A variation of constants formula for an abstract functional-differential equation of retarded type

Abstract

In this paper we obtain a variation of constants formula for the Cauchy problem associated to the nonhomogeneous retarded functional differential equation $$ x'(t)=L(x_t)+f(t), \quad t\geq 0 , \tag{*} $$ where $L:C([-r,0];E)\longrightarrow E$ is a bounded linear operator and $E$ is a Banach space. Once the existence and uniqueness theorem is proved, we define a fundamental solution associated with this problem and we use it to derive a variation of constants formula. The known results on the decomposition of the space $C([-r,0];E)$ with respect to the homogeneous equation $x'(t)=L(x_t)$ allow us to obtain a similar decomposition for the solutions of $(*)$. Moreover, the projection of solutions on a certain invariant finite-dimensional subspace is characterized in terms of the semigroup defined by the homogeneous problem.