A characterization of well-posedness for abstract Cauchy problems with finite delay

Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay,{u′(t)=Au(t)+Fut,t>0;u(0)=x;u(t)=ϕ(t),−r≤t<0, solely in terms of a strongly continuous one-parameter family {G(t)}t≥0 of bounded linear operators that satisfy the functional equationG(t+s)x=G(t)G(s)x+∫−r0G(t+m)[SG(s+⋅)x](m)dm for all t,s≥0,x∈X. In case F≡0 this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the C0-semigroup generated by A.