Existence of nonuniform exponential dichotomies and a Fredholm alternative

We consider linear equations x^′=A(t)x in a Banach space, with nonuniform exponential dichotomies in and , 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.