A variation of constants formula for an abstract functional differential equation of retarded type (Q1374491)

The authors derive a variation of parameters formula for the equation \(x'(t)=L(x_t)+f(t)\), where \(L:C([-r,0],E)\to E\) is a bounded linear operator, \(E\) is a Banach space and \(x_t(\theta)=x(t+\theta), \theta \in [-r,0]\). They use a decomposition of the space \(C([-r,0],E)\) with respect to the spectrum of a generator of a solution semigroup for the homogeneous problem to obtain a decomposition of the solution and corresponding projectors.