Existence, Uniqueness, and Regularity of Solutions to Semilinear Retarded Differential Equations

where 0 < T <∞; φ ∈ 0 := C([−τ,0];X), τ > 0; the linear operator A, defined from the domain D(A) ⊂ X into X , is such that −A generates a C0 semigroup {S(t) : t ≥ 0} of bounded linear operators in X ; the nonlinear map f is defined from [0,T]× T , T := C([−τ,T];X), into X ; the map h is defined from 0 into 0; and for (t,u) ∈ [0,T]× T , the function ut ∈ 0 is given by ut(s) = u(t + s) for s ∈ [−τ,0]. Here t := C([−τ, t];X) for t ∈ [0,T] is the Banach space of all continuous functions from [−τ, t] into X endowed with the supremum norm