The modulus semigroup for linear delay equations III

Martin Stein, Hendrik Vogt, and Jürgen Voigt Fachrichtung Mathematik, Technische Universität Dresden,

01062 Dresden, Germany

Abstract

In this paper we describe the modulus semigroup of the C_{0}-semigroup associated with the linear differential equation with delay

\begin{cases}u′(t) = Au(t) + Lu_{t} (t ⩾ 0), \\ u(0) = x ∈ X, u_{0} = f ∈ L_{p}(−h, 0; X), \end{cases}

in the Banach lattice X × L_{p}(−h, 0; X), where X is a Banach lattice with order continuous norm. The progress with respect to previous papers is that A may be an unbounded generator of a C_{0}-semigroup possessing a modulus semigroup.

MSC 2000: 47D06, 47B60 Keywords: functional differential equation, delay equation, domination, modulus semigroup, Banach lattice

Introduction

In the present paper the results of the papers [3], [11] are further generalised. The main object is to obtain the modulus semigroup for a C_{0}-semigroup arising in the study of the initial value problem for a linear differential equation with delay,

\begin{cases}u′(t) = Au(t) + Lu_{t} (t ⩾ 0), \\ u(0) = x, u_{0} = f , \end{cases}  (DE)

in the L_{p}-context, for 1 ⩽ p < ∞, with initial values x ∈ X, f ∈ L_{p}(−h, 0; X). Here, X is a Banach lattice with order continuous norm, and h = 1 or h = ∞, corresponding to finite or infinite delay. Further, A is the (possibly unbounded) generator of a C_{0}-semigroup on X—the unboundedness of A is the important new feature in this paper—, and L: C([−h, 0]; X) → X is the bounded linear operator given by

mstein@math.tu-dresden.de, vogt@math.tu-dresden.de, voigt@math.tu-dresden.de

Lf := ∫_{[−h,0]} dη(ϑ)f(ϑ) (f ∈ C([−h, 0]; X)),

where η: [−h, 0] → L(X) is a function of bounded variation with no mass in zero. Also, for a function u: (−h, ∞) → X, we recall the notation

u_{t}(ϑ) := u(t + ϑ) (−h < ϑ < 0),

for t ⩾ 0.

It is shown in [1] that the delay equation (DE) is equivalent to an abstract Cauchy problem

\begin{cases}U′(t) = AU(t) & (t ⩾ 0), \\ U(0) = ( \begin{matrix}x \\ φ \end{matrix} ) \end{cases}

on the space X × L_{p}(−h, 0; X), where A is given by

A := (\begin{matrix}A & L \\ 0 & \frac{d}{dϑ} \end{matrix}) ,

with domain

D(A) := {(x, φ) ∈ D(A) × W_{p}^{1}(−h, 0; X); φ(0) = x} .

From [1], [4], [5], [8] it is known that the operator A generates a C_{0}-semigroup T := (e^{tA})_{t⩾0} on the Banach lattice X × L_{p}(−h, 0; X).

Next, assume that the C_{0}-semigroup generated by A possesses a modulus semigroup, i.e., a smallest semigroup dominating (e^{tA})_{t⩾0}, whose generator will be denoted by A^{#}. Also, assume that η is ‘of finite regular variation’ (see Section 1.3 for details), which implies that the operator L possesses a modulus. It is the object of the paper to show that then the C_{0}-semigroup generated by

Ã:= (\begin{matrix}A^{#} & |L| \\ 0 & \frac{d}{dϑ} \end{matrix}) ,

with domain

D(Ã) := {(x, φ) ∈ D(A^{#}) × W_{p}^{1}(−h, 0; X); φ(0) = x} ,

is the modulus semigroup of the C_{0}-semigroup generated by A.

This result is shown in [3] for the case X = ℝ^{n}, where necessarily the generator A is a bounded operator. In [11] the result is generalised to the case of a Banach lattice X with order continuous norm, but still with a bounded generator A.

The first result on the subject is contained in [2], where the case X = ℝ^{n} is treated in the framework of continuous functions (instead of X × L_{p}(−h, 0; X)). We will also generalise this result to the case where X is a Banach lattice with order continuous norm; cf. Section 4.

For motivation why it is interesting to investigate modulus semigroups we refer to [2], [7], [11].

In Section 1 we recall certain notions and present some results needed in the sequel. We prove a ‘domination lemma’, and we introduce the delay semigroups in more detail. In particular, in the second part of Section 1.2 we indicate a new (simplified) method to treat the perturbed delay equation in the case p = 1.

In Section 2 we apply the ‘domination lemma’ of Section 1.1 in order to 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.

In Section 3 we show the main result. Besides the new ideas prepared in Section 2 the proof relies heavily on results contained in [11].

In Section 4 we transfer the result to the framework of continuous functions, using consistent semigroups.

1 Preliminaries

1.1 The domination lemma

For use in Section 2 we single out the following ‘domination lemma’. A version of this technical result was already used in [10; proof of Proposition 1.2].

1.1 Lemma. Let X be a Banach lattice. Let T, S be C_{0}-semigroups on X, S positive, and assume that R: [0, 1] → L(X) satisfies

\frac{1}{t}∥R(t)∥ → 0 (t → 0).

Assume that

|T (t)x| ⩽ S(t)|x| + |R(t)x|  (1.1)

for all x ∈ X, 0 ⩽ t ⩽ 1.

Then T is dominated by S, i.e., |T (t)x| ⩽ S(t)|x| (x ∈ X, t ⩾ 0).

Proof. Let x ∈ X. By induction, inequality (1.1) yields

|T (t)^{n}x| ⩽ S(t)^{n}|x| + ∑_{m=1}^{n} S(t)^{n−m}|R(t)T (t)^{m−1}x|

for all 0 ⩽ t ⩽ 1, n ∈ ℕ. Replacing t by t/n we obtain

|T (t)x| ⩽ S(t)|x| + ∑_{m=1}^{n} S(\frac{n−m}{n}t) |R(\frac{t}{n})T (\frac{m−1}{n}t)x|  (1.2)

for all n ∈ ℕ, 0 ⩽ t ⩽ n. With c_{t} := sup_{0⩽s⩽t} ∥S(s)∥, d_{t} := sup_{0⩽s⩽t} ∥T (s)∥ the last term in inequality (1.2) can be estimated as

∥∑_{m=1}^{n} S(\frac{n−m}{n}t) |R(\frac{t}{n})T (\frac{m−1}{n}t)x|∥ ⩽ nc_{t}∥R(\frac{t}{n})∥d_{t}∥x∥ = c_{t}d_{t} t \frac{n}{t}∥R(\frac{t}{n})∥∥x∥.

Since this tends to zero as n → ∞, inequality (1.2) yields the assertion.

1.2 The delay semigroup

In this part we fix our assumptions concerning the delay semigroup. Assume that X is a Banach space. We assume that the operator A in X is the generator of a C_{0}-semigroup T. We assume that h ∈ {1, ∞}, we choose p ∈ [1, ∞), and we denote by S the C_{0}-semigroup of left translation on L_{p}(−h, 0; X),

S(t)φ(ϑ) := \begin{cases}φ(t + ϑ) & for − h < ϑ < −t, \\ 0 & for − t < ϑ < 0. \end{cases}

We recall that the operator A_{0} in X × L_{p}(−h, 0; X),

A_{0} := (\begin{matrix}A & 0 \\ 0 & \frac{d}{dϑ} \end{matrix}) , D(A_{0}) := \{(x, φ) ∈ D(A) × W_{p}^{1}(−h, 0; X); φ(0) = x\},

is the generator of a C_{0}-semigroup T_{0} which is given by

T_{0}(t) = (\begin{matrix}T (t) & 0 \\ T_{t} & S(t) \end{matrix}) .  (1.3)

Here, T_{t} ∈ L(X, L_{p}(−h, 0; X)) denotes the operator defined by

T_{t}x(ϑ) := \begin{cases}0 & for − h < ϑ < −t, \\ T (t + ϑ)x & for − t < ϑ < 0. \end{cases}

For these statements we refer to [1; Proposition 3.1].

Next, let η: [−h, 0] → L(X) be a function of bounded variation (where, in the case of h = ∞, [−h, 0] denotes the one point compactification of (−∞, 0]). Then one can define an operator L ∈ L(C([−h, 0]; X), X) by

Lφ := ∫ dη(ϑ)φ(ϑ) (φ ∈ C([−h, 0]; X));

we refer to [11; Section 2] for details. We assume that η is left continuous, i.e.,

η(ϑ) = lim_{ϑ′→ϑ−} η(ϑ′)  (1.4)

for all ϑ ∈ (−h, 0]. For ϑ ∈ (−h, 0), this can always be achieved by redefining η, without changing L. For ϑ = 0, however, this means that η does not give rise to mass at zero; we refer to [8; beginning of Section 2] for a short discussion concerning this assumption. We recall that, as a consequence, the variation

|η|([−α, 0]) := sup { ∑_{j=1}^{n} ∥η(ϑ_{j}) − η(ϑ_{j−1})∥; −α = ϑ_{0} < … < ϑ_{n} = 0, n ∈ ℕ},

of η on [−α, 0] tends to zero as α → 0+; cf. [11; Lemma 2.1].

We are going to show that B := (\begin{matrix}0 & L \\ 0 & 0 \end{matrix}) is a small Miyadera perturbation

(cf. [9], [12], [6], [8]) of A_{0}, for any p ∈ [1, ∞), if the norm on X × L_{p}(−h, 0; X) is chosen suitably. For 1 < p < ∞ this is known (and true for any of the norms on the product), whereas for p = 1, this is a new observation (making part of the paper [8] obsolete). We recall the following estimate from [1; Example 3.1]: With M := sup_{0⩽s⩽1} ∥T (s)∥, \frac{1}{p′} + \frac{1}{p} = 1, one has

∫_{0}^{t} ∥L(T_{s}x + S(s)f)∥ ds ⩽ tM|η|([−t, 0])∥x∥ + t^{1/p′}|η|([−h, 0]) ∥f∥_{p} , (1.5)

for all (x, f) ∈ D(A_{0}), 0 ⩽ t ⩽ 1. Note that L(T_{s}x+S(s)f) is the first component of BT_{0}(s)( \begin{matrix}x \\ f \end{matrix}) (the second component being zero). If 1 < p < ∞, then the coefficients of ∥x∥ and ∥f∥_{p} tend to zero as t → 0, and therefore B is an infinitesimally small Miyadera perturbation of A_{0}. For p = 1, however, we choose a norm

∥(x, f)∥_{c} := ∥x∥ + c ∥f∥_{1} ,

with c > |η|([−h, 0]). Then (1.5) shows

∫_{0}^{t} ∥BT_{0}(s)( \begin{matrix}x \\ f \end{matrix})∥_{c} ds ⩽ tM|η|([−t, 0])∥x∥ + |η|([−h, 0]) ∥f∥_{1}
⩽ max(tM|η|([−t, 0]), |η|([−h, 0])/c) ∥(x, f)∥_{c} ,

for all (x, f) ∈ D(A_{0}), 0 ⩽ t ⩽ 1, where max(tM|η|([−t, 0]), |η|([−h, 0])/c) < 1 for small t, i.e., B is a small Miyadera perturbation of A_{0}. These statements

imply that A := (\begin{matrix}A & L \\ 0 & \frac{d}{dϑ} \end{matrix}), with D(A) = D(A_{0}), is the generator of a C_{0}-semi-

group T , for all 1 ⩽ p < ∞. The semigroup T is associated with the Cauchy problem (DE).

1.3 The dominating delay semigroup

Additionally to the assumptions of Section 1.2 we now assume that X is a Banach lattice with order continuous norm, and that the C_{0}-semigroup T possesses a modulus semigroup T^{#}, with generator A^{#}. Applying the assertions of Sec-

tion 1.2 we obtain that Ã_{0} := (\begin{matrix}A^{#} & 0 \\ 0 & \frac{d}{dϑ} \end{matrix}), with domain D(Ã_{0}) := \{(x, φ) ∈

D(A^{#}) × W_{p}^{1}(−h, 0; X); φ(0) = x}, is the generator of a C_{0}-semigroup T̃_{0}.

We assume that the function η is ‘of bounded regular variation’, i.e., η takes its values in the regular operators,

η̃(t) := sup { ∑_{j=1}^{n} |η(ϑ_{j}) − η(ϑ_{j−1})|; −h = ϑ_{0} < ⋯ < ϑ_{n} = t, n ∈ ℕ}

exists for all −h ⩽ t ⩽ 0, and η̃ is of bounded variation. It has been shown in [11; Lemma 3.1] that then the function η̃ is left continuous, in particular

η̃(0) = lim_{ϑ→0} η̃(ϑ).

Also, it has been shown in [11; Proposition 2.5] that the operator associated with the function η̃ is the modulus |L| ∈ L(C([−h, 0]; X), X) of L. Again, the operator

Ã:= (\begin{matrix}A^{#} & |L| \\ 0 & \frac{d}{dϑ} \end{matrix}), with D(Ã) := D(Ã_{0}), generates a C_{0}-semigroup T̃.

1.2 Remarks. (a) From the expression (1.3) for the semigroup T_{0}, and the corresponding expression for the semigroup T̃_{0}, it is immediate that T̃_{0} dominates T_{0}.

(b) Arguing as in [3; Lemma 2.1] one shows that T̃ dominates T . (In fact, in view of the second part of Section 1.2 it is no longer necessary to treat the case p = 1 separately.)

(c) The Banach lattice X × L_{p}(−h, 0; X) has order continuous norm. Therefore it follows from part (a) and [2; Theorem 2.1] that T_{0} possesses a modulus semigroup T_{0}^{#}, and T_{0}^{#}(t) ⩽ T̃_{0}(t) (t ⩾ 0). In the same way, the C_{0}-semigroup T possesses a modulus semigroup T^{#}, and T^{#}(t) ⩽ T̃(t) (t ⩾ 0).

2 Domination of unperturbed and perturbed delay semigroups

In the present section we assume that X is a Banach lattice, and that A, L, 1 ⩽ p < ∞, A_{0}, T_{0}, A, T are as in Section 1.2.

The following result is the main tool for helping to identify the domain of the generator of the modulus semigroup for the delay semigroup; cf. Section 3.

2.1 Proposition. Let the notation be as above, and assume that T is dominated by a C_{0}-semigroup S on X × L_{p}(−h, 0; X). Then T_{0} is dominated by S as well.

Recall from Section 1.2 that B := (\begin{matrix}0 & L \\ 0 & 0 \end{matrix}) is a (small) Miyadera perturbation

of A. Therefore, T can be represented as

T (t) = T_{0}(t) + R_{1}(t),  (2.1)

with

R_{1}(t)( \begin{matrix}x \\ φ \end{matrix}) = ∫_{0}^{t} T (t − s)BT_{0}(s)( \begin{matrix}x \\ φ \end{matrix}) ds ((x, φ) ∈ D(A_{0})).

The procedure of the proof of Proposition 2.1 is similar to [10; proof of Proposition 1.2]. The method consists in finding parts in the representation (2.1) of T allowing to estimate T_{0}, and other parts allowing an estimate needed for the application of Lemma 1.1. The difference to [10] is that in that paper one has to use an iterated form of (2.1).

2.2 Lemma. There exists c ⩾ 0 such that

∥R_{1}(t)( \begin{matrix}x \\ 0 \end{matrix} )∥ ⩽ ct|η|([−t, 0])∥x∥,

for all x ∈ D(A), 0 ⩽ t ⩽ 1.

Proof. This inequality is shown in the same way as [3; inequality (2.4) in Lemma 2.2(a)].

2.3 Remark. In the proof of Proposition 2.1 we will need the following general fact about delay semigroups. For f ∈ L_{p}(−h, 0; X), 0 ⩽ t < h one has

1_{(−h,−t)}P_{2}T (t)( \begin{matrix}0 \\ f \end{matrix}) = S(t)f

(where P_{2} is the projection onto the second component of X ×L_{p}(−h, 0; X)), and this implies

|T_{0}(t)( \begin{matrix}0 \\ f \end{matrix})| = |( \begin{matrix}0 \\ S(t)f \end{matrix})| ⩽ |T (t)( \begin{matrix}0 \\ f \end{matrix})|.

Proof of Proposition 2.1. Let (x, f) ∈ X × L_{p}(−h, 0; X). We estimate (using Remark 2.3 in the second estimate)

|T_{0}(t)( \begin{matrix}x \\ f \end{matrix})| ⩽ |T (t)( \begin{matrix}x \\ 0 \end{matrix} )| + |(T_{0}(t) − T (t))( \begin{matrix}x \\ 0 \end{matrix} )| + |T_{0}(t)( \begin{matrix}0 \\ f \end{matrix})|

⩽ S(t)|( \begin{matrix}x \\ 0 \end{matrix} )| + |R_{1}(t)( \begin{matrix}x \\ 0 \end{matrix} )| + S(t)|( \begin{matrix}0 \\ f \end{matrix})|

= S(t)|( \begin{matrix}x \\ f \end{matrix})| + |R(t)( \begin{matrix}x \\ f \end{matrix})|,

where R(t) := R_{1}(t)( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ). By Lemma 2.2 we have ∥R(t)( \begin{matrix}x \\ f \end{matrix})∥ ⩽ ct|η|([−t, 0])∥x∥, and thus

\frac{1}{t}∥R(t)∥ → 0 (t → 0).

(Recall that |η|([−t, 0]) → 0 because η is assumed to induce no mass at 0.) By Lemma 1.1 we obtain that T_{0} is dominated by S.

3 The modulus semigroup

In this section we assume that X is a Banach lattice with order continuous norm.

3.1 Theorem. Let X, A, and Ã be as introduced in Section 1.3. Then A^{#} = Ã.

The following result will serve as a final preparation for the proof.

3.2 Proposition. With the previous hypotheses and notations, we have: (a) T_{0}^{#}(t) ⩽ T^{#}(t) ⩽ T̃(t), for all t ⩾ 0.

(b) A_{0}^{#} = Ã_{0}.

Proof. (a) The first inequality follows from Proposition 2.1 since T^{#} is a C_{0}-semi-group dominating T . The second inequality was mentioned in Remark 1.2(c).

(b) From Section 1.2 we recall the representation (1.3), and correspondingly,

T̃_{0}(t) = (\begin{matrix}T^{#}(t) & 0 \\ (T^{#})_{t} & S(t) \end{matrix}) .  (3.1)

The inequalities |T_{0}(t)| ⩽ T_{0}^{#}(t) ⩽ T̃_{0}(t) (for the second of these inequalities we refer to Remark 1.2(c)) show that T_{0}^{#} is of the form

T_{0}^{#}(t) = (\begin{matrix}T_{11}^{#}(t) & 0 \\ V (t) & S(t) \end{matrix}) ,

with positive operators T_{11}^{#}(t), V (t) satisfying

|T (t)| ⩽ T_{11}^{#}(t) ⩽ T^{#}(t),  (3.2)
|T_{t}| ⩽ V (t) ⩽ (T^{#})_{t} ,  (3.3)

for all t ⩾ 0.

From the semigroup property of T_{0}^{#} one obtains that (T_{11}^{#}(t))_{t⩾0} is a C_{0}-semi-group on X, and therefore (3.2) implies T_{11}^{#} = T^{#}.

Inequality (3.3) implies spt V (t)x ⊆ [−t, 0), for all x ∈ X. Let (x, φ) ∈ D(A_{0}^{#}), 0 < t < h. Then T_{0}^{#}(t)( \begin{matrix}x \\ φ \end{matrix}) = ( T^{#}(t)x ; φ̃_{t}), where

φ̃_{t}(ϑ) = \begin{cases}φ(t + ϑ) & for −h < ϑ ⩽ −t, \\ V (t)x(ϑ) & for −t < ϑ < 0. \end{cases}

The existence of lim_{t→0} \frac{1}{t}(T_{0}^{#}(t)( \begin{matrix}x \\ φ \end{matrix}) − ( \begin{matrix}x \\ φ \end{matrix})) shows x ∈ D(A^{#}), φ ∈ W_{p}^{1}(−h, 0; X), and we obtain A_{0}^{#}( \begin{matrix}x \\ φ \end{matrix}) = ( \begin{matrix}A^{#}x \\ φ′ \end{matrix}). Since T_{0}^{#} leaves D(A_{0}^{#}) invariant, we also obtain φ̃_{t} ∈ W_{p}^{1}(−h, 0; X), and hence

φ(0) = φ̃_{t}(−t) = lim_{ϑ→(−t)+} φ̃_{t}(ϑ) = lim_{ϑ→(−t)+} V (t)x(ϑ) = x,

where the last equality holds because of (3.3). (In fact, this last equality is first shown for x ⩾ 0, and then carries over to general x.)

Thus we have shown A_{0}^{#} ⊆ Ã_{0}. Since both of these operators are generators we conclude A_{0}^{#} = Ã_{0}.

Proof of Theorem 3.1. From Proposition 3.2(a) and [10; Proposition A.1] we obtain D(Ã) (= D(A_{0}^{#})) ⊆ D(A^{#}). For (x, φ) ∈ D(Ã)_{+} we have

T_{0}^{#}(t)( \begin{matrix}x \\ φ \end{matrix}) ⩽ T^{#}(t)( \begin{matrix}x \\ φ \end{matrix}) ⩽ T̃(t)( \begin{matrix}x \\ φ \end{matrix}) (t ⩾ 0),

and this implies

(\begin{matrix}A^{#}x \\ φ′ \end{matrix}) = A_{0}^{#} (\begin{matrix}x \\ φ \end{matrix}) ⩽ A^{#} (\begin{matrix}x \\ φ \end{matrix}) ⩽ Ã(\begin{matrix}x \\ φ \end{matrix}) = (\begin{matrix}A^{#}x + |L|φ \\ φ′ \end{matrix}) .  (3.4)

We define

L^{#}φ := P_{1}A^{#}( \begin{matrix}φ(0) \\ φ \end{matrix}) − A^{#}φ(0)

(where P_{1} is the projection onto the first component of X ×L_{p}(−h, 0; X)). Then (3.4) implies

A^{#} (\begin{matrix}x \\ φ \end{matrix}) = (\begin{matrix}A^{#}x + L^{#}φ \\ φ′ \end{matrix})

and

0 ⩽ L^{#}φ ⩽ |L|φ.  (3.5)

If additionally φ(0) = 0 then we obtain

L^{#}φ = P_{1}A^{#}( \begin{matrix}0 \\ φ \end{matrix}) = lim_{t→0} \frac{1}{t}P_{1}T^{#}(t)( \begin{matrix}0 \\ φ \end{matrix}).  (3.6)

Now, let ψ ∈ W_{p}^{1}(−h, 0; X), ψ(0) = 0. Then (0, ψ) ∈ D(A), |ψ| ∈ W_{p}^{1}(−h, 0; X) (by [11; Theorem 1.1]), and therefore (0, |ψ|) ∈ D(A^{#}). Now (3.6), the corresponding equality for L and T , and |T (t)( \begin{matrix}0 \\ ψ \end{matrix})| ⩽ T^{#}(t) |( \begin{matrix}0 \\ ψ \end{matrix})| (t ⩾ 0) imply

|Lψ| ⩽ L^{#}|ψ|.  (3.7)

We are going to show that (3.5), (3.7) imply equality in (3.5). First observe that, since η̃ does not give rise to mass at 0, there exists a sequence (φ_{k}) in W_{p}^{1}(−h, 0; X), φ_{k}(0) = 0, 0 ⩽ φ_{k} ⩽ φ (k ∈ ℕ) such that |L|φ_{k} → |L|φ (k → ∞). For k ∈ ℕ the application of [11; Theorem 1.1 and Remark 1.2] yields

|L|φ_{k} = sup{|Lψ|; ψ ∈ W_{p}^{1}(−h, 0; X), |ψ| ⩽ φ_{k}}
⩽ sup{L^{#}|ψ|; ψ ∈ W_{p}^{1}(−h, 0; X), |ψ| ⩽ φ_{k}} ⩽ L^{#}φ_{k} ⩽ L^{#}φ.

For k → ∞ we conclude |L|φ ⩽ L^{#}φ.

Having established equality in (3.5) we have shown Ã ⊆ A^{#}. Since both of these operators are generators we obtain Ã = A^{#}.

4 The modulus semigroup in the space of continuous functions

We assume that all the quantities are as in Section 1.3. We want to treat the delay semigroup in the space of continuous functions and to show properties analogous to those of the preceding section.

For convenience, we only treat the case h = 1 and refer to Remark 4.3(c) for the necessary modifications for h = ∞. The delay semigroup T_{C} in C([−1, 0]; X), associated with the Cauchy problem (DE) is generated by the operator A_{C},

D(A_{C}) := {φ ∈ C^{1}([−1, 0]; X); φ(0) ∈ D(A), φ′(0) = Aφ(0) + Lφ},
A_{C}φ := φ′;

cf. [6; Chap. VI, Sec. 6].

For the remainder of this section we fix 1 ⩽ p < ∞. The operator J_{p}: C([−1, 0]; X) → X × L_{p}(−h, 0; X), J_{p}φ := (φ(0), φ), is continuous. For φ ∈ D(A_{C}), the function u(t) := T_{C}(t)φ (t ⩾ 0) is the unique solution of the Cauchy problem for the delay differential equation

u′(t) = Au(t) + Lu_{t} , u_{0} = φ.

It is easy to see that this implies that t ↦ J_{p}u(t) =: u_{p}(t) is a solution of

u′_{p}(t) = A_{p}u_{p}(t), u_{p}(0) = (φ(0), φ),

and therefore T_{p}(t)( \begin{matrix}φ(0) \\ φ \end{matrix}) = u_{p}(t). These considerations show the following result.

4.1 Proposition. (a) The semigroups T_{C} and T_{p} are consistent, in the sense that J_{p}T_{C}(t) = T_{p}(t)J_{p} (t ⩾ 0).

(b) The semigroups T̃_{C} and T̃_{p} are consistent.

For the proof of Theorem 4.2 below we recall how the modulus semigroup T_{p}^{#} can be obtained. We denote by Γ the set of all subdivisions of 1 by positive reals,

Γ = {γ ∈ (0, 1]^{n}; γ_{1} + ⋯ + γ_{n} = 1, n ∈ ℕ}.

For t ⩾ 0, γ = (γ_{1}, …, γ_{n}) ∈ Γ we define

(T_{p})_{γ}(t) := |T_{p}(γ_{n}t)| ⋯ |T_{p}(γ_{1}t)|

and, for (x, φ) ∈ X_{+} × L_{p}(−h, 0; X)_{+}, obtain

T_{p}^{#}(t)( \begin{matrix}x \\ φ \end{matrix}) = sup_{γ∈Γ}(T_{p})_{γ}(t)( \begin{matrix}x \\ φ \end{matrix}) = lim_{γ∈Γ}(T_{p})_{γ}(t)( \begin{matrix}x \\ φ \end{matrix}).

These statements are proved in [2; proof of Theorem 2.1].

4.2 Theorem. T̃_{C} is the modulus semigroup of T_{C}.

Proof. The property that T̃_{p} dominates T_{p} clearly shows that T̃_{C} dominates T_{C} as well. Assume that S is a C_{0}-semigroup on C([−1, 0]; X) dominating T_{C}.

Let φ ∈ C([−1, 0]; X)_{+}. Then S(s)φ ⩾ |T_{C}(s)ψ| for all s ⩾ 0, ψ ∈ C([−1, 0]; X), |ψ| ⩽ φ. This shows J_{p}S(s)φ ⩾ |T_{p}(s)|J_{p}φ for all s ⩾ 0. Let t ⩾ 0. Then, for γ = (γ_{1}, …, γ_{n}) ∈ Γ, we obtain

J_{p}S(t)φ = J_{p}S(γ_{n}t) ⋯ S(γ_{1}t)φ ⩾ |T_{p}(γ_{n}t)|J_{p}S(γ_{n−1}t) ⋯ S(γ_{1}t)φ
⩾ ⋯ ⩾ |T_{p}(γ_{n}t)| ⋯ |T_{p}(γ_{1}t)|J_{p}φ = (T_{p})_{γ}(t)J_{p}φ.

Taking the supremum over γ ∈ Γ we conclude

J_{p}S(t)φ ⩾ T_{p}^{#}(t)J_{p}φ = T̃_{p}(t)J_{p}φ = J_{p}T̃_{C}(t)φ,

and therefore S(t)φ ⩾ T̃_{C}(t)φ.

So we have shown that S dominates T̃_{C}. This shows the assertion.

4.3 Remarks. (a) For the case X = ℝ^{n}, the result of Theorem 4.2 was shown in [2; Proposition 3.3].

(b) The result of Theorem 4.2 is less general than one might hope to show. Namely, in the space of continuous functions, the delay semigroup can be defined under weaker conditions than assumed in the present paper. Indeed, instead of being defined by a function η of bounded variation, one may just assume L: C([−1, 0]; X) → X to be continuous, in order to obtain the C_{0}-semigroup T ; cf. [6; Chap. VI, Sec. 6]. Then, assuming L to have a modulus |L|, and assuming that L and |L| do not have mass at zero, one obtaines that the corresponding C_{0}-semigroup T̃_{C} dominates T_{C}; cf. [7], [2; Proposition 3.2]. Our method of proof does not yield the conjectured result that also in this case the modulus semigroup of T_{C} is given by T̃_{C}.

(c) In the case of h = ∞ we note that results corresponding to Proposition 4.1 and Theorem 4.2 can be shown in the space

C_{0}((−∞, 0]; X) = {φ ∈ C((−∞, 0]; X); lim_{ϑ→−∞} φ(ϑ) = 0}.

In this case the mapping J_{p} used above does no longer exist. However, on the dense subspace C_{c}((−∞, 0]; X) (= {φ ∈ C((−∞, 0]; X); spt φ compact}) the mapping J_{p} exists, and the restriction of J_{p} to subspaces C_{0}((ϑ_{0}, 0]; X) is continuous with respect to the supremum norm, for all ϑ_{0} ∈ (−∞, 0). Also, T_{C}(t)(C_{0}((ϑ_{0}, 0]; X)) ⊆ C_{0}((ϑ_{0}−t, 0]; X) for all t ⩾ 0. These observations can be used to carry out the proof in an analogous way as for the case of h = 1.

References

[1] A. Bátkai and S. Piazzera: Semigroups and linear partial differential equations with delay. J. Math. Anal. Appl. 264, 1–20 (2001).

[2] I. Becker and G. Greiner: On the modulus of one-parameter semigroups. Semigroup Forum 34, 185–201 (1986).

[3] S. Boulite, L. Maniar, A. Rhandi, and J. Voigt: The modulus semigroup for linear delay equations. Positivity 8, 1–9 (2004).

[4] W. Desch and W. Schappacher: On relatively bounded perturbations of linear C_{0}-semigroups. Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. IV XI, 327–341 (1984).

[5] K.-J. Engel: Spectral theory and generator property of one-sided coupled operator matrices. Semigroup Forum 58, 267–295 (1999).

[6] K.-J. Engel and R. Nagel: One-parameter semigroups for linear evolution equations. Springer, New York, 1999.

[7] W. Kerscher and R. Nagel: Positivity and stability for Cauchy problems with delay. In: Lect. Notes Math. 1324, 216–235 (1988).

[8] L. Maniar and J. Voigt: Linear delay equations in the L_{p}-context. Evolution Equations (G. R. Goldstein, R. Nagel, and S. Romanelli, eds.), Lecture Notes in Pure and Applied Mathematics, vol. 234, Marcel Dekker, 2003, pp. 319– 330.

[9] I. Miyadera: On perturbation theory for semi-groups of operators. Tôhoku Math. J. 18, 299–310 (1966).

[10] M. Stein and J. Voigt: The modulus of matrix semigroups. Arch. Math. 82, 311–316 (2004).

[11] J. Voigt: The modulus semigroup for linear delay equations II. Note Mat., to appear.

[12] J. Voigt: On the perturbation theory for strongly continuous semigroups. Math. Ann. 229, 163–171 (1977).