Existence of nonuniform exponential dichotomies and a Fredholm alternative

We consider linear equations x′=A(t)xx′=A(t)x in a Banach space, with nonuniform   exponential dichotomies in R+R+ and R−R−, and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations. We also obtain a Fredholm alternative for the existence of bounded solutions under nonlinear perturbations.