THE MODULUS SEMIGROUP FOR LINEAR DELAY EQUATIONS

S. BOULITE, L. MANIAR, A. RHANDI, AND J. VOIGT

Dedicated to our friend Rainer Nagel on the occasion of his 60^{th} birthday

Abstract. In this note, we describe the generator of the modulus semigroup of the C_{0}-semigroup associated with the delay equation

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

in the Banach lattice ℝ^{n} × L^{p}(−h, 0; ℝ^{n}).

MCS 2000: 34K06

Keywords: functional differential equation, modulus semigroup, perturbation theory

1. Introduction

The starting point of our considerations is the Cauchy problem for the linear delay equation,

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

in the L^{p}-context, for 1 ⩽ p < ∞, with initial values x ∈ ℝ^{n}, f ∈ L^{p}(−h, 0; ℝ^{n}). Here, h = 1 or h = ∞, A = (A_{ij}) ∈ ℝ^{n×n}, and L: C([−h, 0]; ℝ^{n}) → ℝ^{n} is the bounded linear operator given by

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

where η: [−h, 0] → ℝ^{n×n} is a function of bounded variation with no mass in zero. (Here we have used the notation

C([−∞, 0]; X) := {f ∈ C((−∞, 0]; X); f(−∞) := lim_{τ→−∞} f(τ) exists}.)

Also, for a function u: (−h, ∞) → ℝ^{n}, we recall the notation

u_{t}(θ) := u(t + θ) (−h < θ < 0),

for t ⩾ 0.

It is shown in [2] 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 \\ f \end{matrix} ) \end{cases}

on the space X := ℝ^{n} × L^{p}(−h, 0; ℝ^{n}), where A is given by

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

2 S. BOULITE, L. MANIAR, A. RHANDI, AND J. VOIGT

with domain

D(A) := \{( \begin{matrix}x \\ f \end{matrix} ) ∈ ℝ^{n} × W^{1,p}(−h, 0; ℝ^{n}); f(0) = x\} .

From [2], [4], [5], [7] it is known that the operator A generates a C_{0}-semigroup T := (e^{tA})_{t⩾0} on the Banach lattice X.

The aim of this paper is to show that the semigroup T is dominated, i.e., there is a positive semigroup T̃ := (et̃A)_{t⩾0} such that

|e^{tA}( \begin{matrix}x \\ f \end{matrix} )| ⩽ e^{tÃ}|( \begin{matrix}x \\ f \end{matrix} )| for all ( \begin{matrix}x \\ f \end{matrix} ) ∈ X ,

to show the existence of the modulus semigroup (the smallest dominating positive semigroup ) T_{#} = (e^{tA_{#}})_{t⩾0} of T , and to describe its generator A_{#}.

In [3], I. Becker and G. Greiner gave the generator of the modulus semigroup of the semigroup solution of equation (DE) on the space C([−h, 0]; ℝ^{n}).

Acknowledgement. This paper was started during a stay of J. Voigt at Marrakesh, Morocco, which was supported by the DAAD as well as the Cadi Ayyad University, Marrakesh.

2. Main result

In this section we use the same notation as in the introduction. In the representation of A = A_{0} + B as the sum of

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

with domain D(A_{0}) = D(A), and the (generally unbounded) operator

B := (\begin{matrix}0 & L \\ 0 & 0 \end{matrix}) ,

it is shown in [2], [7] that B is a small Miyadera perturbation of A_{0}. Hence the semigroup (e^{tA})_{t⩾0} is given by the Dyson-Phillips series

e^{tA} := ∑_{k=0}^{∞} R_{k}(t) (t ⩾ 0),

where R_{0}(t) := e^{tA_{0}}, and R_{k+1}(t)( \begin{matrix}x \\ f \end{matrix} ) := ∫_{0}^{t} R_{k}(s)Be^{(t−s)A_{0}}( \begin{matrix}x \\ f \end{matrix} )ds for all ( \begin{matrix}x \\ f \end{matrix} ) ∈ D(A)

(cf. [11], [9; Prop. 2.3], [6; Chap. III, Sec. 3]).

(In fact, the case p = 1 is more complicated; cf. [7]. In this case, B can be decomposed as a sum B = B_{1} + ⋯ + B_{n} such that B_{k} is a small Miyadera perturbation of A_{0} + B_{1} + ⋯ + B_{k−1}, for k = 1, …, n.)

The C_{0}-semigroup (e^{tA_{0}})_{t⩾0} is given (cf. [2]) by

e^{tA_{0}} := (\begin{matrix}e^{tA} & 0 \\ S_{t} & S_{0}(t) \end{matrix}) , (2.1)

where (S_{0}(t))_{t⩾0} is the left shift semigroup on L^{p}(−h, 0; ℝ^{n}) (which is nilpotent in the case h = 1) and S_{t}: ℝ^{n} → L^{p}(−h, 0; ℝ^{n}) is defined by

(S_{t}x)(s) := \begin{cases}e^{(t+s)A}x & for − t ⩽ s < 0, \\ 0 & for − h < s < −t. \end{cases}

THE MODULUS SEMIGROUP FOR LINEAR DELAY EQUATIONS 3

To get the existence of the modulus semigroup of T , it is sufficient to show that the semigroup T is dominated, since the space X has order continuous norm (cf. [3]).

For this purpose, let us denote by A_{#} the matrix in ℝ^{n×n} defined by

(A_{#})_{ij} := \begin{cases}A_{ii} & for i = j, \\ |A_{ij}| & for i ≠ j. \end{cases}

It is known that (e^{tA_{#}})_{t⩾0} is the modulus semigroup of (e^{tA})_{t⩾0} (cf. [8; Ex.C.II.4.19]).

L is a bounded linear operator from C([−h, 0]; ℝ^{n}) to (the finite dimensional space) ℝ^{n}, therefore L is regular (cf. [1; Chap. 1, Thm. 1.10] or [10; Chap. IV, Thm. 1.5]), its modulus is defined by

|L|f := sup{|Lg|; g ∈ C([−h, 0]; ℝ^{n}), |g| ⩽ f} for 0 ⩽ f ∈ C([−h, 0]; ℝ^{n}),

and |L|: C([−h, 0]; ℝ^{n}) → ℝ^{n} is a positive bounded linear operator. In fact, L can be considered as defined by a matrix of measures on [−h, 0], and the measures do not have mass at zero. Then |L| is associated with the matrix of the variations of the respective measures, which again do not have mass at zero. This implies that |L| is of the same form as L, and that the operator Ã defined by D(Ã) = D(A),

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

also generates a C_{0}-semigroup (e^{tÃ})_{t⩾0}. As for the operator A the latter semigroup can be written as

e^{tÃ} := ∑_{k=0}^{∞} R̃_{k}(t) (t ⩾ 0).

The terms R̃_{k}(t) of this series are determined as for the operator A by replacing A_{0} by

Ã_{0} := (\begin{matrix}A_{#} & 0 \\ 0 & \frac{d}{dθ} \end{matrix}), with D(Ã_{0}) := D(A), and B by B̃:= (\begin{matrix}0 & |L| \\ 0 & 0 \end{matrix}).

We can now state the first result of this section.

Lemma 2.1. (e^{tÃ})_{t⩾0} is a positive C_{0}-semigroup dominating (e^{tA})_{t⩾0}.

Proof. It is easy to see that the C_{0}-semigroup (e^{tÃ_{0}})_{t⩾0} is positive, and by induction we have that all terms of the Dyson-Phillips series of (e^{tÃ})_{t⩾0} are positive.

To show the second assertion we just have to verify that

|R_{k}(t)( \begin{matrix}z \\ φ \end{matrix} )| ⩽ R̃_{k}(t) |( \begin{matrix}z \\ φ \end{matrix} )| (2.2)

for all ( \begin{matrix}z \\ φ \end{matrix} ) ∈ D(A) and for every k. Let us verify (2.2) for k = 0. From (2.1), we have

|e^{tA_{0}}( \begin{matrix}z \\ φ \end{matrix} )| = ( \begin{matrix}| e^{tA}z | \\ | S_{t}z + S_{0}(t)φ | \end{matrix} )
⩽ ( \begin{matrix}| e^{tA}z | \\ | S_{t}z | + | S_{0}(t)φ | \end{matrix} ) .

In view of the positivity of (S_{0}(t))_{t⩾0}, and the fact that (e^{tA_{#}})_{t⩾0} is the modulus semigroup of (e^{tA})_{t⩾0}, we obtain

|e^{tA_{0}}( \begin{matrix}z \\ φ \end{matrix} )| ⩽ e^{tÃ_{0}} |( \begin{matrix}z \\ φ \end{matrix} )| (2.3)

4 S. BOULITE, L. MANIAR, A. RHANDI, AND J. VOIGT

for all ( \begin{matrix}z \\ φ \end{matrix} ) ∈ D(A). From (2.3) and the fact that |Lφ| ⩽ |L||φ|, we obtain the assertion by induction.

(As noted before, the case p = 1 is more involved. In this case we observe that the decomposition of the perturbations B, B̃ mentioned above can be achieved simultaneously, and by induction one obtains that (e^{t(A_{0}+B_{1}+···+B_{k})})_{t⩾0} is dominated by (e^{t(Ã_{0}+B̃_{1}+···+B̃_{k})})_{t⩾0}, for k = 1, …, n.) □

The generator of the modulus semigroup T_{#} of (e^{tA})_{t⩾0} (which exists because (e^{tA})_{t⩾0} is dominated and the Banach lattice X has order continuous norm; cf. [3]) will be denoted by A_{#}.

The following lemma will be used to obtain the generator A_{#} explicitly. In the statement, |η|([−t, 0]) will denote the variation of η on [−t, 0],

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

Furthermore, let P_{1} denote the canonical projection from X onto the first component ℝ^{n}.

Lemma 2.2. (a) There exists C ⩾ 0 such that

∥e^{tA}( \begin{matrix}z \\ 0 \end{matrix} ) − e^{tA_{0}}( \begin{matrix}z \\ 0 \end{matrix} ) ∥⩽ C t |η|([−t, 0])∥z∥ (2.4)

for all z ∈ ℝ^{n}, 0 < t < 1.

(b) \frac{1}{t} (P_{1}e^{tA}( \begin{matrix}z \\ 0 \end{matrix} ) − z) → Az, \frac{1}{t} (P_{1}e^{tÃ}( \begin{matrix}z \\ 0 \end{matrix} ) − z) → A_{#}z (t → 0) for all z ∈ ℝ^{n}.

(c) \frac{1}{t}P_{1}e^{tA}( \begin{matrix}0 \\ φ \end{matrix} ) → Lφ, \frac{1}{t}P_{1}e^{tÃ}( \begin{matrix}0 \\ φ \end{matrix} ) → |L|φ (t → 0) for all φ ∈ W^{1,p}(−h, 0; ℝ^{n}).

Proof. (a) In order to show (2.4) we fix z ∈ ℝ^{n}. For 0 < α < 1 − t, we define φ_{α} ∈ W^{1,p}(−h, 0; ℝ^{n}) by

φ_{α}(θ) := \begin{cases}\frac{(θ+α)}{α}z & for − α ⩽ θ < 0, \\ 0 & otherwise. \end{cases}

Then ( \begin{matrix}z \\ φ_{α} \end{matrix} ) ∈ D(A_{0}), therefore

e^{tA}( \begin{matrix}z \\ φ_{α} \end{matrix} ) − e^{tA_{0}}( \begin{matrix}z \\ φ_{α} \end{matrix} ) = ∫_{0}^{t} e^{(t−s)A}Be^{sA_{0}}( \begin{matrix}z \\ φ_{α} \end{matrix} )ds,

and with C_{1} := sup_{0⩽t⩽1} ∥e^{tA}∥, C_{2} := sup_{0⩽t⩽1} ∥e^{tA}∥, we estimate

∥e^{tA}( \begin{matrix}z \\ φ_{α} \end{matrix} ) − e^{tA_{0}}( \begin{matrix}z \\ φ_{α} \end{matrix} )∥ ⩽ ∫_{0}^{t} ∥e^{(t−s)A}( \begin{matrix}L(S_{s}z+S_{0}(s)φ_{α}) \\ 0 \end{matrix} )∥ds
⩽ C_{1} ∫_{0}^{t} ∥ ∫_{[−h,0]} dη(θ)(S_{s}z(θ) + S_{0}(s)φ_{α}(θ))∥ds
⩽ C_{1} ∫_{0}^{t} C_{2}|η|([−s − α, 0])∥z∥ds
⩽ C_{1}C_{2} t |η|([−t − α, 0])∥z∥.

For α → 0 we obtain (2.4).

(b) First, it is easy to verify that

\frac{1}{t} (P_{1}e^{tA_{0}}( \begin{matrix}z \\ 0 \end{matrix} ) − z) → Az (t → 0) (2.5)

for all z ∈ ℝ^{n}. Second, we can write

\frac{1}{t} (P_{1}e^{tA}( \begin{matrix}z \\ 0 \end{matrix} ) − z) = \frac{1}{t}(P_{1}e^{tA}( \begin{matrix}z \\ 0 \end{matrix} ) − P_{1}e^{tA_{0}}( \begin{matrix}z \\ 0 \end{matrix} )) + \frac{1}{t} (P_{1}e^{tA_{0}}( \begin{matrix}z \\ 0 \end{matrix} ) − z) .

Then, in view of (2.4) and (2.5), we obtain

\frac{1}{t} (P_{1}e^{tA}( \begin{matrix}z \\ 0 \end{matrix} ) − z) → Az (t → 0)

for all z ∈ ℝ^{n}. As a consequence, we also obtain

\frac{1}{t} (P_{1}e^{tÃ}( \begin{matrix}z \\ 0 \end{matrix} ) − z) → A_{#}z (t → 0)

for all z ∈ ℝ^{n}.

(c) follows directly by using (b) and

P_{1}e^{tA}( \begin{matrix}0 \\ φ \end{matrix} ) = P_{1}e^{tA}( \begin{matrix}φ(0) \\ φ \end{matrix} ) − P_{1}e^{tA}( \begin{matrix}φ(0) \\ 0 \end{matrix} ),

and since ( \begin{matrix}φ(0) \\ φ \end{matrix} ) ∈ D(A). In the same way, \frac{1}{t}P_{1}e^{tÃ}( \begin{matrix}0 \\ φ \end{matrix} ) → |L|φ (t → 0). □

By using Lemma 2.2, we give some first information on the modulus semigroup (e^{tA_{#}})_{t⩾0} and its generator A_{#}.

Lemma 2.3. (a) D(A) ⊆ D(A_{#}).

(b) There is a sequence (t_{k}) ⊆ (0, ∞), t_{k} → 0, such that

lim_{k→∞} \frac{1}{t_{k}} (P_{1}e^{t_{k}A_{#}}( \begin{matrix}z \\ 0 \end{matrix} ) − z) = A_{#}z for all z ∈ ℝ^{n},

and

lim_{k→∞} \frac{1}{t_{k}} P_{1}e^{t_{k}A_{#}}( \begin{matrix}0 \\ φ \end{matrix} ) = |L|φ for all φ ∈ W^{1,p}(−h, 0; ℝ^{n}).

Proof. From the definition of the modulus semigroup, we have the inequalities

e^{tA}( \begin{matrix}z \\ φ \end{matrix} ) ⩽ e^{tA_{#}}( \begin{matrix}z \\ φ \end{matrix} ) ⩽ e^{tÃ}( \begin{matrix}z \\ φ \end{matrix} ) (2.6)

for all ( \begin{matrix}z \\ φ \end{matrix} ) ∈ X_{+}, t ⩾ 0.

(a) Take 0 ⩽ ( \begin{matrix}z \\ φ \end{matrix} ) ∈ D(Ã) = D(A). Then, in view of (2.6), we have

\frac{1}{t} (e^{tA} − I) ( \begin{matrix}z \\ φ \end{matrix} ) ⩽ \frac{1}{t} (e^{tA_{#}} − I) ( \begin{matrix}z \\ φ \end{matrix} ) ⩽ \frac{1}{t} (e^{tÃ} − I) ( \begin{matrix}z \\ φ \end{matrix} ).

As t → 0, the first and the last terms of the inequality converge to A( \begin{matrix}z \\ φ \end{matrix} ) and Ã( \begin{matrix}z \\ φ \end{matrix} ), respectively.

We want to show that there exists a null sequence (t_{k}) ⊆ (0, 1] such that the sequence (\frac{1}{t_{k}} (e^{t_{k}A_{#}} − I) ( \begin{matrix}z \\ φ \end{matrix} ))_{k} converges weakly. Defining

u(t) := \frac{1}{t} (e^{tA} − I) ( \begin{matrix}z \\ φ \end{matrix} ),
u_{#}(t) := \frac{1}{t} (e^{tA_{#}} − I) ( \begin{matrix}z \\ φ \end{matrix} ),
ũ(t) := \frac{1}{t} (e^{tÃ} − I) ( \begin{matrix}z \\ φ \end{matrix} ) (0 < t ⩽ 1),

we obtain 0 ⩽ u_{#}(t) − u(t) ⩽ ũ(t) − u(t) (0 < t ⩽ 1) and ũ(t) − u(t) → A( \begin{matrix}z \\ φ \end{matrix} ) − Ã( \begin{matrix}z \\ φ \end{matrix} )

(t → 0). Now [1; Chap.4, Thm.13.8] implies that the solid hull of {ũ(t) − u(t); 0 < t ⩽ 1} is relatively weakly compact. Since {u_{#}(t) − u(t); 0 < t ⩽ 1} is a subset of this hull there exists a null sequence (t_{k}) such that (u_{#}(t_{k}) − u(t_{k}))_{k} converges weakly (recall [1; Chap.3, Thm.10.13]), and therefore (u_{#}(t_{k}))_{k} converges weakly.

Observe that, for k ∈ ℕ, the pair (\frac{1}{t_{k}} ∫_{0}^{t_{k}} e^{sA_{#}}( \begin{matrix}z \\ φ \end{matrix} )ds, u_{#}(t_{k})) belongs to the graph of A_{#}. Since A_{#} is a closed operator the graph of A_{#} is weakly closed. Hence ( \begin{matrix}z \\ φ \end{matrix} ) ∈ D(A_{#}).

Finally, D(Ã) = D(Ã)_{+} − D(Ã)_{+} ⊆ D(A_{#}) shows the assertion.

For the first assertion of (b), let 0 ⩽ z ∈ ℝ^{n}. From (2.6) we have

\frac{1}{t} (P_{1}e^{tA}( \begin{matrix}z \\ 0 \end{matrix} ) − z) ⩽ \frac{1}{t} (P_{1}e^{tA_{#}}( \begin{matrix}z \\ 0 \end{matrix} ) − z) ⩽ \frac{1}{t} (P_{1}e^{tÃ}( \begin{matrix}z \\ 0 \end{matrix} ) − z) . (2.7)

In view of Lemma 2.2 (b), the first and the third term of (2.7) converge to Az and A_{#}z as t → 0, respectively.

Denote by (e_{1}, e_{2}, …, e_{n}) the canonical basis of ℝ^{n}. The previous statements show that the set \{\frac{1}{t} (P_{1}e^{tA_{#}}( \begin{matrix}e_{i} \\ 0 \end{matrix} ) − e_{i}); 0 < t ⩽ 1\} is a bounded subset of ℝ^{n}, for i = 1, …, n. Hence there exists a null sequence (t_{k})_{k} ⊆ (0, 1] such that the sequence (\frac{1}{t_{k}} (P_{1}e^{t_{k}A_{#}}( \begin{matrix}e_{i} \\ 0 \end{matrix} ) − e_{i}))_{k} is convergent for all i = 1, …, n. Consequently,

Âz := lim_{k→∞} \frac{1}{t_{k}} (P_{1}e^{t_{k}A_{#}}( \begin{matrix}z \\ 0 \end{matrix} ) − z)

exists for all z ∈ ℝ^{n}, and Â is represented by a matrix Â ∈ ℝ^{n×n}. For 0 ⩽ z ∈ ℝ^{n}, we obviously have

Az ⩽ Âz ⩽ A_{#}z. (2.8)

By the definition of A_{#}, we conclude

Â_{ii} = A_{ii} (i = 1, …, n). (2.9)

Let p_{i} be the canonical projection onto the i^{th} coordinate. Then, for j ≠ i, we have

|p_{i}P_{1}e^{t_{k}A}( \begin{matrix}e_{j} \\ 0 \end{matrix} )| ⩽ p_{i}P_{1}e^{t_{k}A_{#}}( \begin{matrix}e_{j} \\ 0 \end{matrix} ),
\frac{1}{t_{k}} |p_{i}P_{1}e^{t_{k}A}( \begin{matrix}e_{j} \\ 0 \end{matrix} )| ⩽ \frac{1}{t_{k}} p_{i}P_{1}e^{t_{k}A_{#}}( \begin{matrix}e_{j} \\ 0 \end{matrix} ).

We have from above that the terms of the last inequality converge to |A_{ij}| and Â_{ij}, respectively. So we deduce that

|A_{ij}| ⩽ Â_{ij} for i, j ∈ {1, …, n} , i ≠ j. (2.10)

Exploiting (2.8), (2.9) and (2.10), we obtain Â = A_{#}.

Finally, we are going to show the second assertion of (b). Let φ ∈ W^{1,p}(−h, 0; ℝ^{n}). Since ( \begin{matrix}φ(0) \\ φ \end{matrix} ) ∈ D(A) ⊆ D(A_{#}) we obtain the existence of the limit

L_{#}φ := lim_{k→∞} \frac{1}{t_{k}} P_{1}e^{t_{k}A_{#}}( \begin{matrix}0 \\ φ \end{matrix} )
= lim_{k→∞} \frac{1}{t_{k}} P_{1} (e^{t_{k}A_{#}} ( \begin{matrix}φ(0) \\ φ \end{matrix} ) − ( \begin{matrix}φ(0) \\ φ \end{matrix} )) − lim_{k→∞} \frac{1}{t_{k}} (P_{1}e^{t_{k}A_{#}} ( \begin{matrix}φ(0) \\ 0 \end{matrix} ) − φ(0))
= P_{1}A_{#} ( \begin{matrix}φ(0) \\ φ \end{matrix} ) − A_{#}φ(0).

By the definition of L_{#}, we obtain that L_{#}: W^{1,p}(−h, 0; ℝ^{n}) → ℝ^{n} is a positive linear operator, and by (2.6) it satisfies

|Lφ| ⩽ L_{#}φ ⩽ |L|φ

for all 0 ⩽ φ ∈ W^{1,p}(−h, 0; ℝ^{n}). In order to conclude that L_{#}φ = |L|φ it remains to show that 0 ⩽ φ ∈ W^{1,p}(−h, 0; ℝ^{n}), g ∈ C([−h, 0]; ℝ^{n}), |g| ⩽ φ implies |Lg| ⩽ L_{#}φ. To show this choose a sequence (g_{k}) ⊆ W^{1,p}(−h, 0; ℝ^{n}), ∥g_{k} − g∥_{∞} → 0 (k → ∞), and define g̃_{k} := (g_{k} ∧ φ) ∨ (−φ) (k ∈ ℕ). Then g̃_{k}, |g̃_{k}| ∈ W^{1,p}(−h, 0; ℝ^{n}), |g̃_{k}| ⩽ φ (k ∈ ℕ), ∥g̃_{k} − g∥_{∞} → 0 (k → ∞). Moreover,

|Lg̃_{k}| ⩽ |L(g̃_{k}^{+})| + |L(g̃_{k}^{−})|
⩽ L_{#}(g̃_{k}^{+}) + L_{#}(g̃_{k}^{−}) = L_{#}|g̃_{k}|
⩽ L_{#}φ (k ∈ ℕ).

This implies |Lg| ⩽ L_{#}φ. □

Now, we are able to state our main result.

Theorem 2.4. (e^{tÃ})_{t⩾0} is the modulus semigroup of (e^{tA})_{t⩾0}, i.e., A_{#} = Ã.

Proof. Let ( \begin{matrix}z \\ φ \end{matrix} ) ∈ D(A)_{+}. By Lemma 2.3 (a), we have

A_{#}( \begin{matrix}z \\ φ \end{matrix} ) = lim_{t→0} \frac{1}{t}(e^{tA_{#}}( \begin{matrix}z \\ φ \end{matrix} ) − ( \begin{matrix}z \\ φ \end{matrix} )).

Hence, for (t_{k}) from Lemma 2.3 (b), we have

P_{1}A_{#}( \begin{matrix}z \\ φ \end{matrix} ) = lim_{k→∞} \frac{1}{t_{k}} (P_{1}e^{t_{k}A_{#}}( \begin{matrix}z \\ φ \end{matrix} )) − z)

= lim_{k→∞} \frac{1}{t_{k}} (P_{1}e^{t_{k}A_{#}}( \begin{matrix}z \\ 0 \end{matrix} ) − z) + lim_{k→∞} \frac{1}{t_{k}} P_{1}e^{t_{k}A_{#}}( \begin{matrix}0 \\ φ \end{matrix} )

= A_{#}z + |L|φ.

By (2.6) we also have

A( \begin{matrix}z \\ φ \end{matrix} ) ⩽ A_{#}( \begin{matrix}z \\ φ \end{matrix} ) ⩽ Ã( \begin{matrix}z \\ φ \end{matrix} ),

i.e.,

( \begin{matrix}Az+Lφ \\ φ′ \end{matrix} ) ⩽ A_{#}( \begin{matrix}z \\ φ \end{matrix} ) ⩽ ( \begin{matrix}A_{#}z+|L|φ \\ φ′ \end{matrix} ).

Therefore, we have

P_{2}A_{#}( \begin{matrix}z \\ φ \end{matrix} ) = φ′,

where P_{2} is the canonical projection from ℝ^{n} × L^{p}(−h, 0; ℝ^{n}) onto L^{p}(−h, 0; ℝ^{n}).

Thus, we have shown for all 0 ⩽ ( \begin{matrix}z \\ φ \end{matrix} ) ∈ D(A)_{+} that

A_{#}( \begin{matrix}z \\ φ \end{matrix} ) = ( \begin{matrix}A_{#}z+|L|φ \\ φ′ \end{matrix} ) = Ã( \begin{matrix}z \\ φ \end{matrix} ).

To show the equality for all ( \begin{matrix}z \\ φ \end{matrix} ) ∈ D(A) = D(Ã) it is sufficient to note that D(Ã) = D(Ã)_{+} − D(Ã)_{+}. Therefore A_{#} ⊇ Ã, and as A_{#} and Ã are generators, we have A_{#} = Ã.

□

References

[1] C. D. Aliprantis and O. Burkinshaw: Positive Operators. Academic Press, New York, 1985.

[2] A. Bátkai and S. Piazzera: Semigroups and linear partial differential equations with delay. J. Math. Anal. Appl., to appear

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

[4] W. Desch and W. Schappacher: On relatively bounded perturbations of linear C_{0}-semigroups. Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. IV - Vol. 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. GTM 194 Springer-Verlag, New York, 1999.

[7] L. Maniar and J. Voigt: Linear delay equations in the L_{p}-context. Preprint.

[8] R. Nagel (ed.): One-parameter Semigroups of Positive Operators. LNM 1184, Springer-Verlag, Berlin, 1986.

[9] A. Rhandi: Dyson-Phillips expansion and unbounded perturbations of linear c_{0}-semigroups. J. Computational Appl. Math. 44, 339–349 (1992).

[10] H. H. Schaefer: Banach Lattices and Positive Operators. Springer-Verlag, Berlin, 1974.

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

Said Boulite, Lahcen Maniar, Abdelaziz Rhandi Cadi Ayyad University Faculty of Sciences Semlalia, B.P. 2390 Marrakesh, Morocco

e-mail: sboulite@ucam.ac.ma
e-mail: maniar@ucam.ac.ma
e-mail: rhandi@ucam.ac.ma

Jürgen Voigt Technische Universität Dresden Fachrichtung Mathematik 01062 Dresden, Germany

e-mail: voigt@math.tu-dresden.de