Fréchet differentiability of the solutions of a semilinear abstract Cauchy problem

Sufficient conditions are found under which the solutions z(t ;q) of a semilinear abstract Cauchy problem of the form d dt z(t) = A(q)z(t) + F(q, t,z(t)) are Fréchet differentiable with respect to the parameter q. An explicit form is provided for the sensitivity equation satisfied by the Fréchet derivative Dq z(t ;q).  2005 Elsevier Inc. All rights reserved