, the translated function T τ x * is an almost periodic solution to (12) with respect to ω * · τ . Fix now an ω ∈ Ω and choose τ n such that ω * · τ n → ω.…”

“…On the other hand, standard arguments apply to show that the function x(t) = x(ω · t) solves the limiting equation (12). Sinceẋ…”

“…Before stating the result, let us notice that the set of the continuity points of any solution of (11) cannot be arbitrary: because of the uniqueness of the initial value problem associated to (12), it has to be a subset of Ω which is invariant for the flow induced by Ψ . Moreover, let us point out that we are interested in discontinuities which are in some sense unavoidable: if y is a function on Ω, and C ⊂ Ω is the set where y is continuous, we will say that y has nonremovable discontinuities when it cannot be modified outside C, obtaining a continuous function.…”