  In this paper we prove a variation of constants formula for a non autonomous
and non homogeneous Cauchy problems whenever the linear part is not densely
defined and is not a Hille-Yosida operator. By using this variation of
constants formula we derive a necessary and sufficient conditions for the
existence of exponential dichotomy for the evolution family generated by the
associated non autonomous homogeneous problem. We also prove a persistence
result of the exponential dichotomy for small perturbations. Finally we
illustrate our result by consider a parabolic equation with non local and non
autonomous boundary conditions.
