Advances in Pure Mathematics, 2012, 2, 211-215
http://dx.doi.org/10.4236/apm.2012.23030 Published Online May 2012 (http://www.SciRP.org/journal/apm)
α-Times Integrated C-Semigroups
Man Liu, Da-Qing Liao, Qian-Qian Zhu, Fu-Hong Wang
Department of Basic Education, Xuzhou Air Force College, Xuzhou, China
Email: liuman8866@163.com
Received February 3, 2012; revised March 12, 2012; accepted March 20, 2012
ABSTRACT
The α-times integrated C semigroups, α > 0, are introduced and analyzed. The Laplace inverse transformation for α-times
integrated C semigroups is obtained, some known results are generalized.
Keywords: α-Times Integrated C Semigroups; Laplace Inverse Transformation; Pseudo-Resolvent Identity
1. Introduction
Integrated semigroups are more general than strongly
continuous semigroups (i.e., 0 semigroups), cosine
operator functions and exponentially bounded distribu-
tion semigroups. Integrated exponentially bounded semi-
groups were investigated in [1-15]. In this paper, we will
introduce and analyze α-times integrated C semigroups,
. In Theorem 2.6 we give a necessary and suffi-
cient condition for an C
C
R
R
to be the pseudo-resol-
vent of an α-times integrated C semigroups . At the
same time we discuss the Laplace inverse transformation
for α-times integrated C semigroups. The results obtained
are generalizations of the corresponding results for inte-
grated semigroups.

St
Throughout this paper, X is a Banach space,
BX
RA
is
the space of bounded linear operators from X into X,
, ,

DA

K
A denote the domain, range, core
of operator A respectively, .

CBX
2. Definitions and Properties of α-Times
Integrated C Semigroups
For 0
,
,

denote the integral part and deci-
mal part of α respectively.

1sx
is well known Gamma
function, and

0d
s
xe x


 
1s
1
, .

ss
For


:0,jR


, we definite the function ,
and

1
t
jt
 1
j
denotes 0-point Dirac meas-
ure 0
.
For continuous function
f
, 1
 , the definition
of convolution product is as following

 


01
tts fs
jft



d, 1
,1
s
ft


0
.
At first we introduce the fractional differential and in-
tegral of function.
For arbitrary ,
-order differential of function
u denotes



1
00
n
Du tt
0
.
For arbitrary ,
-times cumulative integral of
function u denotes

1
I
ujut


R
.
Definition 2.1. Let , a strongly continuous fam-
ily
St BX
1
0t is called α-times integrated C-
semigroups, if
StCCSt, and (V)00S
2
 
;
(V)


1
1
0
1d
d, ,0
st
t
s
StSsxts rSrCxr
tsrSrCxr ts

 
(2.1)
If
nn N


0t
St
, then
is called n-times
integrated C semigroups.
If nn N



0t
St
0
, and C = I, then is called
n-times integrated semigroups.
0Stx
If
,
(t) implies , then α-
times integrated C semigroups is non-degen-
erated.
00x


0t
St
R
If there exists M > 0,
, such that t
St Me
,
0t
0t
St
0
is called exponentially bounded. , then
Definition 2.2. Let
, a strongly continuous fam-
ily
0t
St BX
is called α-times exponentially
bounded integrated C semigroups generated by A, if
00S
, and there exists , 0M0
, such that
,
A

 , t
St Me
0t, , and for arbitrary
C
opyright © 2012 SciRes. APM
M. LIU ET AL.
212
x
X

,d
t
eStxt


0t0
, we have
,
 
 
1
0
C
RAx ACx

 . (2.2)
Proposition 2.3. Let A be the generator of an α-times
integrated C semigroups ,
St
. Then
1) For all

x
DA and t, 0
 
x StAx
 
,Stx DA ASt (2.3)
  
0d
tSsAxs
 
dxs DA
1
t
Stx Cx

 (2.4)
2)
, for all
0
tSs
x
X0t

, and and
 
0d
t
1
t
A
SsxsStx
Cx


d
t
eStxt

Re
(2.5)
Proof. Letting ,

0
C
Rx

 

, ,
,d,
C
ARuAx
uAStxt
Fix , then

uA
 
0
0
,d
t
CC
t
C
eStRuAxt R
eR



for all Re
, and
x
X

,,0At

. By the uniqueness theo-
rem it follows that

,,
CC
RuASt StRuAu (2.6)
This implies (2.3). Let
x
DRe
A
, then for all

,

 



1
0
1
00
1
00
11
00
d
1
,,
d(
t
CC
tt
t
tt
t
tt
t
CxeCx t
RAxRAAx
eStxteS
eStxted
eStxte

 
 
 
 





 

 

 










0
0
)d
dd
ddd.
tAxt
SsAxs
SsAxst
Then (2.4) follows from the uniqueness theorem.
In order to prove (2.5), let
x
X
, and , 0Ret


,
then by (2.3), (2.4), (2.6) we have




00
0
0
d,
,d
,
tt
C
t
C
t
C
C
C SsxsRASsx
RASs
RASs
RAStx




d
,d
1
s
Axs
xs
tCx






,,RAAx
 
(2.7)
Noting that C
Hence,
, and by (2.7), (2.5) follows.

C
RAxCx


 
dSsxsDA
R
0
t
Corolla ry 2 .4. Let . Then Stx DA for
all
x
X
0t
and . Then is right differenti-
able in if . In that case
0t

St

Sx
 
x DA
1
d,0, .
d
t
StxAStxCxtx X
t
 

:
Proposition 2.5. Let
A
DA X be closed linear
operator, when
uA
, we have
,
1) The pseudoresolvent identity
 
,, ,,
CC CC
RACRAC RARA


(2.8)
2)
 
1
d,1!,
d
nn
n
n
CC
nRACnRA

 

1, 2,n

 
 
 
1
1
1
,,
,
,,
,,,
CC
C
CC
CCC
RACAARAC
ACARA
(2.9)
Proof. 1)
A
CCRA RA
RACRARA




 
 
 

It follows that
 
,, ,,
CC CC
RACRAC RARA


1n
2) We apply the mathematical induction when
,
by (2.8)
 
2
d,,
dCC
RACRA



nk
, (2.9) is complete. i.e., we suppose
 
1
d,1!,
d
kk
k
k
CC
kRAC kRA

 

then
 

 
1
1
1
1
2
1
ddd
,,
d
dd
d1! ,
d
11!,
kk
kk
CC
kk
k
k
C
k
k
C
RACRACC
kRA C
kRA





 

 

1nk
i.e., it follows . The proof is complete.
Theorem 2.6. Let
St be a stongly continuous op-
erator function, and
t
St Me
0t
 
0d
t
C
ReStxt


,
, , letting
Re

. Then
Re
C
R

satisfies the pseudoresolvent

CC
RR
CC
RCRC



(2.10)
if and only if
St satisfies (2
V).
hcondition. Proof. One can easily prove te necessary
prove it is sufficiet. Let us thatn
Letting Re
, Reu
, and u
. Then the re-
solvent equation implies
Copyright © 2012 SciRes. APM
M. LIU ET AL. 213
 

 


CC CC
R CRC
uu





C
CC
R R
u
RCu
RCuRC
uuuu




 

 





(2.11)
  
d
t
StSst
00
CC tu
RR ee
u




 (2.12)
 













00
00
00
0
0
00
dd
1
dd
1
dd
1
dd
1
dd
CC
ut
t
C
t
ut us
ut us
ut us
t
us t
t
t
tus
RCuRC
uu
eStCteu
u
eeSsCst
eeSsCst
u
eeSsCst
u
eeSsCst
u
eeSstCst
u


11
dd
RuCt
u
 













 












Noting that 1dvu
0
uv
ev

Then
 





 

 
1ddd
ddd
ddd
vCvst
stCvst
SrCvst

 

 








 


(2.13)
Moreover,
00 0
1
000
1
00
CC
us v
t
r
tur
ts
tur
t
RCuRC
uu
e
eSst
rs
ee S
tsr
ee



 
 


 

 
1
1
ddd
ddd
ddd
ddd
uv
ut us
ur
t
ts
rt
tu
r
t
tr
ut r
t
C
e
ee
SsC vvst
e
eSsCrts rst
rt s
eeS sCrst
trs
ee SsCrst
 

 
 





 
 
 

(2.14)
and
00 0
00
1
00
1
0
00


 
C
Ru
uu

 


1
00 0
1
0
00
1
00 0
ddd
ddd
() ddd
C
v
ut us
tr
ut r
t
t
ut r
RC
uu
e
eeSsC vvst
trs
e eSsCrst
trs
ee SsCrst

 







 


Using (2.14) and (2.15), we obtain

(2.15)


 
1
00 0ddd
C
t
ut r
RCu
uu
trs
ee SsCrst







(2.16)
Assertion (2
V) follows from (2.13) and (2.16
uniqueness of the Laplace transformation.
3. Laplace Inverse Transformation for
α-Times Integrated C-Semigroups
Lemma 3.1. [16] Let0
) and the
, :,X

 ,
F
F
is

0d
t
Laplace-type expression:

F
ett

,

00
, and


th
tht Mhe

 , ,0th, then

 
1d
,
2π
it
i
teF
i




Theorem 3.2. Let 0
, then the following condi-
tioalns are equivent:
1) A generates an α-times exponentially bounded inte-
grated semigroups
St ;
0t
2) There exists 0
, such that

,
A


for all u
, and
, A generates an

uA C
expone
unded semigroups
ntially
bo
Tt , and
t0
 



1
T t
.
Proo If A generates an α-times exponentially
bounded integrated semigroups


0t
St , then
 

d
St u Aj
t
 

d

f. 1)

01
d
tt
A
Wrxr WtxCx

,

oup g

By ([17], Proposition 3.7(a)),

0t
Wt is an

uAC
semigrenerated by
A
is the exten-
tion of A, By ([17],
A
Proposition 3.11), A
.
mbnheorem 3.4]

2) Coig [18] with [17, T, we can prove
 
01
t
d
t
A
Srxr StxCx
, an
 d the space of op-
Copyright © 2012 SciRes. APM
M. LIU ET AL.
214
erator is exchangeable, by Proposition 2.3, This ends the
proof .
Theorem 3.3. Let A be closed linear operator on X,

A

,

A

, an α-times exponentially bounded
integrated C semigroups
0t wititesimal
gene
St h infin
rator A, and

,
t
Me St
0
,
, then

for
x
DA,
 

0
,
CAx
1d
d
ti
t
R
Ssxse
2π
1d
i
it
i
Ie R
,
2πC
iAx
i




(3.1)

Proof. Let
 
0d
t
tSss,
 
1
() ,
ACx
F
xDA

by Lemma 3.1
  
1
d
t
AC



0
00
00
dd
dd
t
t
t
t
x
F
eS
ex Sss
eSsxst
txt






So

F

satisfies Lemma 3.1,
  
,
1d
,
2π
ti
tC
i
RAx
s e
i
0dSsx



On the other hand, by Theorem 3.2 A generates
uAC
exponentially bounroups ded semig
0t
Tt .
So for
x
DA, we have 



 
0
1
1
1
d
1d
2π
1d
,
2π
d
2π
t
it
i
it
i
Tsxs
eAuACx
i
eARuACx
i
RCx
,it
i
uAeA
i
 








AIT t
.


It follows that
 
St u
Whence



 


 




00
0
1
,d
2π
1d
,.
2π
t
it
i
it
C
i
Ss
AI T
RuA
uAIeACx
i
Ie RAx
i
ddxsuAITsxs
dusxs
tt






And the integral on the right converges uniformly on
any bounded intervals.
Corollary 3.4. The conditions are same as Theorem 3.3,
then for
x
X,




,
1d
2π
1,d
2π
itC
i
it
C
i
RAx
Stx e
i
Ie RAx
i




(3.2)
rem 3.3



Proof. by Theo

0
,
1d
d2π
i
tC
i
RAx
Ssxse
i
t


 (3.3)
Then
 
0
,
1d
d2π
ti
tC
i
RAAx
ASsxs e
i


 .
By (2.5) and noting that

1
1d
12π
it
i
te
i

 


Therefore

 




1
,
1d
1d
2π
,
1d
2π
,
1d
,
1d
2π
i
it
i
iC
t
i
iC
t
i
RAAx
eCx
i
RAAx
e
i
RAxAA
RAx
e
i

12π
C
t
i
StxCx e
i
2π
iC
t
ie
i
t













ining Theorem 3.3 we can prove the next pa
ry 3.5. The conditions are same as The
3.

Combrt.
Corolla o rem
3, then for
x
X
,


2


0
2
,
1d
d2π
1d
,
2π
ti
tC
i
it
C
i
RAx
Ssxse
i
Ie RAx
i






(3.4)
Proof. by Theorem 3.3
 
0
,
1d
d2π
ti
tC
i
RAx
Ss se
i


 (3.5)
integrating (3.5) from 0 to t, i.e.,
 


00
2
,
1d
dd
2π
,
1d
1
2π
tti
tC
i
itC
i
RAx
tsSsxses
i
RAx
e
i







Noting that
,
1d
0
2π
iC
i
RAx
i



Copyright © 2012 SciRes. APM
M. LIU ET AL.
Copyright © 2012 SciRes. APM
215
Consequently,
 
2
d2π
tC
i
Ssx e
i
0
,
1d
ti
RAx
ts s

.
The next part is easy to prove.
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