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In this paper we prove a variation of constants formula for a non autonomousand non homogeneous Cauchy problems whenever the linear part is not denselydefined and is not a Hille-Yosida operator. By using this variation ofconstants formula we derive a necessary and sufficient conditions for theexistence of exponential dichotomy for the evolution family generated by theassociated non autonomous homogeneous problem. We also prove a persistenceresult of the exponential dichotomy for small perturbations. Finally weillustrate our result by consider a parabolic equation with non local and nonautonomous boundary conditions.

Variation of constants formula and exponential dichotomy for non autonomous non densely defined Cauchy problems