Advances in Pure Mathematics, 2013, 3, 680-684
Published Online November 2013 (http://www.scirp.org/journal/apm)
http://dx.doi.org/10.4236/apm.2013.38091
Open Access APM
On Maximal Regularity and Semivariation of
α-Times Resolvent Families*
Fubo Li, Miao Li
Department of Mathematics, Sichuan University, Chengdu, China
Received October 15, 2013; revised November 15, 2013; accepted November 21, 2013
Copyright © 2013 Fubo Li, Miao Li. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Let and A be the generator of an -times resolvent family 1 α <<2 α
()
{
}
0t
St
α
≥ on a Banach space X. It is shown
that the fractional Cauchy problem ,
()() ()
tutAutf t
α
=+D
(
]
0,tr∈; has maximal regularity
on
() (
0,uu
′
)( )
0DA∈
[
]
(
0, ;Cr
)
X if and only if is of bounded semivariation on
()
⋅S
α
[
]
0, r.
Keywords: -Times Resolvent Family; Maximal Regularity; Semivariation
α
1. Introduction
Many initial and boundary value problems can be re-
duced to an abstract Cauchy problem of the form
()() ()
[
]
()( )
,0,
0
utAutft tr
uxDA
′=+ ∈
=∈ (1.1)
where A is the generator of a C0-semigroup. One says that
(1.1) has maximal regularity on
[
]
(
0, ;CrX
)
if for every
[
]
(
0, ;
)
f
CrX∈ there exists a unique
[
]
(
10,;rX
)
uC∈
satisfying (1.1). From the closed graph theorem it
follows easily that if there is maximal regularity on
[
]
()
0, ;CrX, then there exists a constant C > 0 such that
[] () [] () [] ()
0, ;0, ;0, ;. CrXCrX CrX
uAuf
′+≤
Travis [1] proved that the maximal regularity is equi-
valent to the 0-semigroup generated by C
A
being of
bounded semivariation on
[
]
0, r.
Chyan, Shaw and Piskarev [2] gave similar results for
second order Cauchy problems. More precisely, they
showed that the second order Cauchy problem
()() ()(
]
()()( )
,0,
0,0,,
utAutfttr
uxu yxyD
′′ =+ ∈
′
==∈A
(1.2)
has maximal regularity on
[
]
0, r if and only if the cosine
opeator function generated by A is of bounded semivaria-
tion on
[
]
0, r.
In this paper, we will consider the maximal regularity
for fractional Cauchy problem
()() ()(
]
()()( )
,0,
0,0,,
utAutf ttr
uxu yxyD
α
=+ ∈
′
==∈
t
D
1, α ∈
A
(1.3)
where ,
()
2
A
is the generator of an -times
resolvent family (see Definition 2.2) and t is under-
stood in the Caputo sense. We show that (1.3) has ma-
ximal regularity on
α
u
α
D
[
]
(
0, ;CrX
)
if and only if the
corresponding -times resolvent family is of bounded
semivariation on
α
[
]
0, r.
2. Preliminaries
Let , 12 α <<
() ()
0:
g
t δ =t and
() () ()
1
:0
t
gt
β
β
β
β
−
=>
Γ
for . Recall the Caputo fractional derivative of
order
0
0 α >
()
t>
( )() []
2
22
0
d
:d
d
t
t
f
,0,tgtsfsstr
s
α −
=− ∈

D
α
for
[
]
(
;
)
20,
f
CrX∈. The condition that *The authors are partially supported by the NSFC of China (Grant No.
11371263), and the second author is also supported by Program for
N
ew Century Excellent Talents in University of China.
[
]
()
20, ;
f
CrX∈

F. B. LI, M. LI 681
can be relaxed to
[
]
()
10, ;
f
CrX∈ and
() () ()
[
]
()
2
22
00 0,;
g
fffgCrX
α −′
∗− −∈,
for details and further properties see [3] and references
therein. And in the above we denote by
()
()()()
0d
t
g
ft gtsfss
ββ
∗= −

the convolution of
g
β
with f. Note that gg g
α
β
α
β
+
∗= .
Consider a closed linear operator A densely defined in
a Banach space X and the fractional evolution Equation
(1.3).
Definition 2.1 A function
[
]
(
0, ;uC rX∈
)
is called a
strong solution of (1.3) if
[
]
() ()
[
]
()
() () [] ()
1
2
2
0, ;0, ;,
0, ;
uCrDACrX
gutxtyCrX
α −
∈
∗−−∈

and (1.3) holds on
[
]
0, r.
[
]
(
0, ;uC rX∈
)
is called a
mild solution of (1.3) if
()
g
uDA
α
∗∈ and
() () () () ()
utx tyAgu tgft
αα
−− =∗+∗
for
[
]
0,tr∈.
Definition 2.2 Assume that A is a closed, densely de-
fined linear operator on X. A family
()
{
}
()
St BX
α
⊂
0t≥
is called an α-times resolvent family generated by
A
if
the following conditions are satisfied:
(a) is strongly continuous on and
;
()
S
α
⋅
SI=
(
S t
+

()
()
0
α
(b) and
) ()()
DA DA
α
⊂
()
A
StxStAx
αα
=
for all
()
,0
x
DA t∈≥
()
;
(c) For all
x
DA∈ and , 0t≥
x
() ) ()
Stxx tA
ααα
=+
(
g S∗.
Remark 2.3 Since A is closed and densely defined, it
is easy to show that for all
x
X∈,
() ()( )
g
StxDA
αα
∗∈
and
() ()
.
A
gStxS∗=xx−
αα α
The α-times resolvent families are closely related to
the solutions of (1.3). It was shown in [3] that if A ge-
nerates an α-times resolvent family , then (1.3) has
()
S
α
⋅
a unique strong solution given by .
()( )
0d
t
Stx Ssys
αα
+
Next, we recall the definition of functions of bounded
semivariation (see e.g. [4]). Given a closed interval
[
]
,ab of the real line, a subdivision of
[
]
,ab is a finite
sequence 01. Let :n
dadd db=<<< =
[
]
,Dab de-
note the set of all subdivisions of
[
]
,ab .
Definition 2. 4 For
[
]
()
:,GabBX→ and
[
]
,dDab∈,
define
[
]
()( )
1
1
sup:, 1
d
n
iiiii
n
SV G
GdGdxxXx
−
=


=−∈




and
[
]
[
]
[
]
{
}
sup: ,
d
SVGSVGdDa b=∈
.
We say G is of bounded semivariation if
[
]
SV G<∞.
3. Main Results
We begin with some properties on -times resolvent
families which will be needed in the sequel.
α
Proposition 3.1 Let and 1 α <<2
()
{
}
0t≥ be the
-times resolvent family with generator
St
α
α
A
. Define
() () ()
()()
1
1
0d, ,
t
PtxgS tx
g
tsSsxs xX
ααα
αα
−
−
=∗
=− ∈

then the following statements are true.
x
X∈, and
()()
0d
tPsxs DA
α
∈

()()
t
(a) For every
0d;
A
PsxsS txx=−
αα

(b) For every
x
X∈, , 0,ab t≤≤
() (
d
b
a
)
s
Pt sxxDA
α
−∈

and
()()()
()
d
d;
b
a
b
a
A
sPtsx saStaxbStbx
Stsxs
ααα
α
−=−−−
+−


(c) For every
x
X∈,
()() (
0d
t
)
g
tssPsxsDA
αα
−∈

and
()()
()
() () ()
0d
;
t
AgtssPsxs
g
StxtPtx
αα
αα α
α
−
=− ∗+

(d) If
[
]
()
0, ;
f
CrX∈, then
()
g
SfDA
αα
∗∗∈
()
and
()
1.
A
gS fSf
αα α
∗∗ =−∗ (3.1)
Proof. (a) follows from the fact that
()
()
()
() ()()
11
0d
tPsxsg gStx
g
StxDA
ααα
αα
−
=∗ ∗
=∗ ∈

and
() () ()
A
gStxStxx
αα α
∗=− by Remark 2.3.
(b) By integration by parts we have
() ()
() ()
() () () ()
() () () ()
() ()
0
dd d
d
d
d,
bbs
s
aa
b
s
a
b b
aa
b
a
sPtsx ssPtx
sgStsx
s
gStsx gStsxs
ag Staxbg Stbx
gStsxs
αα
αα
αα αα
αα αα
αα
ττ

−= −



=∗−

=−∗ −+∗ −
=∗−−∗−
+∗−




≤
Open Access APM

F. B. LI, M. LI 682
since
() ()()
d
g
StxsDA
αα
∗∈ by Remark 2.3, opera-
ting
A
on both sides of the above identity gives (b).
(c) follows from the fact that
()()
()()() ()()
()()
() ()
() () () ()
() () () ()
() () () ()
0
00
1
0
1
11 1
1
d
d
d
t
tt
t
gtssPsxs
d
g
tsstPsxstgtsPsxs
gtsPsxstgPtx
gP txtgPtx
g
g Stxtgg Stx
ggStxtg gStx
αα
αααα
αα αα
αα αα
ααα ααα
αααα αα
α
α
α
α
+
+
+− −
−
−
=−−+ −
=−−+ ∗
=−∗+ ∗
=−∗ ∗+∗ ∗
=−∗∗+∗ ∗



belongs to and
()
DA
()()
()
() () () () ( ()
()
()
() ()
()
()
() () ()
() () ()
() () ()
0
1
1
1
1
d
11
.
t
AgtssPsxs
)
g
Ag StxtgAg Stx
gStxtg Stx
gStx gtx
tgSttgtx
gStxtPtx
αα
αααα αα
αααα
αα α
αα α
αα α
α
α
αα
α
−
−
+
−
−
=−∗∗+∗ ∗
=−∗−+∗ −
=− ∗+
+∗−
=− ∗+

(d) (3.1) is true for step functions, and then for con-
tinuous functions by the closedness of
A
.
The following two lemmas can be proved similarly as
that in [1,2].
Lemma 3.2 If
[
]
(
0, ;
)
f
CrX∈
()
St
α
and the -times
resolvent family is of bounded semivariation on
α
[
]
0, r, then and
() ()( )
t DA∗∈Pf
α
() ()()( )
0d
t
s.
A
Pft Stsfs
αα
∗=− −



Lemma 3.3 If
[
]
(
0, ;
)
f
CrX∈
()
St
α
and the -times
resolvent family is of bounded semivariation on
α
[
]
0, r, then is continuous in t on
()(
0d
t
sSts fs α
−


)
[
]
0, r.
We next turn to the solution of
()() ()(
]
() ()
,0,
00,00,
tutAutf ttr
uu
α
=+ ∈
′
==
D,
(3.2)
where
A
is the generator of an -times resolvent
family. If is a mild solution of (3.2), then by De-
finition 2.1
α
()
vt
() ()( )
g
vt DA∗∈
α
and
() () () () ()
vtAgvtgft
αα
=∗+∗.
It then follows from the properties of -times re-
solvent family that
α
()
()
()
()
()
1
,
vSAgSv
SvSAgv
SvAgv
Sg f
ααα
ααα
αα
αα
∗= −∗∗
=∗−∗∗
=∗−∗
=∗∗
which implies that
g
Sf
αα
∗∗ is differentiable and
() () ()
() ()
()
()
1
d
d
.
vtgSf t
t
g
Sft
Pft
αα
αα
α
−
=∗∗
=∗∗
=∗
Therefore, the mild solution of (1.3) is given by
()()( ) () ()
0d.
t
utS txSsysPft
αα α
=+ +∗
 (3.3)
Proposition 3.4 Let A be the generator of an
α
-times
resolvent family , and let
()
S
α
⋅
[
]
()
;0,
f
CrX∈ and
()
,
x
yDA∈. Then the following statements are equi-
valent:
(a) (1.3) has a strong solution;
(b)
() ()
[
]
()
10,;SfC rX
α
∗⋅∈ ;
(c) for 0 ≤ t ≤ r and
() ()( )
Pft DA
α
∗∈
() ()
A
Pft
α
∗
is continuous in on t
[
]
0,
()
ut
r.
Proof. (a) If is a strong solution of (1.3), then
is given by (3.3) since every strong solution is a mild
solution. Therefore, by the definition of strong solutions,
u
[
]
()
2
21 0, ;
g
PfgSfCrX
αα α −∗∗=∗∗∈ ;
it then follows that
[
]
(
10,;SfC rX
α
∗∈
)
, this is (b).
(b)(c). Suppose that
[
]
(
0, ;SfC rX
α
∗∈
)
1. Since
1
g
PfgS f
ααα
∗∗=∗∗, by Proposition 3.1(d),
()
1
g
PfDA
α
∗∗∈
and
()( )()
11.
A
gP fAgSfSf
αααα
∗∗ =∗∗ =−∗ (3.4)
Since
A
is closed and
[
]
(
10, ;SfC rX
α
∗∈
)
, we
have and
()
PfDA
α
∗∈
()()
A
PfS ff
αα
∗= ∗ −
′ is
continuous.
(c)(a). By (3.4),
()( )()
11
1
g
APfAgPfSf
ααα
∗∗=∗∗=−∗,
therefore is differentiable and thus
Sf
α
∗
21
g
PfgS f
αα α −∗∗=∗∗
is in
[
]
(
20, ;CrX
)
. It is easy to check that de-
fined by (3.3) is a strong solution of (1.3).
()
ut
Now we are in the position to give the main result of
this paper. The proof is similar to that of Proposition 3.1
in [1] or Theorem 4.2 in [2], we write it out for com-
pleteness.
Open Access APM

F. B. LI, M. LI
Open Access APM
683
position 3.4(c) , it thus follows from the
closedness of A that
()
LfD A∈
[
]
()
:0,;
A
LCr XX→ is bounded.
Theorem 3.5 Suppose that
A
generates an
α
-times
resolvent family
()
{
}
0t≥. Then the function (3.3) is a
strong solution of the Cauchy problem (1.3) for every
pair
St
α
)(
,
x
yD∈A and continuous function
f
if and
only if is of bounded semivariation on
()
S
α
⋅
[
]
0, r.
Let be a subdivision of
{}
0
n
ii
d=
[
]
0, r and 0>
such that
{
}
11
min in ii dd
≤≤ −
<−. For with X
i∈x
()
11,2,,1
i
xi n≤= +, define
[
]
()
,0,; d
f
CrX∈
 by
Proof. The sufficiency follows from Lemmas 3.2 and
3.3.
()
()
1
,
11
,
,
,
ii
di
iiii
xd
fd
i
i
d
x
xx dd
τ
τττ
−
++
≤≤ −


=−
+−−≤≤






Conversely, suppose that for
()
,
x
yDA∈ and con-
tinuous function
f
, given by (3.3) is a strong
solution for (1.3). Define the bounded linear operator
()
ut
[
]
()
:0,;LCr XX→ by . By Pro-
() () ()
LfPf r
α
=∗ then
[] ()
,0,;1 dCrX
f≤
. By Proposition 3.1,
()
()()
() ()()
()
()()() ()
()()()()
1
,,
0
11
1
11
1
11
d
dd d
ii i
ii i
r
dd
ndd d
i
ii ii
dd d
i
n
iii ii iii
i
iii iii
AL fA Prs fss
sd
APrsxsAPrsx sAPrsxx x
Srdx SrdxSrdxSrdx
dSrdxxSrd xx
α
αα α
αααα
αα
−
−
++
−−
=
−+
=
++
=−
−

=−+−+ −−


 
=−−−++−+−−
 

 −−+−−−−


 








1 +
()()() ()() () ()
()() () ()
()()()() () ()
11
11 1
1
111
1
11
d
1d
1d
i
i
i
i
i
i
d
iiiiiiiiii
d
nd
iiiii i
d
i
nd
iiiiiiii
d
i
dSrdx xdSrdx xSrsx xs
Srdx SrdxSrsxxs
SrdSrdxSrdxxSrsx xs
αα α
αα α
αα αα
ε
++
−
−+ +
−
=
−++
−
=


+−−+−−−−+−−




 =−−−+−−





=−−− −−−+−−












,

1+
it then follows that
()()
()
()() () ()
1,1
11
1d.
i
i
nn
d
iiid iiii
d
ii
SrdSrdxALfSrdx xSrsx xs
ααα α −++
−
==
 −− −≤+−−−−−


 
1
i
By letting , we obtain that is of bounded
semivariation on
0→S
α
[
]
0, r.
{
()
}
Corollary 3.6 Suppose that 0t
α
≥ is an -
times resolvent family with generator
St α
A
and is
of bounded semivariation on
()
S
α
⋅
[
]
0, r for some . 0>r
Then for
()
()
()
RP tDA
α
⊂
[
]
0,tr∈
()
tAPt and
α
is bounded on
[
]
0, r
Xx ∈
.
Proof. For , consider
() ()
f
tSt
α
α =
() ()()
Pft tPtx
αα
∗=
x. By
Proposition 3.1(c), is a mild solution of (3.2).
Moreover, it follows from Proposition 3.4 that
is a strong solution of (3.2). Since a strong solution must
be a mild solution, we have . Thus
our claim follows from Proposition 3.4.
()
t x
α
tP
Pf
α
∗
Remark 3.7 Let . If 1 α =
A
generates a 0 C-semi-
group , then the condition that is bounded
on
()
T⋅
()
tAT t
[
]
0,
2=
r implies that is analytic (see [5]). When
and A generates a cosine function , then the
condition that is bounded on
()
T⋅
()
t
α
(
⋅
)
C
tAC
[
]
0, r implies that
A
is bounded ([3]). However, since there is no semi-
group properties for
α
-times resolvent family, it is not
clear that one can get the analyticity of from the
local boundedness of .
()
S
α
⋅
()
t
α
tAP
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