Asymptotic equivalence of differential equations and asymptotically   almost periodic solutions
  In this paper we establish asymptotic (biasymptotic) equivalence between spaces of solutions of a given linear homogeneous system and a perturbed system. The perturbations are of either linear or weakly linear characters. Existence of a homeomorphism between subspaces of almost periodic and asymptotically (biasymptotically) almost periodic solutions is also obtained. 
M. U. Akhmet, M.A. Tleubergenova, A. Zafer