The linear equation (E) u(t) = Au(t) + Lutwith an initial condition u0= y, in a Banach space X is considered: here A is a generator of an analytic semigroup in X, and L is a continuous linear operator from a phase space Y of functions on R−with values in D(−A)∞, 0<∞<1. The existence of the resolvent operator for (E), as well as the variation of parameters formula for a mild, strong and strict solution of the nonhomogeneous equation, are obtained. The method is based on the inverse Laplace transform and it also yields sharp stability results.