Bulletin T.CXLI de l’Académie serbe des sciences et des arts − 2010
Classe des Sciences mathématiques et naturelles
Sciences mathématiques, No 35
PERTURBATION THEOREMS FOR CONVOLUTED C-SEMIGROUPS AND
COSINE FUNCTIONS
M. KOSTIĆ1
(Presented at the 3rd Meeting, held on April 23, 2010)
A b s t r a c t. In this paper, we prove several different types of additive
perturbation theorems for (local) convoluted C-semigroups and cosine functions.
AMS Mathematics Subject Classification (2000): 47D06, 47D60, 47D62
Key Words: convoluted C-semigroups, convoluted C-cosine functions, perturbations
1. Introduction and Preliminaries
Convoluted C-semigroups and cosine functions ([6]-[8], [14], [16]-[17], [25])
allow one to consider in a unified treatment the notion of fractionally integrated C-semigroups and cosine functions ([1]-[3], [20], [22], [39], [41]).
We refer the reader to [1]-[2], [5], [9]-[11], [16]-[17], [25], [36] and [42] for
examples of differential operators generating various types of convoluted Csemigroups and cosine functions. In the present paper, we study additive
1
This research was supported by grant 144016 of Ministry of Science and Technological
Development, Republic of Serbia.
26
M. Kostić
perturbation theorems for such classes of operator semigroups and cosine
functions and continue the researches raised in [12], [16], [20], [28]-[29], [31],
[35] and [39]-[40] (cf. also [4], [9]-[10], [21], [26], [32]-[34], [37]-[38], and
[23]-[24], for similar results).
Throughout this paper E denotes a non-trivial complex Banach space, L(E)
denotes the space of bounded linear operators from E into E, A denotes
a closed linear operator acting on E and [D(A)] denotes the Banach space
D(A) equipped with the norm ||x||[D(A)] := ||x||+||Ax||, x ∈ D(A). By R(A)
is denoted the range of operator A. Henceforward L(E) ∋ C is an injective
operator, τ ∈ (0, ∞], K is a complex-valued locally integrable function in
[0, τ ) and K is not identical to zero. Given t ∈ R, set ⌊t⌋ := sup{k ∈
Z : k ≤ t} and ⌈t⌉ := inf{k ∈ Z : k ≥ t}. Set Θ(t) :=
Θ−1 (t) :=
∫t
∫t
K(s)ds and
0
Θ(s)ds, t ∈ [0, τ ); then Θ is an absolutely continuous function
0
in [0, τ ) and Θ′ (t) = K(t) for a.e. t ∈ [0, τ ). Let us recall that a function
K ∈ L1loc ([0, τ )) is called a kernel if for every ϕ ∈ C([0, τ )), the assumption
∫t
K(t − s)ϕ(s)ds = 0, t ∈ [0, τ ), implies ϕ ≡ 0; thanks to the famous
0
Titchmarsh’s theorem, the condition 0 ∈ suppK implies that K is a kernel.
We mainly use the following condition:
(P1) K is Laplace transformable, i.e., it is locally integrable on [0, ∞) and
there exists β ∈ R so that
K̃(λ) = L(K)(λ) := lim
∫b
b→∞ 0
e−λt K(t)dt :=
∫∞
e−λt K(t)dt exists for all
0
λ ∈ C with Reλ > β. Put abs(K) :=inf{Reλ : K̃(λ) exists}.
In Theorem 2.9, we use the following condition:
(P2) K̃(λ) ̸= 0 for all λ ∈ C with Reλ > abs(K).
Definition 1.1. ([14], [16]-[17]) Let A be a closed operator and let 0 <
τ ≤ ∞. If there exists a strongly continuous operator family (SK (t))t∈[0,τ )
(SK (t) ∈ L(E), t ∈ [0, τ )) such that:
(i) SK (t)A ⊆ ASK (t), t ∈ [0, τ ),
(ii) SK (t)C = CSK (t), t ∈ [0, τ ) and
27
Perturbation theorems for convoluted C-semigroups and cosine functions
(iii) for all x ∈ E and t ∈ [0, τ ):
∫t
SK (s)xds ∈ D(A) and
0
∫t
A
SK (s)xds = SK (t)x − Θ(t)Cx,
0
then it is said that A is a subgenerator of a (local) K-convoluted C-semigroup
(SK (t))t∈[0,τ ) . If τ = ∞, then we say that (SK (t))t≥0 is an exponentially
bounded K-convoluted C-semigroup with a subgenerator A if, additionally,
there exist M ≥ 1 and ω ∈ R such that ||SK (t)|| ≤ M eωt , t ≥ 0.
Definition 1.2. ([14], [16]-[17]) Let A be a closed operator and let 0 < τ ≤
∞. If there exists a strongly continuous operator family (CK (t))t∈[0,τ ) such
that:
(i) CK (t)A ⊆ ACK (t), t ∈ [0, τ ),
(ii) CK (t)C = CCK (t), t ∈ [0, τ ) and
(iii) for all x ∈ E and t ∈ [0, τ ):
∫t
(t − s)CK (s)xds ∈ D(A) and
0
∫t
A
(t − s)CK (s)xds = CK (t)x − Θ(t)Cx,
0
then it is said that A is a subgenerator of a (local) K-convoluted C-cosine
function (CK (t))t∈[0,τ ) . If τ = ∞, then we say that (CK (t))t≥0 is an exponentially bounded K-convoluted C-cosine function with a subgenerator A if, additionally, there exist M ≥ 1 and ω ∈ R such that ||CK (t)|| ≤ M eωt , t ≥ 0.
α−1
Plugging K(t) = tΓ(α) in Definition 1.1 and Definition 1.2, where α > 0
and Γ(·) denotes the Gamma function, we obtain the well-known classes
of α-times integrated C-semigroups and cosine functions; a (local) 0-times
integrated C-semigroup, resp. C-cosine function, is defined to be a (local) Csemigroup, resp. C-cosine function. The integral generator of (SK (t))t∈[0,τ ) ,
resp. (CK (t))t∈[0,τ ) , is defined by
{
(x, y) ∈ E × E : SK (t)x − Θ(t)Cx =
∫t
0
}
SK (s)yds, t ∈ [0, τ ) , resp.,
28
M. Kostić
{
(x, y) ∈ E × E : CK (t)x − Θ(t)Cx =
∫t
}
(t − s)CK (s)yds, t ∈ [0, τ ) ,
0
and it is a closed linear operator which is an extension of any subgenerator
of (SK (t))t∈[0,τ ) , resp. (CK (t))t∈[0,τ ) . Suppose that A is a subgenerator of
(SK (t))t∈[0,τ ) , resp. (CK (t))t∈[0,τ ) . By [18, Proposition 1.1], we know that
the integral generator  of (SK (t))t∈[0,τ ) , resp. (CK (t))t∈[0,τ ) , satisfies  =
C −1 ÂC = C −1 AC.
Lemma 1.3. ([17]) Let A be a closed operator and let 0 < τ ≤ ∞. Then
the following assertions are equivalent:
(i) A is a subgenerator of a K-convoluted C-cosine function (CK (t))t∈[0,τ )
in E.
I )
is a subgenerator of a Θ-convoluted CA 0
( C 0 )
semigroup (SΘ (t))t∈[0,τ ) in E × E, where C ≡
.
0 C
(ii) The operator A ≡
In this case:




SΘ (t) = 
∫t
( 0
CK (s)ds
0
CK (t) − Θ(t)C
∫t
0

(t − s)CK (s)ds 
∫t
CK (s)ds

 , 0 ≤ t < τ,

0
and the integral generators of (CK (t))t∈[0,τ ) and (SΘ (t))t∈[0,τ ) , denoted re( 0 I )
spectively by B and B, satisfy B =
.
B 0
Definition 1.4. Let 0 < α ≤ π2 and let (SK (t))t≥0 be a K-convoluted
C-semigroup. Then we say that (SK (t))t≥0 is an analytic K-convoluted Csemigroup of angle α, if there exists an analytic function SK : Σα → L(E)
which satisfies
(i) SK (t) = SK (t), t > 0 and
(ii)
lim
z→0, z∈Σγ
SK (z)x = 0 for all γ ∈ (0, α) and x ∈ E.
It is said that (SK (t))t≥0 is an exponentially bounded, analytic K-convoluted
C-semigroup of angle α, if for every γ ∈ (0, α), there exist Mγ ≥ 0 and
ωγ ≥ 0 such that ||SK (z)|| ≤ Mγ eωγ Rez , z ∈ Σγ .
Perturbation theorems for convoluted C-semigroups and cosine functions
29
Since there is no risk for confusion, we shall also write SK for SK .
2. Perturbations
The following rescaling result for subgenerators of (local) convoluted Csemigroups extends [16, Proposition 3.2].
Theorem 2.1. Suppose z ∈ C, K and F satisfy (P1), there exists a ≥ 0
such that
K̃(λ) − K̃(λ + z)
=
K̃(λ + z)
∫∞
e−λt F (t)dt, Reλ > a, K̃(λ + z) ̸= 0,
(1)
0
and A is a subgenerator (the integral generator) of a (local) K-convoluted
C-semigroup (SK (t))t∈[0,τ ) . Then A − z is a subgenerator (the integral generator) of a (local) K-convoluted C-semigroup (SK,z (t))t∈[0,τ ) , where
−tz
SK,z (t)x := e
∫t
SK (t)x +
F (t − s)e−zs SK (s)xds, x ∈ E, t ∈ [0, τ ). (2)
0
Furthermore, in the case τ = ∞, (SK,z (t))t≥0 is exponentially bounded provided that F and (SK (t))t≥0 are exponentially bounded.
P r o o f. It is clear that (SK,z (t))t∈[0,τ ) is a strongly continuous operator
family that commutes with C and A − z. Let x ∈ E be fixed. Then we
obtain
∫t
∫t
∫s
(A − z) SK,z (s)xds = (A − z) [e−zs SK (s)x + F (s − r)e−zr SK (r)xdr]ds
0
0
∫t
−zt
= (A − z)[e
0
∫s
0
0
SK (s)xds + z e−sz
0
+(A − z)
∫t
∫t ∫s
SK (r)xdrds]
F (s − r)e−zr SK (r)xdrds
0 0
∫t
∫t
= e−zt [SK (t)x − Θ(t)Cx] − ze−zt SK (s)xds + z e−sz [SK (s)x − Θ(s)Cx]ds
0
−z
∫t
2
0
∫s
−sz
e
0
0
∫t
∫s
SK (r)xdrds + (A − z) F (t − s) e−zr SK (r)xdrds
0
= e−zt [SK (t)x − Θ(t)Cx] − ze
0
∫t
−zt
0
SK (s)xds
30
M. Kostić
∫t
∫t
∫s
0
0
0
+z e−sz [SK (s)x − Θ(s)Cx]ds − z 2 e−sz
SK (r)xdrds
∫t
∫s
∫s
∫r
0
0
0
0
+ F (t − s)(A − z)[e−zs SK (r)xdr + z e−zr
SK (v)xdvdr]ds
= SK,z (t)x − f1 (t) − f2 (t)Cx, x ∈ E, where:
∫t
∫t
∫t
∫s
0
0
0
0
f1 (t) = ze−zt SK (s)xds − z e−sz SK (s)xds + z 2 e−sz
∫t
∫s
∫t
∫s
0
0
SK (r)xdrds
+z e−zs F (t − s) SK (r)xdrds − z F (t − s) e−zr [SK (r)x − Θ(r)Cx]drds
0
0
+z
∫t
2
0
∫s
∫r
−zr
0
0
F (t − s) e
SK (v)xdvdrds, t ∈ [0, τ ) and
∫t
f2 (t) = Θ(t)e−zt + z e−zs Θ(s)ds
0
∫t
∫t
∫s
0
0
0
+ F (t − s)e−zs Θ(s)ds − F (t − s) e−zr Θ(r)drds, t ∈ [0, τ ).
Fix a number t ∈ (0, τ ) and define afterwards a function SeK : [0, ∞) → L(E)
by setting:
{
SK (s), s ∈ [0, t],
e
SK (s) =:
SK (t), s > t.
Clearly, (SeK (t))t≥0 is a strongly continuous operator family and there exist
M > 0 and ω ∈ R such that ||SeK (t)|| ≤ M eωt , t ≥ 0. Define f˜1 : [0, ∞) →
L(E) by replacing SK in the representation formula for f1 with SeK . Then
f˜1 extends continuously the function f1 to the whole non-negative real axis,
and moreover, f˜1 is Laplace transformable. Using the elementary operational properties of Laplace transform and (1), one obtains L(f1 (t))(λ) =
L(f2 (t))(λ) = 0 for all sufficiently large real numbers λ. An application of the
uniqueness theorem for the Laplace transform gives that A − z is a subgenerator of a (local) K-convoluted C-semigroup (SK,z (t))t∈[0,τ ) . Suppose now
that A is the integral generator of (SK (t))t≥0 . Then one has C −1 AC = A
and this implies that C −1 (A − z)C = A − z is the integral generator of
(SK,z (t))t∈[0,τ ) . Finally, the exponential boundedness of (SK,z (t))t≥0 simply
31
Perturbation theorems for convoluted C-semigroups and cosine functions
follows from (2) and the exponential boundedness of F and (SK (t))t≥0 .
(λ)
Suppose K = L−1 ( ppmk (λ)
), where L−1 denotes the inverse Laplace transform,
pk and pm are polynomials of degree k and m, respectively, and k > m. Then
the condition (1) holds for a suitable exponentially bounded function F. Supα−1
pose now α > 0 and K(t) = tΓ(α) , t > 0. Then there exists a sufficiently large
positive real number a such that
L(
∞ ( ) n n−1
∑
α z t
n=1
n (n−1)! )(λ),
obtain |
∞ ( ) n n−1
∑
α z t
= (1 + λz )α − 1 =
(α)
λ > a, where 1α = 1. Since sup |
n∈N
n (n−1)! |
n=1
K̃(λ)−K̃(λ+z)
K̃(λ+z)
n
∞ ( ) n
∑
α z
n=1
n λn
=
| =: L0 < ∞, we
≤ L0 |z|e|z|t , t ≥ 0. With Theorem 2.1 and this ob-
servation in view, one obtains the following extension of [20, Proposition
2.4(b)] and [28, Proposition 3.3]; for the global case, see [21].
Corollary 2.2. Suppose z ∈ C, α > 0 and A is a subgenerator, resp. the
integral generator, of a (local, global exponentially bounded) α-times integrated C-semigroup (Sα (t))t∈[0,τ ) . Then A − z is a subgenerator, resp. the
integral generator, of a (local, global exponentially bounded) α-times integrated C-semigroup (Sα,z (t))t∈[0,τ ) , which is given by:
−zt
Sα,z (t)x = e
Sα (t)x +
( )
∫t ∑
∞
α z n tn−1
0 n=1
n (n − 1)!
e−zs Sα (s)xds, t ∈ [0, τ ), x ∈ E.
The following perturbation theorem generalizes [16, Theorem 4.1].
Theorem 2.3. Suppose B ∈ L(E), K is a kernel and satisfies (P1), A is a
subgenerator (the integral generator) of a (local) K-convoluted C-semigroup
(SK (t))t∈[0,τ ) , BA ⊆ AB, BC = CB and there exists a > 0 such that the
following conditions hold:
(i) For every n ∈ N, there is a function Kn satisfying (P1) and
g
K
n (λ) = K̃(λ)
Put Θn (t) :=
∫t
|Kn (s)|ds, t ≥ 0, n ∈ N.
0
(ii)
∞
∑
n=1
dn ( 1 )
(λ), λ > a, K̃(λ) ̸= 0.
dλn K̃
Θn (t) < ∞, t ≥ 0.
32
M. Kostić
Then A+B is a subgenerator (the integral generator) of a (local) K-convoluted
B (t))
C-semigroup (SK
t∈[0,τ ) , which satisfies for every x ∈ E and t ∈ [0, τ ) :
B
SK
(t)x
tB
= e SK (t)x +
∞ ∑
i
∑
Bi
i=1 n=1
i!
( ) ∫t
n
(−1)
i
n
Kn (t − s)si−n SK (s)xds. (3)
0
Furthermore, the following holds:
B (t) − etB S (t)|| ≤ e||B|| max ||S (s)||
(a) ||SK
K
K
s∈[0,t]
∞
∑
n=1
Θn (t)et||B|| , t ∈ [0, τ ).
(b) Suppose τ = ∞, (SK (t))t∈[0,τ ) is exponentially bounded and there exist
constants M > 0 and ω ≥ 0 such that
∞
∑
Θn (t) ≤ M eωt , t ≥ 0.
(4)
n=1
B (t))
Then (SK
t∈[0,τ ) is also exponentially bounded.
P r o o f. First of all, notice that the commutation of B with C and A implies
that the function u1 , resp. u2 , given by u1 (t) :=
resp. u2 (t) :=
∫t
∫t
SK (s)Bxds, t ∈ [0, τ ),
0
BSK (s)xds, t ∈ [0, τ ), is a solution of the initial value
0
problem

1

 u ∈ C([0, τ ) : [D(A)]) ∩ C ([0, τ ) : E),


u′ (t) = Au(t) + Θ(t)CBx, t ∈ [0, τ ),
u(0) = 0.
Since K is a kernel, we have the uniqueness of solutions of the preceding
problem ([14], [17]) and this implies BSK (t)x = SK (t)Bx, t ∈ [0, τ ), x ∈ E.
Then we obtain:
B
||SK
(t) −etB SK (t)|| ≤ maxs∈[0,t] ||SK (s)||
∞
∑
Θn (t)
∞ ∑
i
∑
i=1 n=1
i ( )
∑ ∥B∥ i ∑
i −n
t
= maxs∈[0,t] ||SK (s)||
Θn (t)
i!
n t
n=1
n=1
i≥1
∞
i
i
∑
i (t+1)
≤ maxs∈[0,t] ||SK (s)||
Θn (t) ||B||
i! t
ti
n=1
∞
∑
= e||B|| maxs∈[0,t] ||SK (s)||
Θn (t)et||B|| , t ∈ (0, τ ).
n=1
∞
∑
n=1
||B||i ( i ) i−n
i!
n t
i
33
Perturbation theorems for convoluted C-semigroups and cosine functions
Hence, the assertion (a) holds. The previous computation also shows that
B (t))
(SK
t∈[0,τ ) is a strongly continuous operator family that commutes with
A+B and C. Let x ∈ E be fixed. Then the dominated convergence theorem,
the closedness of A and integration by parts, as well as the argumentation
B (t) − etB S (t)||, imply:
used in the estimation of term ||SK
K
∫t
∫t
B (s)xds = (A + B) esB S (s)xds
(A + B) SK
K
0
∞ ∑
i
∑
+
Bi
i=1 n=1
i!
0
(−1)
( )
n i
n
(A + B)
∫t ∫s
Kn (s − r)ri−n SK (r)xdrds
0 0
∫t
= etB [SK (t)x − Θ(t)Cx] + B esB Θ(s)Cxds
0
∞
∑
+
i
∑
i=1 n=1
( )
Bi
n i
i! (−1) n (A
∫t
∫s
+ B) Kn (t − s) ri−n SK (r)xdrds
0
[
0
]
∫t
= etB SK (t)x − Θ(t)Cx + B esB Θ(s)Cxds
0
∫t
0
Bi
( )
n i
i! (−1) n
i=1 n=1
[
∫s
s) si−n SK (r)xdr − (i −
0
+
× Kn (t −
∞ ∑
i
∑
(A + B) ×
∫s
∫r
0
0
]
n) ri−n−1 SK (v)xdvdr ds
∫t
= etB [SK (t)x − Θ(t)Cx] + B esB Θ(s)Cxds
0
+
∞ ∑
i
∑
B i+1
i!
i=1 n=1
[
∫s
(−1)
( ) ∫t
n i
n
Kn (t − s) ×
0
∫s
∫r
0
0
]
× si−n SK (r)xdr − (i − n) ri−n−1 SK (v)xdvdr ds
0
∞ ∑
i
∑
+
Bi
i=1 n=1
+
∞ ∑
i
∑
i=1 n=1
i!
(−1)
( )
Bi
n i (n
(−1)
i!
n
=
B (t)x
SK
( ) ∫t
n i
n
Kn (t − s)si−n [SK (s)x − Θ(s)Cx]ds
0
∫t
∫s
0
0
− i) Kn (t − s) ri−n−1 [SK (r)x − Θ(r)Cxdr]ds
− f1 (t) − f2 (t)Cx, t ∈ [0, τ ), where:
( ) ∫t
B i+1
n i
Kn (t − s) ×
(−1)
i!
n
i=1 n=1
0
∫s
∫s
∫r
×[si−n SK (r)xdr − (i − n) ri−n−1 SK (v)xdvdr]ds
0
0
0
f1 (t) =
∞ ∑
i
∑
34
M. Kostić
+
∞ ∑
i
∑
i=1 n=1
( )
Bi
n i
i! (−1) n (n
∫t
∫s
0
0
− i) Kn (t − s) ri−n−1 SK (r)xdrds, t ∈ [0, τ )
and
∫t
( ) ∫t
Bi
n i
(−1)
Kn (t − s)si−n Θ(s)ds
i!
n
i=1 n=1
0
∫t
∫s
i) Kn (t − s) ri−n−1 Θ(r)drds, t ∈ [0, τ ).
0
0
f2 (t) = etB Θ(t) − B esB Θ(s)ds +
0
+
∞
∑
i
∑
i=1 n=1
( )
Bi
n i
i! (−1) n (n
−
∞ ∑
i
∑
Then the partial integration implies:
( ) ∫t
∫s
∫r
B i+1
n i
Kn (t − s) ri−n SK (r)xdrds
i! (−1) n
i=1 n=1
0
0
0
t
s
r
∞ ∑
i
(
)
∫
∫
∫
∑
i
B
n i
i−n−1 S (r)xdrds, t ∈
+
K
i! (−1) n (n − i) Kn (t − s) r
i=1 n=1
0
0
0
f1 (t) =
∞ ∑
i
∑
[0, τ ).
The coefficient of B i , i ≥ 2 in the expression of f1 (t) equals
i−1
∑
·
(−1)
n
(n − i
n=1
i!
∫t
∫s
Kn (t − s)
0
( )
(
)
∫r
r
i−n−1
0
( )
(
i
1
i−1 )
+
n
(i − 1)!
n
SK (r)xdrds = 0,
0
)
i−1
i
1
because n−i
= 0. Thereby, f1 (t) = 0, t ∈ [0, τ ). On the
i! n + (i−1)! n
other hand, the usual series arguments imply that the coefficient of B i in
the expression of f2 (t) equals Θ(t), t ≥ 0 if i = 0, and
f2,i (t) :=
+
i
∑
n=1
+
i
∑
n=1
ti
i! Θ(t)
(
1
n i
i! (−1) n
( )
1
n i
i! (−1) n (n
) ∫t
−
∫t
0
si−1
(i−1)! Θ(s)ds
Kn (t − s)si−n Θ(s)ds
0
∫t
∫s
0
0
− i) Kn (t − s) ri−n−1 Θ(r)drds, t ≥ 0,
if i ≥ 1. Proceeding as before, one obtains, as a consequence of the condition
(iii), that the function t 7→ f2,i (t), t ≥ 0 satisfies (P1) and that there exists
Perturbation theorems for convoluted C-semigroups and cosine functions
35
a′′ > 0 such that
L(f2,i (t))(λ) = i!1 (−1)i
)
K̃(·) (i)
(λ)
·
−
(
)(i−1)
1 1
i−1 K̃(·)
(λ)
λ (i−1)! (−1)
·
(
)(n)
( )
1
1
i i K̃(λ)
(−1)
(λ) λ1 K̃ (i−n) (λ)
i!
n
K̃(·)
n=1
(
)(i)
(
)(i−1)
1 1
1
i K̃(·)
i−1 K̃(·)
(λ)
−
(λ)
(−1)
(−1)
i!
·
λ (i−1)!
·
+
=
(
i
∑
i
= i!1 (−1)i
(
(
)
(−1)
1
+ K̃(λ)
− K̃(λ)
K̃ (i) (λ)
λ
i!
)
K̃(·) (i)
(λ)
·
−
)(i−1)
(
1 1
i−1 K̃(·)
(λ)
λ (i−1)! (−1)
·
i+1
+ (−1)i!
K̃ (i) (λ)
λ
= 0,
for all λ > a′′ with K̃(λ) ̸= 0. This enables one to deduce that f2 (t) =
B (t))
Θ(t), t ∈ [0, τ ) and that (SK
t∈[0,τ ) is a (local) K-convoluted C-semigroup
with a subgenerator A + B. The proof of (b) follows from a simple computation; furthermore, the supposition that A is the integral generator of
(SK (t))t∈[0,τ ) implies that C −1 AC = A and that C −1 (A + B)C = A + B is
B (t))
the integral generator of (SK
t∈[0,τ ) . This completes the proof of theorem.
Remark 2.4.
(i) The assumption (i) of Theorem 2.3 is satisfied for the function K =
L−1 ( pka(λ) ), where pk is a polynomial of degree k ∈ N and a ∈ C \ {0}.
Then n0 = k and Kn ≡ 0, n ≥ k +1. In this case, we have the existence
of positive real numbers M and ω such that (4) holds.
(ii) ([16]) Let n > 1 and let P be an analytic function in the right half
plane {λ ∈ C : Reλ > λ0 } for some λ0 ≥ 1. Suppose that P (λ) ̸=
0, Reλ > λ0 , and that there exist C > 0 and r ∈ (0, 1] with:
|P (λ)| ≥ C|λ|n , Reλ > λ0 ,
|
di
P (λ)| ≤ C|λ|−ir |P (λ)|, Reλ > λ0 , i ∈ N,
dλi
P (j)
∈ LT (C), j ≤ 1/r, j ∈ N,
P
where LT (C) denotes the set of all Laplace transforms of exponentially
bounded functions. Then the condition (i) of Theorem 2.3 holds for
36
M. Kostić
the function K = L−1 (1/P ) and there exist M > 0 and ω ≥ 0 such
that (4) holds.
(iii) The conditions (ii) and (iii) quoted in the formulation of Theorem 2.3
can be replaced with:
(ii)’ there exist M1 ≥ 1 and ω1 ≥ 0 such that
∞ ∑
i
∑
||B||i
( ) ∫t
i!
i=1 n=1
i
n
|Kn (t − s)|si−n ds ≤ M1 eω1 t , t ≥ 0
0
and
(iii)’ to every i ∈ N, there exists ai > 0 such that the function
t 7→ max |Θ(s)|e−ai t
s∈[0,t]
L1 ([0, ∞) : R).
i
∑
n=1
(2t+2)i
Θn (t),
i!
t ≥ 0 belongs to the space
Notice only that one can prove that f1 ≡ 0 by direct computation of
coefficient of B i , i ∈ N and that the condition (iii)’ is necessary in
our striving to show that, for every i ∈ N, the function t 7→ f2,i (t),
t ≥ 0 satisfies (P1); it is also clear that (iii)’ holds provided that Θ is
exponentially bounded and that, for every n ∈ N, Θn is exponentially
bounded. Let us prove now that (ii)’ and (iii)’ hold for the function
σ
K = L−1 (e−λ ), where σ ∈ (0, 1). First of all, we know that K is
σ
an exponentially bounded, continuous kernel. Let f (λ) = eλ , λ ∈
C \ (−∞, 0]. Then the mapping λ 7→ f (λ), λ ∈ C \ (−∞, 0] is analytic,
f ′ (λ) = σλσ−1 f (λ) and
f
(n)
(λ) =
n−1
∑
i=0
(
)
n − 1 ( σ−1 )(n−i−1)
·
(λ)f (i) (λ), λ ∈ C \ (−∞, 0]. (5)
i
Using (5), one concludes inductively that, for every n ∈ N, there exist
real numbers pi,n (σ), 1 ≤ i ≤ n such that, for every t ≥ 0 :
g
K
n (λ) =
n
∑
pi,n (σ)λiσ−n , Reλ > 0 and Θn (t) ≤
i=1
n
∑
|pi,n (σ)tn−iσ |
i=1
Put p0,n (σ) := 0, n ∈ N. By the foregoing, we have
(
e·
σ
)(n)
σ
(λ) = eλ
n
∑
i=1
pi,n (σ)λiσ−n
Γ(n + 1 − iσ)
.
37
Perturbation theorems for convoluted C-semigroups and cosine functions
and
(
e·
σ
)(n+1)
σ
(λ) = eλ
n+1
∑(
)
pi,n (σ)(iσ − n) + σpi−1,n (σ) λiσ−(n+1) ,
i=1
for all n ∈ N and λ ∈ C with Reλ > 0. Hence, p1,n (σ) = σ(σ − 1) · · ·
(σ − (n − 1)), n ∈ N \ {1}, pn,n (σ) = σ n , n ∈ N and
pi,n+1 (σ) = pi,n (σ)(iσ − n) + σpi−1,n (σ), n ∈ N, 2 ≤ i ≤ n.
(σ )
Clearly, Lσ := sup |
n
n∈N0
n≥2:
| < ∞. Applying (6) we infer that for every
n+1
∑
≤ σ(σ − 1) · · · (σ − n) +
(
)
n−1
∑
≤ Lσ σ + n n! + nσ
(6)
i!|pi,n+1 (σ)|
i=1
n [
∑
(
)
]
σi!|pi−1,n (σ)| + n σ + 1 i!|pi,n (σ)| + (n + 1)!
i=2
(
)∑
n
i!|pi,n (σ)| + n σ + 1
i=1
i!|pi,n (σ)| + (n + 1)!.
i=2
The preceding inequality implies that, for every ζ ≥ 2 + 4σ + 2Lσ , the
following holds:
n
∑
i!|pi,n (σ)| ≤ ζ n n! for all n ∈ N.
(7)
i=1
Denote by ζσ the minimum of all numbers satisfying (7). Then a
simple computation shows that, for every x ∈ E :
∞ ∑
i
∑
||B||i
i=1 n=1
i!
i
n
||Kn (t − s)si−n SK (s)x|| ds
0
≤ max ||SK (s)x||
s∈[0,t]
( ) ∫t
∞
∑
i=1
n
i ∑
||B||i ζσi ∑
ti+1−lσ i!
, t ≥ 0.
i!
Γ(i + 2 − lσ)l!
n=1 l=1
(8)
On the other hand, it is easily verified that:
i ∑
n
∑
n=1 l=1
i!
≤ i2(2−σ)i , i ∈ N.
Γ(i + 2 − lσ)l!
(9)
38
M. Kostić
Combining (8)-(9), it follows that, for every t ∈ [0, min(1, τ )),
2−σ
B
max ||SK (s)||
SK (t) − etB SK (t) ≤ t||B||ζσ 22−σ e||B||ζσ 2
s∈[0,t]
and, in case τ > 1,
2−σ
B
SK (t)−etB SK (t) ≤ t2 ||B||ζσ 22−σ e||B||ζσ 2 t max ||SK (s)||, t ∈ [1, τ ),
s∈[0,t]
proving the condition (ii)’; furthermore, in case that τ = ∞ and that
B (t))
(SK (t))t≥0 is exponentially bounded, then (SK
t≥0 is also exponentially bounded. It is clear that (iii)’ holds and that the above conσ
clusions remain true in case K = L−1 (e−aλ ), where σ ∈ (0, 1) and
a > 0.
(iv) Suppose α > 0, K(t) =
tα−1
Γ(α) ,
( α)
t > 0, L0 := sup |
n∈N
n
| and A is a subgen-
erator of a (local, global exponentially bounded) α-times integrated Csemigroup (Sα (t))t∈[0,τ ) . Then Kn (t) = α(α−1)···(α−n+1)
tn−1 , L0 < ∞,
(n−1)!
(α) n
Θn (t) = | n |t , t ≥ 0, n ∈ N and this implies that the condition
(iii) of Theorem 2.3 does not hold if α ∈
/ N. Nevertheless, the series
B (t) − etB S (t)|| ≤
appearing in (3) still converges, the estimate ||SK
K
2t||B||
L0 max ||SK (s)||e
, t ∈ [0, τ ) follows analogically and the proof of
s∈[0,t]
Theorem 2.3 can be repeated verbatim. Having in mind these observations, we obtain the next important generalization of [16, Corollary
4.5] and [35, Theorem 2.3] (cf. also [21, Theorem 3.5]):
Theorem 2.5. Suppose α > 0, A is a subgenerator, resp. the integral
generator, of a (local, global exponentially bounded) α-times integrated Csemigroup (Sα (t))t∈[0,τ ) , B ∈ L(E), BA ⊆ AB and BC = CB. Then A+B is
a subgenerator, resp. the integral generator, of a (local, global exponentially
bounded) α-times integrated C-semigroup (SαB (t))t∈[0,τ ) , which satisfies, for
every x ∈ E and t ∈ [0, τ ),
SαB (t)x
tB
= e Sα (t)x +
∞ ∑
i
∑
Bi
i=1 n=1
i!
( )( ) ∫t
n
(−1) n
i
n
α
n
0
(t − s)n−1 si−n Sα (s)xds.
Perturbation theorems for convoluted C-semigroups and cosine functions
39
Notice ([36]) that the previous formula can be rewritten in the following
form:
SαB (t)x
( )
∞
∑
α
tB
= e Sα (t)x+
(−B)
i
i=1
∫t
i
0
(t − s)i−1 Bs
e S(s)xds, x ∈ E, t ∈ [0, τ ).
(i − 1)!
(10)
The following perturbation theorem for generators of exponentially bounded,
analytic integrated C-semigroups is applicable on a class of (differential)
operators analyzed by R. deLaubenfels in [10, Section XXI, Section XXIV].
Theorem 2.6. Suppose r > 0, α ∈ (0, π2 ], A is a subgenerator, resp. the
integral generator, of an exponentially bounded, analytic r-times integrated
C-semigroup (Sr (t))t≥0 of angle α, B ∈ L(E), BA ⊆ AB and BC = CB.
Then A + B is a subgenerator, resp. the integral generator, of an exponentially bounded, analytic r-times integrated C-semigroup (SrB (t))t≥0 of angle
α, where
SrB (z)x
zB
:= e
Sr (z)x+
( )
∞
∑
α
i=1
∫z
(−B)i
i
0
(z − s)i−1 Bs
e Sr (s)xds, x ∈ E, z ∈ Σα .
(i − 1)!
(11)
(r )
P r o o f. Clearly, L0 = sup |
n
n∈N
| < ∞. Notice that, for every z ∈ Σα ,
the series appearing in (11) is absolutely convergent and that, for every
γ ∈ (−α, α) such that |γ| > arg(z), we have the following:
||SrB (z) − ezB Sr (z)|| ≤
∑
L0 ||B||i
i≥1
Re
∫ z |z|i−1
0
||B|||z| M eωγ Rez ds
γ
(i−1)! e
≤ RezMγ L0 ||B||e(2||B||+ωγ )Rez .
(12)
This implies that (Sr (z))z∈Σα is a strongly continuous operator family satisfying the conditions (i) and (ii) stated in the formulation of Definition 1.4.
It remains to be shown that the mapping
z 7→
( )
∞
∑
α
i=1
i
∫z
i
(−B)
0
(z − s)i−1 Bs
e Sr (s)ds, z ∈ Σα
(i − 1)!
(13)
40
M. Kostić
is analytic. By standard arguments, the mapping f0 (z) =
∫z
e−Bs Sr (s)ds, z ∈
0
Σα is analytic and f0′ (z) = e−Bz Sr (z), z ∈ Σα . This simply yields that, for
every i ∈ N, the mapping fi (t) =
∫z
0
(z−s)i−1 Bs
(i−1)! e Sr (s)ds,
z ∈ Σα is analytic
and that fi′ (z) = fi−1 (z), z ∈ Σα . Furthermore, the series in (11) is locally
uniformly convergent; this follows from the next obvious estimate:
( )
∫z
α
i (−B)i
0
(
(z−s)i−1 Bs
(i−1)! e Sr (s)ds
)i
≤ Mγ sup |z|||B||
z∈K
e
(||B||+ω) sup |z|
z∈K
(i−1)!
,
where K is an arbitrary compact subset of Σα and γ is chosen so that
K ⊆ Σγ . An application of the Weierstrass theorem completes the proof of
theorem.
The following theorem extends [30, Theorem 3.8] and [35, Theorem 2.4,
Theorem 2.5, Corollary 2.6] (cf. also [40, Theorem 2.3]). The proof is
omitted since it follows by the use of the argumentation given in [35], [29,
Section 10] and [40].
Theorem 2.7. Suppose n ∈ N, A is a subgenerator, resp. the integral
generator, of a (local, global exponentially bounded) n-times integrated Csemigroup (S(t))t∈[0,τ ) , B ∈ L(E), R(B) ⊆ C(D(An )) and BCx = CBx, x ∈
D(A). Then A+B is a subgenerator, resp. the integral generator, of a (local,
global exponentially bounded) n-times integrated C-semigroup (SB (t))t∈[0,τ )
which satisfies the following integral equation:
∫t
SB (t)x = S(t)x +
dn
S(t − s)C −1 BSB (s)xds, t ∈ [0, τ ), x ∈ E.
dtn
0
With Theorem 2.7 in view, one can prove the following extension of [29,
Theorem 10.1] that is comparable with [35, Theorem 4.6] and [39, Theorem
3.1]; notice only that the assertions related to the study of unbounded perturbations of generators of integrated C-semigroups (cf. [35, Theorem 3.1,
Theorem 3.2] and [23, Theorem 3.1]) and cosine functions can be proved
similarly. It seems possible to prove the assertions of Theorem 2.7 and Theorem 2.8 in the case of (local) fractionally integrated C-semigroups and cosine
functions.
Perturbation theorems for convoluted C-semigroups and cosine functions
41
Theorem 2.8. Suppose n ∈ N, A is a subgenerator, resp. the integral generator, of a (local, global exponentially bounded) n-times integrated
n+1
C-cosine function (C(t))t∈[0,τ ) , B ∈ L(E), R(B) ⊆ C(D(A⌊ 2 ⌋ )) and
BCx = CBx, x ∈ D(A). Then A + B is a subgenerator, resp. the integral generator, of a (local, global exponentially bounded) n-times integrated
C-cosine function (CB (t))t∈[0,τ ) .
The following theorem mimics an interesting perturbation result of C. Kaiser
and L. Weis ([12]-[13]) which can be additionally refined if the Fourier type
of the space E ([1], [12]) is also taken into consideration.
Theorem 2.9. Suppose K satisfies (P1), (P2) and there exists β ∈ (abs(K), ∞)
such that for every ε > 0 there exists Cε > 0 with
1
≤ Cε eε|λ| , λ ∈ C, Reλ > β.
|K̃(λ)|
(14)
(i) Suppose A generates an exponentially bounded K-convoluted semigroup
(SK (t))t≥0 such that ||SK (t)|| ≤ M1 eωt , t ≥ 0 for some M1 > 0 and
ω ≥ 0. Let B be a linear operator such that D(A) ⊆ D(B) and that
there exist M ∈ (0, 1) and λ0 ∈ (max(β, ω), ∞) satisfying ||BR(λ :
A)|| ≤ M, λ ∈ C, Reλ = λ0 . Then, for every α > 1, the operator A+B
α−1
generates an exponentially bounded, (K ∗0 tΓ(α) )-convoluted semigroup.
(ii) Suppose A generates an exponentially bounded K-convoluted semigroup
(SK (t))t≥0 such that ||SK (t)|| ≤ M1 eωt , t ≥ 0 for some M1 > 0 and
ω ≥ 0. Let B be a densely defined linear operator such that there
exist M ∈ (0, 1) and λ0 ∈ (max(β, ω), ∞) satisfying ||R(λ : A)Bx|| ≤
M ||x||, x ∈ D(B), λ ∈ C, Reλ = λ0 . Then there exists a closed
extension D of the operator A + B such that, for every α > 1, the
α−1
operator D generates an exponentially bounded, (K ∗0 tΓ(α) )-convoluted
semigroup. Furthermore, if A and A∗ are densely defined, then D is
the part of the operator (A∗ + B ∗ )∗ in E.
(iii) Suppose A generates an exponentially bounded K-convoluted cosine
function (CK (t))t≥0 such that ||CK (t)|| ≤ M1 eωt , t ≥ 0 for some
M1 > 0 and ω ≥ 0. Let B be a linear operator such that D(A) ⊆
D(B) and that there exist M > 0 and λ0 ∈ (max(β, ω), ∞) satisfying
M
||BR(λ2 : A)|| ≤ |λ|
, λ ∈ C, Reλ = λ0 . Then, for every α > 1,
the operator A + B generates an exponentially bounded, (K ∗0
convoluted cosine function.
tα−1
Γ(α) )-
42
M. Kostić
(iv) Suppose A generates an exponentially bounded K-convoluted cosine
function (CK (t))t≥0 such that ||CK (t)|| ≤ M1 eωt , t ≥ 0 for some
M1 > 0 and ω ≥ 0. Let B be a densely defined linear operator such
that there exist M > 0 and λ0 ∈ (max(β, ω), ∞) satisfying ||R(λ2 :
M
A)Bx|| ≤ |λ|
||x||, x ∈ D(B), λ ∈ C, Reλ = λ0 . Then there exists a
closed extension D of the operator A+B such that, for every α > 1, the
α−1
operator D generates an exponentially bounded, (K ∗0 tΓ(α) )-convoluted
cosine function. Furthermore, if A and A∗ are densely defined, then
D is the part of the operator (A∗ + B ∗ )∗ in E.
P r o o f. We will prove only (iii) and (iv). By [17, Theorem 3.1], we
have that {λ2 : λ ∈ C, Reλ > max(β, ω)} ⊆ ρ(A) and that ||R(λ2 : A)|| ≤
M1
, λ ∈ C, Reλ > max(β, ω). Suppose z ∈ C and Rez > λ0 .
|λ||K̃(λ)|(Reλ−ω)
Put λ = λ0 + iImz and notice that
(
)
||BR(z 2 : A)|| = BR(λ2 : A) I + (λ2 − z 2 )R(z 2 : A) (
≤ BR(λ2 : A) 1 + |λ − z||λ + z|||R(z 2 : A)||
≤
M
|λ|
(
M1
1 + |λ − z||λ + z| |z||K̃(z)|(
Rez−ω)
≤
M
|λ|
≤M
≤M
(
(
(
1
1 + |λ + z| |z||M
K̃(z)|
1
|λ|
1
λ0
+ (1 +
+
+
)
)
|z|
M1
|λ| ) |z||K̃(z)|
M1
|z||K̃(z)|
)
)
M1
λ0 |K̃(z)|
)
.
(15)
Consider now the function h : {z ∈ C : Rez ≥ 0} → L(E) defined by
h(z) := zBR((z + λ0 )2 : A), z ∈ C, Rez ≥ 0. Then ||h(it)|| ≤ M, t ∈ R
and, owing to (14) and (15), we have that, for every ε > 0, there exists Cε > 0
such that ||h(z)|| ≤ Cε eε|z| for all z ∈ C with Rez ≥ 0. An application of
the Phragmén-Lindelöf type theorems (cf. for instance [1, Theorem 3.9.8,
p. 179]) gives that ||h(z)|| ≤ M for all z ∈ C with Rez ≥ 0. This, in turn,
implies that there exists a > λ0 such that ||BR(λ2 : A)|| < 12 , λ2 ∈ ρ(A + B)
and that, for every λ ∈ C with Reλ > a :
λR(λ2 : A + B) = λR(λ2 : A)(I − BR(λ2 : A))−1 ≤
1
.
|K̃(λ)|
The proof of (iii) follows by making use of [17, Theorem 3.1] and [36,
Theorem 1.12] while the proof of (iv) is a consequence of [12, Lemma 3.2]
and a similar reasoning.
Perturbation theorems for convoluted C-semigroups and cosine functions
43
By the proof of Theorem 2.9, we immediately obtain the following corollary.
Corollary 2.10.
(i) Suppose A generates a cosine function (C(t))t≥0 satifying ||C(t)|| ≤
M eωt , t ≥ 0 for appropriate M > 0 and ω ≥ 0. If B is a linear
operator such that D(A) ⊆ D(B) and that there exist M ′ > 0 and
M
, λ ∈ C, Reλ = λ0 , then,
λ0 ∈ (ω, ∞) satisfying ||BR(λ2 : A)|| ≤ |λ|
for every α > 1, the operator A+B generates an exponentially bounded,
α-times integrated cosine function.
(ii) Suppose A generates a cosine function (C(t))t≥0 satifying ||C(t)|| ≤
M eωt , t ≥ 0 for appropriate M > 0 and ω ≥ 0. Let B be a densely
defined linear operator such that there exist M ′ > 0 and λ0 ∈ (ω, ∞)
M
||x||, x ∈ D(B), λ ∈ C, Reλ = λ0 .
satisfying ||R(λ2 : A)Bx|| ≤ |λ|
Then there exists a closed extension D of the operator A+B such that,
for every α > 1, the operator D generates an exponentially bounded, αtimes integrated cosine function. Furthermore, if A and A∗ are densely
defined, then D is the part of the operator (A∗ + B ∗ )∗ in E.
We close the paper with the following illustrative example.
Example 2.11.
(i) ([22]) Let E := C0 (R) ⊕ C0 (R) ⊕ C0 (R), C(f, g, h) := (f, g, sin(·)h(·)),
f, g, h ∈ C0 (R) and A(f, g, h) := (f ′ + g ′ , g ′ , (χ[0,∞) − χ(−∞,0] )h),
(f, g, h) ∈ D(A) = {(f, g, h) ∈ E : f ′ ∈ C0 (R), g ′ ∈ C0 (R), h(0) =
0}. Arguing as in [22, Example 8.1, Example 8.2], one gets that
A is the integral generator of an exponentially bounded once integrated C-semigroup and that A is not a subgenerator of any local
C-semigroup. Suppose now mi ∈ C 1 (R), i = 1, 2, the mappings
t 7→ |t|mi (t), t ∈ R and t 7→ |t|m′i (t), t ∈ R are bounded for i = 1, 2;
C(R) ∋ m3 is bounded and satisfies m3 (0) = 0. Put B(f, g, h) :=
∫·
∫·
0
0
(m1 (·) f (s)ds, m2 (·) g(s)ds, sin(·)m3 (·)h(·)), f, g, h ∈ C0 (R). Then
one can simply verify B ∈ L(E), R(B) ⊆ C(D(A)) and BC(f, g, h) =
CB(f, g, h), (f, g, h) ∈ E. By Theorem 2.7, one obtains that A + B
is the integral generator of an exponentially bounded once integrated
C-semigroup.
(ii) Let E := L1 (R) and let D := d/dx with maximal distributional domain. Then it is well known (cf. also [12, Corollary 3.4, Example
44
M. Kostić
7.1]) that E has the Fourier type 1, and in particular, that E is not a
B−convex Banach space. Furthermore, A := D2 = d2 /dx2 generates
a bounded cosine function (C(t))t≥0 given by
(
)
C(t)f (x) :=
)
1(
f (x + t) + f (x − t) , t ≥ 0, x ∈ R, f ∈ L1 (R),
2
and Sobolev imbedding theorem implies D(A) = W 1,2 (R) ⊆ C(R) ∩
L∞ (R). Suppose g ∈ L1 (R) \ L∞ (R) and define a linear operator B :
L1 (R)∩L∞ (R) → L1 (R) by Bf (x) := f (x)g(x), f ∈ L1 (R)∩L∞ (R).
In general, B cannot be extended to a bounded linear operator from
L1 (R) into L1 (R) and R(B) * D(A). Clearly,
||B(2λR(λ2 : A)f )|| =
≤
∫∞
−∞
∫∞
−∞
∫∞
|g(x)||
∫∞
e−λt (f (x + t) + f (x − t))dt|dx
0
|g(x)| (|f (x + t)| + |f (x − t)|)dtdx
0
≤ 2||g||||f ||, λ ∈ C, Reλ > 0, f ∈ L1 (R),
and this implies that all assumptions quoted in the formulation of
Corollary 2.10(i) holds with λ0 = 1. Hence, A + B generates an exponentially bounded α-times integrated cosine function for every α > 1;
let us also point out that it is not clear whether there exists β ∈ [0, 1)
such that A + B generates a (local) β-times integrated cosine function
although one can simply prove that there exist a > 0 and M > 0 such
M
, λ ∈ C, Reλ > a.
that ||λR(λ2 : A + B)|| ≤ Re
λ
(iii) Suppose A generates a (local) α-times integrated cosine function for
some α > 0, B ∈ L(E) and BA ⊆ AB. Then the proof of [15, Theorem
4.3] and the analysis given in [17, Example 7.3] imply that, for every
s ∈ (1, 2), ±iA generate global K1/s -semigroups and that ±iA generate
σ
local K1/2 -semigroups, where Kσ (t) = L−1 (e−λ )(t), t ≥ 0, σ ∈ (0, 1).
By Theorem 2.3 and Remark 2.4(iii), we have that ±i(A+B) generate
global K1/s -semigroups for every s ∈ (1, 2) and that ±i(A+B) generate
local K1/2 -semigroups. Therefore, a large class of differential operators
(cf. [2], [11] and [42]) generating integrated cosine functions can be
used to provide applications of Theorem 2.3.
(iv) ([19], [16]) Let s > 1,
}
{
E :=
∥f (p) ∥∞
<∞ ,
f ∈ C [0, 1] | ∥f ∥ := sup
p!s
p≥0
∞
45
Perturbation theorems for convoluted C-semigroups and cosine functions
and
A := −d/dx, D(A) := {f ∈ E : f ′ ∈ E, f (0) = 0}.
It is well known that there exist positive real numbers m and M such
1
that {λ ∈ C : Reλ ≥ 0} ⊆ ρ(A) and ∥R(λ : A)∥ ≤ M em|λ| s , Reλ ≥ 0
1
1
([19]). Since |e−ξλ s | ≤ e−ξ|λ| s cos( 2s ) , ξ > 0, λ ∈ C, Reλ > 0, we
have that A generates a global exponentially bounded Ka, 1 -convoluted
π
s
semigroup for every a >
m
π ,
cos( 2s
)
0. Let n ∈ N and let Bf (x) :=
1
where Ka, 1 (t) = L−1 (e−aλ s )(t), t ≥
s
n ∫x
∑
(x−s)n−1
i=1 0
(n−1)!
f (s)ds, x ∈ [0, 1], f ∈
E. Then it is checked at once that B ∈ L(E) and that BA ⊆ AB.
Owing to Theorem 2.3 and Remark 2.4(iii), we easily infer that A + B
generates a global exponentially bounded Ka, 1 -convoluted semigroup.
s
REFERENCES
[1] 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.
[2] W. A r e n d t, H. K e l l e r m a n n, Integrated solutions of Volterra integrodifferential
equations and applications, Volterra integrodifferential equations in Banach spaces
and applications, Proc. Conf., Trento/Italy 1987, Pitman Res. Notes Math. Ser. 190
(1989), 21–51.
[3] 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 jumps of regularity, J. Math. Anal. Appl. 186 (1994), 572–595.
[4] W. A r e n d t, C. J. K. B a t t y, Rank-1 perturbations of cosine functions and
semigroups, J. Funct. Anal. 238 (2006), 340–352.
[5] R. B e a l s, On the abstract Cauchy problem, J. Funct. Anal. 10 (1972), 281–299.
[6] I. C i o r ă n e s c u, Local convoluted semigroups, in: Evolution Equations (Baton
Rauge, LA, 1992), 107–122, Dekker New York, 1995.
[7] I. C i o r ă 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 (1995), 1273–1278.
[8] I. C i o r ă 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.
[9] J. C h a z a r a i n, Problémes de Cauchy abstraites et applications á quelques problémes
mixtes, J. Funct. Anal. 7 (1971), 386-446.
[10] R. d e L a u b e n f e l s, Existence Families, Functional Calculi and Evolution
Equations, Lecture Notes in Mathematics 1570, Springer 1994.
46
M. Kostić
[11] M. H i e b e r, Integrated semigroups and differential operators on Lp spaces, Math.
Ann. 291 (1995), 1-16.
[12] C. K a i s e r, L. W e i s, Perturbation theorems for α-times integrated semigroups,
Arch. Math. 81 (2003), 215–228.
[13] C. K a i s e r, Integrated semigroups and linear partial differential equations with
delay, J. Math. Anal. Appl. 292 (2004), 328–339.
[14] M. K o s t i ć, Convoluted C-cosine functions and convoluted C-semigroups, Bull. Cl.
Sci. Math. Nat. Sci. Math. 28 (2003), 75–92.
[15] M. K o s t ić, P. J. M i a n a, Relations between distribution cosine functions and
almost-distribution cosine functions, Taiwanese J. Math. 11 (2007), 531–543.
[16] M. K o s t i ć, S. P i l i p o v i ć, Global convoluted semigroups, Math. Nachr. 280
(2007), 1727–1743.
[17] M. K o s t ić, S. P i l i p o v i ć, Convoluted C-cosine functions and semigroups.
Relations with ultradistribution and hyperfunction sines, J. Math. Anal. Appl. 338
(2008), 1224–1242.
[18] M. K o s t ić, Convoluted C-groups, Publ. Inst. Math., Nouv. Sér 84(98) (2008),
73–95.
[19] P. C. K u n s t m a n n, Stationary dense operators and generation of non-dense
distribution semigroups, J. Operator Theory 37 (1997), 111–120.
[20] M. L i, Q. Z h e n g, α-times integrated semigroups: local and global, Studia Math.
154 (2003), 243–252.
[21] Y. - C. L i, S. - Y. S h a w, Perturbation of non-exponentially-bounded α-times
integrated C-semigroups, J. Math. Soc. Japan 55 (2003), 1115–1136.
[22] Y. - C. L i, S. - Y. S h a w, N -times integrated C-semigroups and the abstract Cauchy
problem, Taiwanese J. Math. 1 (1997), 75–102.
[23] C. L i z a m a, J. S á n c h e z, On perturbation of K−regularized resolvent families,
Taiwanese J. Math. 7 (2003), 217–227.
[24] C. L i z a m a, V. P o b l e t e, On multiplicative perturbation of integral resolvent
families, J. Math. Anal. Appl. 327 (2007), 1335–1359.
[25] 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.
[26] T. M a t s u m o t o, S. O h a r u, H. R. T h i e m e, Nonlinear perturbations of a
class of integrated semigroups, Hiroshima Math. J. 26 (1996), 433–473.
[27] I. M i y a d e r a, M. O k u b o, N. T a n a k a, On integrated semigroups which are
not exponentially bounded, Proc. Japan Acad. 69 (1993), 199-204.
[28] J. M. A. M. v a n N e e r v e n, B. S t r a u b, On the existence and growth of
mild solutions of the abstract Cauchy problem for operators with polynomially bounded
resolvent, Houston J. Math. 24 (1998), 137-171.
[29] S. - Y. S h a w, Cosine operator functions and Cauchy problems, Conferenze del Seminario Matematica dell’Universitá di Bari. Dipartimento Interuniversitario Di Matematica Vol. 287, ARACNE, Roma, 2002, 1–75.
Perturbation theorems for convoluted C-semigroups and cosine functions
47
[30] S. - Y. S h a w, C. - C. K u o, Generation of local C-semigroups and solvability of
the abstract Cauchy problems, Taiwanese J. Math. 9 (2005), 291–311.
[31] S. - Y. S h a w, C. - C. K u o, Y. - C. L i, Perturbation of local C-semigroups,
Nonlinear Analysis, Nonlinear Anal. 63 (2005), 2569-2574.
[32] N. T a n a k a, On perturbation theory for exponentially bounded C-semigroups,
Semigroup Forum 41 (2003), 215–236.
[33] N. T a n a k a, Perturbation theorems of Miyadera type for locally Lipschitz continuous
integrated semigroups, Studia Math. 41 (1990), 215–236.
[34] H. R. T h i e m e, Positive perturbations of dual and integrated semigroups, Ado.
Math. Sci. Appl. 6 (1996), 445–507.
[35] S. W. W a n g, M. Y. W a n g, Y. S h e n, Perturbation theorems for local integrated
semigroups and their applications, Studia Math. 170 (2005), 121–146.
[36] T. - J. X i a o, J. L i a n g, The Cauchy Problem for Higher-Order Abstract Differential
Equations, Springer, 1998.
[37] T. - J. X i a o, J. L i a n g, Perturbations of existence families for abstract Cauchy
problems, Proc. Amer. Math. Soc. 130 (2002), 2275–2285.
[38] T. - J. X i a o, J. L i a n g, F. L i, A perturbation theorem of Miyadera type for local
C-regularized semigroups, Taiwanese J. Math. 10 (2006), 153–162.
[39] J. Z h a n g, Q. Z h e n g, On α-times integrated cosine functions, Math. Japon. 50
(1999), 401–408.
[40] Q. Z h e n g, Perturbations and approximations of integrated semigroups, Acta Math.
Sinica 9 (1993), 252–260.
[41] Q. Z h e n g, Integrated cosine functions, Internat. J. Math. Sci. 19 (1996), 575–580.
[42] Q. Z h e n g, Coercive differential operators and fractionally integrated cosine functions, Taiwanese J. Math. 6 (2002), 59-65.
Faculty of Technical Sciences
University of Novi Sad
Trg D. Obradovića 6
21125 Novi Sad
Serbia
e-mail: marco.s@verat.net