Dichotomy and asymptotic equivalence of bounded solutions for systems in Banach spaces

In this paper we study the asymptotic equivalence of bounded solutions for a linear system and its perturbed system in a Banach space. We give sufficient conditions for the asymptotic equivalence by a dichotomy together with the Lipschitz property of the composition of the Green function and the perturbed term. The conditions permit that the dichotomy has different growth rates on the subspaces of the invariant decomposition and the compositions of the perturbed term and the projections have different Lipschitz constants, which are weaker than the ones in the known literatures even in the finite-dimensional case.