Asymptotically Almost Automorphic Solutions of Dynamic Equations on Time Scales

In the present work, we introduce the concept of asymptotically almost automorphic functions on time scales and study their main properties. We study nonautonomous dynamic equations on time scales given by xΔ(t) = A(t)x(t) + f(t) and xΔ(t) = A(t)x(t) + f(t; x(t)), t ∈ T, where T is an invariant under translations time scale and A ∈ R(T,ℝn×n). We give new criteria ensuring the existence of an asymptotically almost automorphic solution for both equations. © 2019 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University in Toruń.