Asymptotic Behavior of Linear and Almost Periodic Discrete Evolution Systems on Banach Space $\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})$

Abstract

Let $v_{n}$ be the solution of the nonhomogeneous evolution difference equation $v_{n+1} = \mathcal {A}_nv_{n}+ \alpha _{n+1}$, $v_0= 0$ for all $\ n\in \mathbb {Z}_+$, where $\mathcal {A}_n$ is a sequence of almost periodic (possibly unbounded) linear operators on a Banach space $\mathcal {W}$. Let $\mathcal {C}_{00}(\mathbb {Z}_+,\mathcal {W})$ is the space of all $\mathcal {W}$-valued bounded sequences which decays at zero and at infinity and $\mathcal {AP}_0(\mathbb {Z}_+,\mathcal {W})$ is the space of all $\mathcal {W}$-valued almost periodic sequences decaying at zero. We consider the space $\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})$ consisting of all sequences $\alpha (n)$ with relatively compact ranges for which there exists $\beta (n)\in \mathcal {AP}_0(\mathbb {Z}_+,\mathcal {W})$ and $\gamma (n) \in \mathcal {C}_{00}(\mathbb {Z}_+,\mathcal {W})$ such that $\alpha (n) = \beta (n) + \gamma (n)$. We prove that $v_{n}$ belongs to $\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})$ if and only if for each $x\in \mathcal {W}$ the discrete evolution family of operators $E = \{\mathcal {E}(n, m): n,m \in \mathbb {Z}_+, \ n\ge m \}$ is uniformly exponentially stable. Our results are based on evolution semigroup approach.