[For part 1, see S. Boulite, L. Maniar, A. Rhandi and J. Voigt, Positivity 8, No. 1, 1–9 (2004; Zbl 1065.34054); for part 2, see J. Voigt, Note Mat. (to appear).]
The authors deal with the \(C_0\)-semigroup associated with the linear differential equation with delay \[ \begin{gathered} u'(t)= Au(t)+ Lu_t\qquad (t\geq 0),\\ u(0)= x\in X,\quad j_0= f\in L_p(-h,0; X)\end{gathered} \] in the Banach lattice \(X\times L_p(-h,0;X)\), where \(X\) is a Banach lattice with order continuous norm and \(A\) is, in general, an unbounded generator of a \(C_0\)-semigroup possessing a modulus semigroup. To this end, they prove a “domination lemma” and introduce the delay semigroups in more detail. Using the “domination lemma”, the authors show that a semigroup dominating the perturbed (by the operator \(L\)) semigroup for the delay equation is also a dominating semigroup for the unperturbed semigroup. Moreover, the authors transfer the result to the framework of continuous functions, using consistent semigroups.