Resumen de Asymptotic Behavior of Linear and Almost Periodic Discrete Evolution Systems on Banach Space AAPr 0(Z+,W)

Akbar Zada, Muhammad Arif, Habiba Khalid

• Let vn be the solution of the nonhomogeneous evolution difference equation vn+1 = Anvn + αn+1, v0 = 0 for all n ∈ Z+, where An is a sequence of almost periodic (possibly unbounded) linear operators on a Banach space W. Let C00(Z+, W) is the space of allW-valued bounded sequences which decays at zero and at infinity and AP0(Z+, W)is the space of allW-valued almost periodic sequences decaying at zero.

  We consider the space AAPr 0(Z+, W) consisting of all sequences α(n) with relatively compact ranges for which there exists β(n) ∈ AP0(Z+, W) and γ (n) ∈ C00(Z+, W) such that α(n) = β(n)+γ (n). We prove that vn belongs to AAPr 0(Z+, W)if and only if for each x ∈ W the discrete evolution family of operators E = {E(n, m) : n, m ∈ Z+, n ≥ m} is uniformly exponentially stable. Our results are based on evolution semigroup approach.