Hölder Regularity for Abstract Fractional Cauchy Problems with Order in (0,1) ()
1. Introduction
Recently there are increasing interests on fractional differential equations due to their wide applications in viscoelasticity, dynamics of particles, economic and science et al. For more details we refer to [1] [2].
Many evolution equations can be rewritten as an abstract Cauchy problem, and then they can be studied in an unified way. For example, a heat equation with different initial data or boundary conditions can be written as a first order Cauchy problem, in which the governing operator generates a C0-semigroup, and then the solution is given by the operation of this semigroup on the initial data. See for instance [3] [4]. Prüss [5] developed the theory of solution operators to research some abstract Volterra integral equations and it was Bajlekova [6] who first use solution operators to discuss the fractional abstract Cauchy problems. If the coefficient operator of a fractional abstract Cauchy problem generates a C0-semigroup, we can invoke an operator described by the C0-semigroup and a probability density function to solve this problem, for more details we refer to [7] [8] [9]. The vector-valued Laplace transform developed in [3] is an important tool in the theory of fractional differential equations.
There are some papers devoted to the fractional differential equations in many different respects: the connection between solutions of fractional Cauchy problems and Cauchy problems of first order [10]; the existence of solution of several kinds of fractional equations [11] [12]; the Hölder regularity for a class of fractional equations [13] [14]; the maximal
regularity for fractional order equations [6]; the boundary regularity for the fractional heat equation [15]; the relation of continuous regularity for fractional order equations with semi-variations [12]. In this paper we are mainly interested in the Hölder regularity for abstract Cauchy problems of fractional order.
Pazy [4] considered the regularity for the abstract Cauchy problem of first order:
(1.1)
where A is the infinitesimal generator of an analytic C0-semigroup. He showed that if
for some
, then
is Hölder continuous with
exponent
in
; if moreover
, then u is Hölder continuous
with the same exponent in
. If in addition f is Hölder continuous, then
Pazy showed that there are some further regularity of
and
. Li [16]
gave similar results for fractional differential equations with order
. In this paper we will extend their results to fractional Cauchy problems with order in
.
Our paper is organized as follows. In Section 2 there are some preliminaries on fractional derivatives, fractional Cauchy problems and fractional resolvent families. In Section 3 we give the regularity of the mild solution under the condition that
. And some further continuity results are given in Section 4.
2. Preliminaries
Let A be a closed densely defined linear operator on a Banach space X. In this paper we consider the following equation:
(2.1)
where u and f are X-valued functions,
, and
is the Caputo fractional derivative defined by
in which for
,
and
is understood as the Dirac measure
at 0. The convolution of two functions f and g is defined by
when the above integrals exist.
The classical (or strong) solution to (2.1) is defined as:
Definition 2.1. If
,
is called a solution of (2.1) if
1)
.
2)
.
3) u satisfies (2.1) on
.
By integration (2.1) for α-times, we are able to define a kind of weak solutions.
Definition 2.2. If
,
is called a mild solution of (2.1) if
for every
and
And it is therefore natural to give the following definition of α-resolvent family for the operator A.
Definition 2.3. A family
is called an α-resolvent family for the operator A if the following conditions are satisfied:
1)
is continuous for every
and
;
2)
and
for all
and
;
3) the resolvent equation
holds for every
.
If there is an α-times resolvent family
for the operator A, then the mild solution of (2.1) is given by the following lemma.
Lemma 2.4. [10] Let A generate an α-times resolvent family
and let
. If (2.1) has a mild solution, then it is given by
For the strong solution of (2.1), we have
Lemma 2.5. [10] Let A generate an α-times resolvent family
and let
,
. If
, then the following statements are equivalent:
(a) (2.1) has a strong solution on
.
(b)
is differentiable on
.
(c)
for
and
is
continuous on
.
If in addition, the α-times resolvent family
admits an analytic extension to some sector
, and
for all
, we will then denote it by
.
If
, then there exists constants C,
and
such that
and
(2.2)
for each
. The α-times resolvent family generated by A can be given by
where
is oriented counter-clockwise. And the corresponding operators
are defined by
Lemma 2.6. Let
and
. We have
(1)
for every
and
for
;
(2) for every
,
and
for
;
(3)
for
,
for any integer
and
for
, where
.
Proof. (1) By the definition of
and (2.2),
Since
taking
, we can obtain that the above integral is bounded by
Analogously one can show the estimate
It thus follows the estimate for
.
(2) By the identity
, we have
since
. Moreover,
By the closedness of the operator A, the assertion of (2) follows.
(3) By the proof of (2) and the closedness of A,
And the second part of (3) can be proved similarly. □
Remark 2.7. Similar results for
were given in [16]. It is obvious that
if
and
if
.
3. Regularity of the Mild Solutions
In this section we consider the mild solution of (2.1) with
. Suppose that the operator A generates an analytic α-resolvent family, then by Lemma 2.4 and Remark 2.7 the mild solution of (2.1) is given by
(3.1)
Theorem 3.1. Let
,
, and
with
.
Then for every
and
,
, where
is given
by (3.1). If moreover
such that
, then
.
Proof. Since
is analytic, we only need to show that
.
Let
and
, then
By Hölder’s inequality and Lemma 2.6,
We remark that the constant C here and in the sequel may be vary line by line, but not depending on t and h. Next, we estimate
. For
,
, first assume that
,
since
; if
, then

from which it follows also that
.
If
with
, then by [[10], Lemma 4.5] we have that
is differentiable and thus Lipschitz continuous. □
If we put more conditions on
, the regularity of
can be raised.
Proposition 3.2. Let
,
and
with
. For every
, we define the function
by

If
exists, then for every
,
. If moreover
, then
.
Proof. If
satisfies the assumption, by [[17], Theorem 13.2] there exists a function
such that
. Thus
![]()
Since
is analytic and bounded,
. It is easy to see
that
is Hölder continuous with index
. This completes the
proof. □
4. Regularity of the Classical Solutions
Motivated by the results in [18] for the C0-semigroups, we first give the following proposition.
Proposition 4.1. Let
and
. Assume
,
and
. Then the mild solution of (2.1) is the classical solution.
Proof. By Lemma 2.5 we only need to show that
for every
and
is continuous on
. We decompose
into two parts:
![]()
where
![]()
belongs to
and is continuous. To prove that
, we define the following functions:
![]()
and
![]()
for
small enough. It is clear that
as
. Moreover.
![]()
for all
, it follows from the fact that for a fix
the map
is a continuous mapping, we conclude that
![]()
By our assumption and Lemma 2.6 there exists a constant
such that
![]()
consequently, the function
is integrable. Hence by the closedness of A we obtain that
![]()
The continuity of the function
follows directly from the fact
![]()
This completes the proof. □
We will then give the regularity of such classical solutions.
Lemma 4.2. Let
with
, and
with
. Define
![]()
Then for any
,
and
.
Proof. For fixed
, since
we have
![]()
By the closedness of A, we obtain
. Thus it remains to show that
. For
and
we have the following decomposition:
![]()
Since
we have
![]()
We can estimate
as follows:
![]()
And it is easy to show that
. Combining all above we have
. □
The following theorem extends [[16], Theorem 4.4] to the case that
.
Theorem 4.3. Let
with
,
with
,
, and u is the classical solution of (2.1). The following assertions hold.
(1) For every
,
,
.
(2) If moreover
, then
and
are continuous on
.
(3) If
, then
and
.
Proof. (1) If u is the classical solution of (2.1) on
, then
. It is only need to prove
We decompose
![]()
By Lemma 4.2,
. Let
. If
, then
![]()
![]()
thus we have
![]()
(2) We only need to show that
is continuous at
. Since
and
is continuous,
![]()
as
.
(3) We show that
. Indeed, this follows from
![]()
5. Conclusion
In this paper, we proved the Hölder regularity of the mild and strong solutions to the α-order abstract Cauchy problem (2.1) with
. Our results are complemental to the existing results of Pazy [18] for the case
and Li [16] for the case that
.