Bulletin T.CXXVII de l’Académie Serbe des Sciences et des Arts - 2003 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 28

CONVOLUTED C-COSINE FUNCTIONS AND CONVOLUTED C-SEMIGROUPS

M. KOSTIĆ

(Presented at the 3rd Meeting, held on May 9, 2003)

A b s t r a c t. Convoluted C-cosine functions and convoluted Csemigroups are introduced and investigated.Then their relations to the wellposedness of second order abstract Cauchy problem and the Θ-convoluted Cregularized Cauchy problem are studied. Generation results for convoluted C-semigroups are given.

AMS Mathematics Subject Classification (2000): 47D05,47D06,47D60. Key Words: convoluted C-cosine functions, convoluted C-semigroups,

well-posedness.

1. Introduction

The classical theory of C_{0}-semigroups of operators is generalized in many different directions. In 1987, Arendt obtained an extension of Widder’s representation theorem for the Laplace transform in a Banach space. This version of Widder’s theorem stimulated the development of the theory of integrated semigroups. Many new types of operator families in Banach spaces are defined on the basis of Arendt’s ideas. Convoluted semigroups were introduced by Cioranescu and Lumer in 1994. In contrast to the case of integrated semigroups, whose resolvents have polynomial growth in some logarithmic regions, in the case of convoluted semigroup the resolvent exists in some smaller region and it may grow not faster than Ce^{M(λ)}, where M(λ) is the associated function of a sequence (M_{n}) which defines a space of ultradifferentiable functions. We also note that convoluted semigroups are closely related to ultradistribution semigroups of Komatsu.

On the other hand, C-semigroups were studied by many authors. In 1990, Tanaka and Okazawa ([18]) defined local C-semigroups and local integrated semigroups. The use of C-semigroups is a powerful method in studying ill-posed abstract Cauchy problems. Local C-cosine functions as well as integrated C-semigroups and integrated C-cosine functions were introduced later ; see [14] , [20] and [21].

Here, we deal with some new types of operator families in Banach spaces. They unify the classes of integrated C-cosine functions and integrated Csemigroups. The corresponding Cauchy problems

(ACP_{2}) : \{ \begin{matrix}u ∈ C([0, τ) : D(A)) ∩ C^{2}([0, τ) : E), \\ u′′(t) = Au(t), \\ u(0) = x, u′(0) = y, \end{matrix}

and

(ΘC) : \{ \begin{matrix}u ∈ C([0, τ) : D(A)) ∩ C^{1}([0, τ) : E), \\ u′(t) = Au(t) + Θ(t)Cx, \\ u(0) = 0, \end{matrix}

are considered. Convoluted C-semigroups are characterized by their asymptotic ΘC-resolvents.

Throughout this paper E denotes a complex Banach space and L(E) denotes the space of bounded linear operators from E into E. For a closed linear operator A, its domain, range and null space are denoted by D(A), R(A) and N(A), respectively. We will assume in the sequel that C ∈ L(E) is an injective operator with CA ⊂ AC. The C resolvent set of A, ρ_{C}(A), is the set of complex numbers λ such that λ−A is injective and R(C) ⊂ R(λ−A).

Convoluted C-cosine functions and convoluted C-semigroups 77

2. Definition and elementary properties of global convoluted C-cosine functions

Throughout this section we will always assume that K ∈ C([0, ∞) : ℂ) is an exponentially bounded function such that K̃(λ) ≠ 0 (Reλ ≥ β), where K̃(λ) is the Laplace transform of K(t).

Notation: Θ(t) := ∫_{0}^{t} K(s)ds, t ≥ 0.

The next definition is the convoluted version of [21], Chapter 1., Definition 4.1.

Definition 2.1. Let w ∈ ℝ. If (w^{2}, ∞) ⊂ ρ_{C}(A) and there exists a strongly continuous operator family (C_{K}(t))_{t≥0}, (C_{K}(t) ∈ L(E), t ≥ 0), such that ∥C_{K}(t)∥ = O(e^{wt}), t ≥ 0, and

λ(λ^{2} − A)^{−1}Cx = \frac{1}{K̃(λ)} ∫_{0}^{∞}e^{−λt}C_{K}(t)xdt, λ > max(w, β), x ∈ E,

then it is said that A is a subgenerator of a K-convoluted C-cosine function (C_{K}(t))_{t≥0}. The operator Ā := C^{−1}AC is called the generator of (C_{K}(t))_{t≥0}.

Remark. Since CA ⊂ AC, one has that Ā is an extension of A. If C = I, a K-convoluted C-cosine function has a unique subgenerator which coincides with its generator. With K(t) = \frac{t^{k−1}}{(k−1)!}, k ∈ ℕ, one obtains the class of k-times integrated C-cosine functions. The generator of a K-convoluted Ccosine function is uniquely determined. From the previous definition it is also clear that (C_{K}(t))_{t≥0} is a nondegenerate family, i.e.,

if C_{K}(t)x = 0 for all t ≥ 0 , then x = 0.

Theorem 2.2. Let (C_{K}(t))_{t≥0} be a strongly continuous, exponentially bounded operator family and let A be a closed operator. Then the statements (i) and (ii) are equivalent, where:

(i) A is the generator of a K-convoluted C-cosine function (C_{K}(t))_{t≥0}, (ii) a) C_{K}(t)C = CC_{K}(t), t ≥ 0,

b) C_{K}(t)A ⊂ AC_{K}(t), t ≥ 0,

c) A ∫_{0}^{t}(t − s)C_{K}(s)xds = C_{K}(t)x − Θ(t)Cx, t ≥ 0, x ∈ E.

P r o o f. (i) ⇒ (ii) Let x ∈ E. Clearly, (λ^{2}−A)^{−1}C^{2}x = C(λ^{2}−A)^{−1}Cx, for all sufficiently large λ. By Definition 2.1, the previous equality implies

\frac{1}{K̃(λ)} ∫_{0}^{∞}e^{−λt}C_{K}(t)Cxdt = \frac{1}{K̃(λ)} ∫_{0}^{∞}e^{−λt}CC_{K}(t)xdt.

Thus, a) follows from the uniqueness theorem for the Laplace transforms. Assume now x ∈ D(A). One has

λ(λ^{2} − A)^{−1}CAx = \frac{1}{K̃(λ)} ∫_{0}^{∞}e^{−λt}C_{K}(t)Axdt, i.e.,

λA(λ^{2} − A)^{−1}Cx = \frac{1}{K̃(λ)} ∫_{0}^{∞}e^{−λt}C_{K}(t)Axdt.

This easily implies

A ∫_{0}^{∞}e^{−λt}C_{K}(t)xdt = ∫_{0}^{∞}e^{−λt}C_{K}(t)Axdt,

and b) follows from [21], Theorem 1.10. In order to prove c), we note that

L (∫_{0}^{t}(t − s)C_{K}(s)xds) (λ) = L(t)(λ)L(C_{K}(t)x)(λ)

= \frac{1}{λ^{2}} K̃(λ)λ(λ^{2} − A)^{−1}Cx
= \frac{K̃(λ)}{λ}(λ^{2} − A)^{−1}Cx,

which implies

A (L (∫_{0}^{t}(t − s)C_{K}(s)xds) (λ)) = K̃(λ)λ(λ^{2} − A)^{−1}Cx − \frac{K̃(λ)}{λ}Cx
= L(C_{K}(t)x − Θ(t)Cx)(λ).

Hence, 3) is a consequence of [21], Theorem 1.10.

(ii) ⇒ (i) Suppose ∥C_{K}(t)∥ ≤ Me^{wt}, t ≥ 0, for some real number w with w ≥ β. Using b) and c), we have

L(C_{K}(t)x)(λ) = \frac{K̃(λ)}{λ}Cx + \frac{1}{λ^{2}} AL(C_{K}(t)x)(λ), i.e.,

(λ^{2} − A)L(C_{K}(t)x)(λ) = λK̃(λ)Cx, λ > w, x ∈ E.  (1)

Hence, R(C) ⊂ R(λ^{2}−A), for λ > w. Let us show that λ^{2}−A is injective if λ satisfies λ > w. Suppose (λ^{2} − A)x = 0. It implies

C_{K}(t)x − Θ(t)x = ∫_{0}^{t}(t − s)C_{K}(s)Axds = λ^{2} ∫_{0}^{t}(t − s)C_{K}(s)xds, t ≥ 0,

and consequently,

L(C_{K}(t)x)(λ) = \frac{K̃(λ)}{λ}Cx+λ^{2}L(t)L(C_{K}(t)x)(λ) = \frac{K̃(λ)}{λ}Cx+L(C_{K}(t)x)(λ),

for all sufficiently large λ. Thus, Cx = 0 and x = 0. By (1), it follows (w^{2}, ∞) ⊂ ρ_{C}(A) and

λ(λ^{2} − A)^{−1}Cx = \frac{1}{K̃(λ)} ∫_{0}^{∞}e^{−λt}C_{K}(t)xdt, λ > w, x ∈ E.

Using the well-known Arendt-Widder theorem (see [3]), we obtain the following Hille-Yoshida type theorem:

Theorem 2.3. (a) The following statements are equivalent:

(ai) A is a subgenerator of an exponentially bounded Θ-convoluted Ccosine function (C_{Θ}(t))_{t≥0} satisfying condition

lim_{σ→0} sup_{h≤σ} \frac{∥C_{Θ}(t + h) − C_{Θ}(t)∥}{h} ≤ Me^{wt}, t ≥ 0, for some M > 0.

(aii) There exists a ≥ w such that (a^{2}, ∞) ⊂ ρ_{C}(A) and

∥\frac{d^{k}}{dλ^{k}} [λK̃(λ)(λ^{2} − A)^{−1}C]∥ ≤ \frac{Mk!}{(λ − w)^{k+1}} , k ∈ ℕ_{0}, λ > a.

(b) Assume that A is densely defined. Then the following statements are equivalent:

(bi) A is a subgenerator of an exponentially bounded K-convoluted Ccosine function (C_{K}(t))_{t≥0} satisfying ∥C_{K}(t)∥ ≤ Me^{wt}, t ≥ 0, for some M > 0.

(bii) There exists a ≥ w such that (a^{2}, ∞) ⊂ ρ_{C}(A) and

∥\frac{d^{k}}{dλ^{k}} [λK̃(λ)(λ^{2} − A)^{−1}C]∥ ≤ \frac{Mk!}{(λ − w)^{k+1}} , k ∈ ℕ_{0}, λ > a.

3. Global convoluted C-semigroups

Definition 3.1. Let β < w < ∞. If (w, ∞) ⊂ ρ_{C}(A) and there exists a strongly continuous operator family (S_{K}(t))_{t≥0} such that ∥S_{K}(t)∥ = O(e^{wt}), t ≥ 0, and

(λ − A)^{−1}Cx = \frac{1}{K̃(λ)} ∫_{0}^{∞}e^{−λt}S_{K}(t)xdt, λ > w, x ∈ E,

then it is said that A is a subgenerator of a K-convoluted C-semigroup (S_{K}(t))_{t≥0}.

Note, Definition 3.1 implies that (S_{K}(t))_{t≥0} is a nondegenerate family. If C = I then Definition 3.1 means that A is the generator of a global exponentially bounded convoluted semigroup (S_{K}(t))_{t≥0}.

Using the similar arguments as in the proofs of Theorem 2.2 and Theorem 2.3, one has the following results:

Theorem 3.2. Let (S_{K}(t))_{t≥0} be a strongly continuous, exponentially bounded operator family and let A be a closed operator. Then the assertions (i) and (ii) are equivalent, where:

(i) A is a subgenerator of K-convoluted C-semigroup (S_{K}(t))_{t≥0}, (ii) a) S_{K}(t)C = CS_{K}(t), t ≥ 0,

b) S_{K}(t)A ⊂ AS_{K}(t), t ≥ 0,

c) A ∫_{0}^{t} C_{K}(s)xds = C_{K}(t)x − Θ(t)Cx, t ≥ 0, x ∈ E.

Theorem 3.3. a) The following assertions are equivalent:

(ai) A is a subgenerator of an exponentially bounded Θ-convoluted Csemigroup (S_{Θ}(t))_{t≥0} satisfying condition

lim_{σ→0} sup_{h≤σ} \frac{∥S_{Θ}(t + h) − S_{Θ}(t)∥}{h} ≤ Me^{wt}, t ≥ 0, for some M > 0.

(aii) There exists a ≥ w such that (a, ∞) ⊂ ρ_{C}(A) and

∥\frac{d^{k}}{dλ^{k}} [\frac{1}{K̃(λ)}(λ − A)^{−1}C]∥ ≤ \frac{Mk!}{(λ − w)^{k+1}} , k ∈ ℕ_{0}, λ > a.

b) Assume that A is densely defined. Then the following statements are equivalent:

(bi) A is a subgenerator of an exponentially bounded K-convoluted Csemigroup (S_{K}(t))_{t≥0} satisfying ∥S_{K}(t)∥ ≤ Me^{wt}, t ≥ 0, for some M > 0.

(bii) There exists a ≥ w such that (a, ∞) ⊂ ρ_{C}(A) and

∥\frac{d^{k}}{dλ^{k}} [\frac{1}{K̃(λ)}(λ − A)^{−1}C]∥ ≤ \frac{Mk!}{(λ − w)^{k+1}} , k ∈ ℕ_{0}, λ > a.

The next proposition relates a K-convoluted C-cosine function and a K-convoluted C-semigroup. Its proof can be obtained as in the case of integrated semigroups. We give it here for the sake of completness.

Proposition 3.4. Suppose that A and −A are subgenerators of a Kconvoluted C-semigroups, S_{K} and V_{K}, respectively. Then A^{2} is a subgenerator of a K-convoluted C^{2}-cosine function.

P r o o f. It is not hard to verify that λ^{2}−A^{2} is injective for all sufficiently large λ ∈ ℝ. Moreover, R(C^{2}) ⊂ R(λ^{2} − A^{2}) and

(λ^{2} − A^{2})^{−1}C^{2}x = [(λ − A)^{−1}C][(λ + A)^{−1}Cx]
= \frac{1}{2λ}[(λ + A)^{−1}Cx + (λ − A)^{−1}Cx]
= (2K̃(λ))^{−1}L(S_{K}(t)x + V_{K}(t)x)(λ),

x ∈ E, λ sufficiently large.

Hence,

λ(λ^{2} − A^{2})^{−1}C^{2}x = \frac{1}{K̃(λ)}L (\frac{1}{2} (S_{K}(t)x + V_{K}(t)x)) (λ),

x ∈ E, λ sufficiently large.

In general, the converse of Proposition 3.4 is not true. For some special integrated semigroups (E = L^{p}(Ω), Ω ⊂ ℝ^{n} open) , the converse has been proved by Hieber.

4. Local K-convoluted C-cosine functions

Using Theorem 2.2, we define the class of local K-convoluted C-cosine functions as follows.

Definition 4.1. Let A be a closed operator and K(·) be a continuous function on [0, τ), 0 < τ ≤ ∞. If there exists a strongly continuous operator family (C_{K}(t))_{t∈[0,τ)} such that:

(i) C_{K}(t)C = CC_{K}(t), t ∈ [0, τ),

(ii) C_{K}(t)A ⊂ AC_{K}(t), t ∈ [0, τ),

(iii) for all x ∈ E and t ∈ [0, τ):

∫_{0}^{t}(t − s)C_{K}(s)xds ∈ D(A), and

A ∫_{0}^{t}(t − s)C_{K}(s)xds = C_{K}(t)x − Θ(t)Cx,

then C_{K} is called a (local) K-convoluted C-cosine function and A is called a subgenerator of C_{K}.

Properties and examples of integrated C-cosine functions, which are not exponentially bounded, can be found in [20]. Clearly, if A is a subgenerator of C_{K}, then for all x ∈ D(A) function t ↦ C_{K}(t)x is continuously differentiable on [0, τ) and

\frac{d}{dt}C_{K}(t)x = ∫_{0}^{t} C_{K}(s)Axds + K(t)Cx, t ∈ [0, τ).

We say that a function t ↦ v(t) belonging to C([0, τ) : E) is a Kconvoluted mild solution of (ACP_{2}) for (x, y) ∈ E^{2} if for all t ∈ [0, τ),

∫_{0}^{t}(t − s)v(s)ds ∈ D(A) and

A ∫_{0}^{t}(t − s)v(s)ds = v(t) − Θ(t)x − ∫_{0}^{t} Θ(s)yds, t ∈ [0, τ).

Using the same arguments as in [20, Theorem 1.5], one has the following proposition.

Proposition 4.2. Let A be a subgenerator of a K-convoluted C-cosine function (C_{K}(t))_{t∈[0,τ)} and x, y ∈ E. If K is a kernel, i.e.

(∀u ∈ C([0, τ) : E)) (∫_{0}^{t} K(t − s)u(s)ds ≡ 0 ⇒ u ≡ 0) ,

then all K-convoluted mild solutions of (ACP_{2}) for (x, y) are unique.

Remark. By Titchmarsh’s theorem ([3],p.[106]), the condition 0 ∈suppK implies that K is a kernel.

Proposition 4.3. Suppose that for all x ∈ R(C) there exists a unique K-convoluted mild solution of (ACP_{2}) for (x, 0). Then A is a subgenerator of a K-convoluted C-cosine function.

P r o o f. Define C_{K}(t)x := v(t), where v(t) is the K-convoluted mild solution of (ACP_{2}) for (Cx, 0). The uniqueness of mild solutions implies that (C_{K}(t))_{t∈[0,τ)} is a strongly continuous family of linear operators satisfying (iii) of Definition 4.1. It is easy to see that t ↦ CC_{K}(t)x is a K-convoluted mild solution of (ACP_{2}) for (C^{2}x, 0). As a consequence, we have (i) of Definition 4.1. The proof of (ii) of Definition 4.1 can be obtained as in [20, Theorem 1.5]. So, we have only to prove that C_{K}(t), t ∈ [0, τ), is a bounded operator. We will follow the proof of [2, Proposition 2.3] with appropriate changes. Consider the mapping Φ : E → C([0, τ) : D(A)), given by

Φ(x)(t) = ∫_{0}^{t}(t − s)C_{K}(s)xds, t ∈ [0, τ) , x ∈ E,

where D(A) is equipped with the graph norm ∥x∥_{A} = ∥x∥ + ∥Ax∥, and C([0, τ) : D(A)) is a Frechet space for the seminorms

p_{n}(v) := sup_{t∈[0,τ− \frac{1}{n}]} ∥v(t)∥_{D(A)}, v ∈ C([0, τ) : D(A)).

Clearly, Φ is a linear mapping. Let us show that Φ has a closed graph.

Suppose x_{n} → x, and sup_{t∈[0,τ− \frac{1}{n}]} ∫_{0}^{t}(t − s)C_{K}(s)x_{n}ds → f(t), n → ∞. It

implies

Af(t) = lim_{n→∞} A ∫_{0}^{t}(t − s)C_{K}(s)x_{n}ds = lim_{n→∞}[C_{K}(t)x_{n} − Θ(t)Cx_{n}], t ∈ [0, τ),

and

lim_{n→∞} C_{K}(t)x_{n} = Af(t) + Θ(t)Cx, t ∈ [0, τ).

Using the dominated convergence theorem, we have

f(t) = lim_{n→∞} ∫_{0}^{t}(t − s)C_{K}(s)x_{n}ds = ∫_{0}^{t}(t − s)[Af(s) + Θ(s)Cx]ds, t ∈ [0, τ).

So, f(0) = f′(0) = 0, f ∈ C^{2}([0, τ) : E) and

Af(t) = f′′(t) − Θ(t)Cx, t ∈ [0, τ).

Hence, A ∫_{0}^{t}(t − s)v(s)ds = v(t) − Θ(t)Cx, t ∈ [0, τ), where v = f′′. This

implies v(t) = C_{K}(t)x, t ∈ [0, τ), and f = Φ(x). Hence, for all sufficiently large n ∈ ℕ there exists c_{n} such that

∥A ∫_{0}^{t}(t − s)C_{K}(s)xds∥ ≤ c_{n}∥x∥, x ∈ E, t ∈ [0, τ − \frac{1}{n}).

Since A ∫_{0}^{t}(t − s)C_{K}(s)xds = C_{K}(t)x − Θ(t)Cx, x ∈ E, t ∈ [0, τ), one can

easily conclude that C_{K}(t) ∈ L(E), t ∈ [0, τ).

Lemma 4.4. Let A be a subgenerator of a K-convoluted C-cosine function (C_{K}(t))_{t∈[0,τ)}, 0 < τ ≤ ∞, and H ∈ L_{loc}^{1}((0, τ) : ℂ). Then A is a subgenerator of (H ∗_{0} K)-convoluted C-cosine function ((H ∗_{0} C_{K})(t))_{t∈[0,τ)}, where ∗_{0} is the convolution like mapping given by

f ∗_{0} g(t) := ∫_{0}^{t} f(t − s)g(s)ds.

P r o o f. Clearly, ((H ∗_{0} C_{K})(t))_{t∈[0,τ)} is a strongly continuous operator family satisfying (i) and (ii) of Definition 4.1. Moreover,

A ∫_{0}^{t}(t − s) ∫_{0}^{s} H(s − r)C_{K}(r)xdrds

= A((id ∗_{0} H ∗_{0} C_{K})(t))x = A((H ∗_{0} id ∗_{0} C_{K})(t))x

= A ∫_{0}^{t} H(t − s) ∫_{0}^{s}(s − r)C_{K}(r)xdrds

= ∫_{0}^{t} H(t − s)[C_{K}(s)x − Θ(s)Cx]ds = (H ∗_{0} C_{K})(t)x − (H ∗_{0} Θ)(t)Cx,

x ∈ E, t ∈ [0, τ).

This completes the proof.

Example 4.5. Let E := L^{p}(ℝ^{n}), 1 < p < ∞. A remarkable result of Hieber says that the Laplacian Δ with domain H^{p,2}(ℝ^{n}) is the generator of an α-times integrated cosine function if and only if α ≥ (n − 1) |\frac{1}{p} − \frac{1}{2} |

(cf. [3], [8]). Hence, the Laplacian Δ is the generator of a (H ∗_{0} \frac{t^{α−1}}{Γ(α)} )- convoluted cosine function on E for any H ∈ L_{loc}^{1}((0, ∞) : ℂ) and α ≥ max \{(n − 1) |\frac{1}{p} − \frac{1}{2} | , 1\}.

Lemma 4.6. Suppose that A is a subgenerator of a K-convoluted Ccosine function (C_{K}(t))_{t∈[0,τ)}. Then for all (x, y) ∈ R(C)×R(C) there exists a K-convoluted mild solution of (ACP_{2}) .

P r o o f. Let x = Cx_{1} and y = Cy_{1}. Using Lemma 4.4, it is easy to

see that the function v(t) := C_{K}(t)x_{1} + ∫_{0}^{t} C_{K}(s)y_{1}ds is a K-convoluted mild

solution of (ACP_{2}) for (x, y).

By the foregoing we have the next proposition (see also [20]).

Proposition 4.7. If K is a kernel then the following statements are equivalent:

a) A is a subgenerator of a K-convoluted C-cosine function.

b) for all (x, y) ∈ R(C) × R(C) there exists a unique K-convoluted mild solution of (ACP_{2}) .

5. Basic properties of K-convoluted C-semigroups

Let 0 < τ ≤ ∞ and let K be a continuous function on [0, τ) with K ≠ 0.

Definition 5.1. A strongly continuous operator family (S_{t})_{t∈[0,τ)} is called a (local) K-convoluted C-semigroup on E if it satisfies:

1) S(0) = 0, S(t)C = CS(t), ∀t ∈ [0, τ).

2) S(t)S(s)x = [ ∫_{0}^{t+s}− ∫_{0}^{t} − ∫_{0}^{s}] K(t + s − r)S(r)Cxdr,

x ∈ E, 0 ≤ s, t, s + t < τ.

S(·) is said to be nondegenerate if S(t)x = 0 for all t ∈ [0, τ) implies x = 0. For a nondegenerate K-convoluted C-semigroup we may define its generator A via

A := {(x, y) ∈ E^{2} : S(t)x − Θ(t)Cx = ∫_{0}^{t} S(s)yds, ∀t ∈ [0, τ)} .

It is a closed linear operator in E.

Let us consider now the abstract (ΘC)-problem.

Definition 5.2. The Θ-convoluted C-regularized Cauchy problem, (ΘC) in short, is well-posed if for all x ∈ E there exists a unique solution of (ΘC). Here D(A) is supplied with the graph norm ∥x∥_{A} = ∥x∥ + ∥Ax∥.

We give now the result analogous to [3, Proposition 2.3]. The proof is similar, so it is omitted.

Proposition 5.3. Let τ < ∞. Suppose (ΘC) is well-posed. Then there exists a unique nondegenerate strongly continuous function S : [0, τ) → L(E)

such that for every x ∈ E, ∫_{0}^{t} S(s)xds ∈ D(A) and

A ∫_{0}^{t} S(s)xds = S(t)x − Θ(t)Cx, t ∈ [0, τ).  (•)

If 0 ≤ t < τ and 0 ≤ s < t we have the following equality:

Θ(s)Θ(t − s) − ∫_{t−s}^{t} K(t − r)Θ(r)dr + ∫_{0}^{s} K(t − r)Θ(r)dr = 0.

It can be proved using the standard arguments. Now we can obtain the results analogous to [13, Proposition 2.4, Theorem 2.5]. The consideration is similar, so it is omitted. We have only to add that the last equality gives that the coeficient of C^{2}x in the proof of [13, Proposition 2.4] equals zero.

Proposition 5.4. Let (S_{t})_{t∈[0,τ)} be a nondegenerate strongly continuous operator family such that S(t)A ⊂ AS(t), CS(t) = S(t)C, t ∈ [0, τ).

Suppose that for every x ∈ E, ∫_{0}^{t} S(s)xds ∈ D(A) and satisfies (•). Then:

1) S(t) is a local K-convoluted C-semigroup whose generator is an extension of A.

2) If K is a kernel then (ΘC) is well-posed.

Proposition 5.5. Suppose that (ΘC) is well-posed. Let S be as in Proposition 5.3. Then:

1) SC = CS, SA ⊂ AS.

2) S(t)S(s) = S(s)S(t), 0 ≤ s, t < τ.

3) S(·) is a local K-convoluted C-semigroup generated by C^{−1}AC.

4) For all λ ∈ ρ_{C}(A) we have

(λ − A)^{−1}CS(t) = S(t)(λ − A)^{−1}C, t ∈ [0, τ).

Remarks. Putting C = I in Proposition 5.4. we have [15, Theorem 1.3.4], which gives the semigroup property for local convoluted semigroups. Even if K(t) = \frac{t^{k−1}}{(k−1)!}, k ∈ ℕ, there exist examples of local integrated Csemigroups whose generators have empty C-resolvents ([13]). Combining Proposition 5.4 and Proposition 5.5, one has the following: if A is a subgenerator of an exponentially bounded K-convoluted C-semigroup (S_{K}(t))_{t≥0} in the sense of Definition 3.1, then (S_{K}(t))_{t≥0} is a K-convoluted C-semigroup in the sense of Definition 5.1, and if additionally K is a kernel, then the generator of (S_{K}(t))_{t≥0} is C^{−1}AC.

The next proposition can be proved as in the case of integrated semigroups.

Proposition 5.6. Let (S_{t})_{t∈[0,τ)} be a (local) K-convoluted C-semigroup generated by A. If x ∈ D(A^{k}) and K ∈ C^{k−1}([0, τ) : ℂ) for some k ∈ ℕ then we have

\frac{d^{k}}{dt^{k}} S(t)x = S(t)A^{k}x + ∑_{i=0}^{k−1} K^{(i)}(t)CA^{k−1−i}x, t ∈ [0, τ).

6. The corresponding (ΘC) problem and generation results

We start with

Proposition 6.1. Suppose K ∈ C^{k}([0, τ) : ℂ), k ∈ ℕ. Assume also that the problem (ΘC) is well-posed for A. Then for all x ∈ D(A^{k+1}) there exists a unique solution of

(ΘC_{k}) : \{ \begin{matrix}u ∈ C^{1}([0, τ) : E) ∩ C([0, τ) : D(A)) \\ u′(t) = Au(t) + \frac{d^{k}}{dt^{k}} K(t)Cx \\ u(0) = ∑_{i=0}^{k−1} K^{(i)}(0)A^{k−1−i}Cx. \end{matrix}

P r o o f. Let (S_{t})_{t∈[0,τ)} be as in Proposition 5.3. One can easily verify that

u(t) := ∫_{0}^{t} S(s)A^{k+1}xds + ∑_{i=0}^{k} Θ^{(i)}(t)A^{k−i}Cx; t ∈ [0, τ), x ∈ D(A^{k+1}),

is the solution of (ΘC_{k}). The uniqueness follows from the well-posedness of (ΘC) at x = 0.

Of course, if a ∈ L_{(loc)}^{1}([0, τ) : ℂ) and if the problem (ΘC) is well-posed then the problem (a ∗_{0} Θ, C) is well-posed. The semigroups S_{K} and S_{a∗_{0}K} satisfy S_{a∗_{0}K} = a ∗_{0} S_{K}.

Theorem 6.2. Let k ∈ ℕ\{1} and let K satisfies K ∈ C^{k+1}([0, τ) : ℂ) and K^{(i)}(0) = 0, 0 ≤ i ≤ k − 2. If A is a closed linear operator with λ_{0} ∈ ρ(A) and if for all x ∈ D(A^{k+1}) there exists a unique solution of (ΘC_{k}) then:

1) (ΘC) is well-posed for A.

2) C^{−1}AC generates a K-convoluted C-semigroup on [0, τ); here τ < ∞.

P r o o f. Let z = Cy, y ∈ D(A^{k+1}). Define u_{1}(t) := (λ_{0} − A) ∫_{0}^{t} u_{y}(s)ds,

t ∈ [0, τ), where u_{y} is the solution of (ΘC_{k}) for y. One can simply verify that u_{1} is a solution of

(ΘC_{k−1}) : \{ \begin{matrix}u ∈ C^{1}([0, τ) : E) ∩ C([0, τ) : D(A)) \\ u′(t) = Au(t) + \frac{d^{k−1}}{dt^{k−1}} K(t)x \\ u(0) = 0, \end{matrix}

for x = (λ_{0} − A)z. Moreover, (ΘC_{k−1}) has a solution for all x ∈ (λ_{0} − A)CD(A^{k+1}).

Similarly, the problem

(ΘC_{k−2}) : \{ \begin{matrix}u ∈ C^{1}([0, τ) : E) ∩ C([0, τ) : D(A)) \\ u′(t) = Au(t) + \frac{d^{k−2}}{dt^{k−2}} K(t)x \\ u(0) = 0 \end{matrix}

has a solution for all x ∈ (λ_{0} − A)^{2}CD(A^{k+1}) (we can take u_{2}(t) := (λ_{0} −

A) ∫_{0}^{t} u_{1}(s)ds).

By induction we obtain a solution u_{k+1} of

(ΘC_{−1}) : \{ \begin{matrix}u ∈ C^{1}([0, τ) : E) ∩ C([0, τ) : D(A)) \\ u′(t) = Au(t) + Θ(t)x \\ u(0) = 0, \end{matrix}

for all x ∈ (λ_{0} − A)^{k+1}CD(A^{k+1}). Since R(C) ⊂ (λ_{0} − A)^{k+1}CD(A^{k+1}) we have proved 1), because the uniqueness follows from the well-posedness of (ΘC_{k}) at x = 0. The rest of proof is clear by Proposition 5.5.

Remarks: 1) We have proved the converse of [13, Remark 2.6, (d)]. Our results in this section also generalize [3, Proposition 3.3]. See also ([21], p. [36]).

Propositions 6.1, 5.4, 5.5 and Theorem 6.2 yield immediately

Theorem 6.3. Let τ, K and k be as in Theorem 6.2. If A is a closed linear operator with λ_{0} ∈ ρ(A), then the following statements are equivalent:

1) (Θ, I) is well-posed for A on [0, τ).

2) (Θ^{(k)}, R(λ_{0} : A)^{k}) is well-posed for A on [0, τ).

If K is a kernel then 1 ) and 2 ) are equivalent to:

1)’ A generates a K-convoluted semigroup on [0, τ). 2)’ A generates a K^{(k)}-convoluted R(λ_{0} : A)^{k}-semigroup on [0, τ), re-

spectively.

For the similar results see [15, Theorem 1.4.8] or [21, Theorem 6.7]. Let τ < ∞. Assume that (ΘC) is well-posed. Define

L_{γ}(λ) := ∫_{0}^{γ} e^{−λs}S(s)ds; γ ∈ [0, τ), λ ∈ [0, ∞),

where S(·) is given by Proposition 5.3. We collect some properties of the operators L_{γ}(λ) in the next proposition. The proof is analogous to [13, Proposition 5.1] and so it can be omitted.

Proposition 6.4. Let x ∈ E. Then

1) The function λ → L_{γ}(λ)x belongs to C^{∞}([0, ∞) : E) and there exists M_{γ} such that

∥\frac{λ^{n}}{(n − 1)!} \frac{d^{n−1}}{dλ^{n−1}} L_{γ}(λ)∥ ≤ M_{γ}, ∀λ ≥ 0, n ∈ ℕ.

2) L_{γ}(λ) commutes with C and A.

3) (λ − A)L_{γ}(λ) = −e^{−λγ}S(γ)x + ∫_{0}^{γ} e^{−λs}K(s)Cxds.

4) L_{γ}(λ)L_{γ}(η) = L_{γ}(η)L_{γ}(λ).

A family {L_{γ}(λ) : γ ∈ [0, τ), λ ≥ 0} ⊂ L(E) is called asymptotic ΘC-resolvent for A if there exists a strongly continuous operator family (V_{t})_{t∈[0,τ)} such that 1), 2) and 4) hold and 3) holds with S(γ) replaced by V (γ).

Using the same arguments as in [13, Theorem 5.2, Corollary 5.3], we have the following results.

Theorem 6.5. Let A be a closed operator. Assume that A has an asymptotic ΘC-resolvent {L_{γ}(λ) : γ ∈ [0, τ), λ ≥ 0}. If K is a kernel then

(∫_{0}^{t} Θ(s)ds, C) is well-posed.

Proposition 6.6. Suppose D(A) is dense in E and K is a kernel. Then (ΘC) is well-posed for A on [0, τ) if and only if A has an asymptotic ΘC-resolvent {L_{γ}(λ) : γ ∈ [0, τ), λ ≥ 0}.

It is clear that the characterizations for local K-convoluted C-semigroups are generalizations of the characterizations for local integrated C-semigroups, and hence, for local C-cosine functions ([13]).

REFERENCES

[1] W. A r e n d t, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327–352.

[2] W. A r e n d t, O. E l - M e n n a o u i, V. K e y a n t u o, Local integrated semigroups: evolution with the jumps of regularity, J. Math. Analysis Appl. 186 (1994), 572-595.

[3] W. A r e n d t, C. J. K. B a t t y, M. H i e b e r, F. N e u b r a n d e r, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, 2001.

[4] I. C i o r a n e s c u, Local convoluted semigroups, in: Evolution Equations (Baton Rouge, LA, 1992), 107–122, Dekker New York, 1995.

[5] I. C i o r a n e s c u, G. L u m e r, Problèmes d’évolution régularisés par un noyan général K(t). Formule de Duhamel, prolongements, théorèmes de génération, C.R. Acad. Sci. Paris Sér. I Math., 319 (12), 1273–1278, 1995.

[6] I. C i o r a n e s c u, G. L u m e r, On K(t)-convoluted semigroups, in: Recent Developments in Evolution Equations (Glasgow, 1994), 86–93. Longman Sci. Tech., Harlow, 1995.

[7] O. E l - M e n n a o u i, V. K e y a n t u o, Trace theorems for holomorphic semigroups and the second order Cauchy problem, Proceedings of the AMS 124 (1996),1445–1458.

[8] V. K e y a n t u o, The Laplace transform and the ascent method for abstract wave equations, Journal of Differential Equations 122 (1995),27–47.

[9] H. K o m a t s u, Ultradistributions. I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20:25–105, 1973.

[10] H. K o m a t s u, Operational calculus and semi-groups of operators, in: Functional Analysis and Related Topics, 1991 (Kyoto), 213–234. Springer, Berlin, 1993.

[11] P. C. K u n s t m a n n, Stationary dense operators and generation of non-dense distribution semigroups, Journal of Operator Theory 37 (1997), 111–120.

[12] P. C. K u n s t m a n n, Banach space valued ultradistributions and applications to abstract Cauchy problems,[in preparation]

[13] M. L i, F a - L u n H u a n g, Q. Z h e n g, Local integrated C-semigroups, Studia Math. 145 (2001), 265–280.

[14] J. C. L i, S. Y. S h o w, N-times integrated C-semigroups and the abstract Cauchy problem, Taiwanese J. Math. 1 (1997), 75–102.

[15] I. V. M e l n i k o v a, A. I. F i l i n k o v, Abstract Cauchy Problems: Three Approaches, Chapman & Hall /CRC, 2001.

[16] F. N e u b r a n d e r, Integrated semigroups and their applications to complete second order Cauchy problems, Semigroup Forum 38 (1989), 233–251.

[17] M. S o v a, Cosine operator functions, Rozprawy Mat. 49, 1966.

[18] N. T a n a k a, N. O k a z a w a, Local C-semigroups and local integrated semigroups, Proc. London Math. Soc. 61 (1990), 63–90.

[19] S. W. W a n g, Mild integrated C-existence families, Studia Math. 112 (1995), 251– 266.

[20] S. W. W a n g, Z. H u a n g, Strongly continuous integrated C-cosine operator functions, Studia Math. 126 (1997), 273–289.

[21] T.-J. X i a o, J. L i a n g, The Cauchy Problem for Higher-Order Abstract Differential Equations, Springer, 1998.

Institute of Mathematics University of Novi Sad Trg Dositeja Obradovića 4 21000 Novi Sad Serbia and Montenegro e-mail: makimare@neobee.net