1. Introduction
Fractional differential equations have recently attracted a lot of attention due to their applications in science and engineering; in particular, they can describe much more nonlocal phenomena in physics, such as fluid mechanics, the diffusion phenomenon, and viscoelasticity. In lots of processes or phenomena with long-range temporal cumulative memory effects and/or long-range spatial interactions, theoretical and numerical results have also shown that fractional differential equations display more prominent advantages than integer order ones. In the past two decades, the theory of fractional differential equations has attracted the attention of researchers all over the world, as in the monographs [
1,
2,
3,
4] and the recent references.
Consider the Cauchy problem of fractional evolution equations on an infinite interval
where
is the Hilfer fractional derivative of order
and type
,
is Riemann–Liouville integral of order
,
A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e.,
semigroup)
in Banach space
X,
is a function to be defined later.
The Hilfer fractional derivative is a natural generalization of Caputo derivative and Riemann–Liouville derivative [
1]. It is obvious that fractional differential equations with Hilfer derivatives include fractional differential equations with a Riemann–Liouville derivative or Caputo derivative as special cases.
The well-posedness of fractional evolution equations is an important research topic of evolution equations, as many types of fractional partial differential equations, such as fractional diffusion equations, wave equations, Navier–Stokes equations, Rayleigh–Stokes equations, Fokker–Planck equations, Schrödinger equations, etc., can be abstracted as fractional evolution equations [
5,
6,
7]. However, it seems that there are few works concerned with fractional evolution equations on an infinite interval. Most of these results involve the existence of solutions for fractional evolution equations on a finite interval
, where
(for example, see [
8,
9,
10,
11]). The Ascoli–Arzelà theorem and various fixed point theorems are widely used to study the existence of solutions. It is well known that the classical Ascoli–Arzelà theorem is powerful technique to give a necessary and sufficient condition for judging the relative compactness of a family of abstract continuous functions, while it is limited to finite closed interval.
In this paper, by using the generalized Ascoli–Arzelà theorem and some new techniques, we prove the existence of mild solutions for the infinite interval problem (
1) when the semigroup is compact as well as noncompact. In particular, we do not need to assume that the
satisfies the Lipschitz condition. The main methods of this paper are based on the generalization of Ascoli–Arzelà theorem on infinite intervals, Schauder’s fixed point theorem, and Kuratowski’s measure of noncompactness.
2. Preliminaries
We first introduce some notations and definitions about fractional calculus, Kuratowski’s measure of noncompactness, and the definition of mild solutions. For more details, we refer to [
1,
2,
12,
13].
Assume that X is a Banach space with the norm . Let and J be an infinite interval of . By we denote the space of all continuous functions from J to X with the norm . We denote by the space of all bounded linear operators from X to X with the usual operator norm .
Definition 1 (see [
2])
. The fractional integral of order λ for a function is defined asprovided the right side is point-wise defined on , where is the gamma function. Definition 2 (Hilfer fractional derivative, see [
1])
. Let and . The Hilfer fractional derivative of order λ and type μ for a function is defined as Remark 1. (i)
In particular, when , , thenwhere is the Riemann–Liouville derivative. (ii)
When , , we havewhere is Caputo derivative.Let
D be a nonempty subset of
X. Kuratowski’s measure of noncompactness
is said to be:
where the diameter of
is given by diam
,
Lemma 1 ([
14])
. Let be a continuous function family. If there exists such thatThen is integrable on , and Definition 3 ([
15])
. The Wright function is defined bywith the following property Lemma 2 ([
8])
. The Cauchy problem (1) is equivalent to the integral equation Lemma 3 ([
8])
. Assume that satisfies integral Equation (2). Thenwhere Due to Lemma 3, we give the following definition of the mild solution of (
1).
Definition 4. By the mild solution of the Cauchy problem (1), we mean that the function which satisfies Suppose that A is the infinitesimal generator of a semigroup of uniformly bounded linear operators on Banach space X. This means that there exists such that .
Lemma 4 ([
4,
8])
. If is a compact operator, then and are also compact operators. Lemma 5. Assume that is a compact operator. Then is equicontinuous.
Lemma 6 ([
8])
. For any fixed , , and are linear operators, i.e., for any and Lemma 7 ([
8])
. If is equicontinuous, then the operators , and are strongly continuous, which means that, for and , we have Let
Then,
is a Banach space with the norm
.
In the following, we state the generalized Ascoli–Arzelà theorem.
Lemma 8 ([
16])
. The set is relatively compact if and only if the following conditions hold:- (a)
for any , the set is equicontinuous on ;
- (b)
uniformly for ;
- (c)
for any , is relatively compact in X.
3. Main Results
We introduce the following hypotheses:
- (H0)
is equicontinuous, i.e., is continuous in the uniform operator topology for .
- (H1)
is Lebesgue measurable with respect to t on . is continuous with respect to y on X.
- (H2)
There exists a function
such that
Then
is a Banach space with the norm
For any
, define an operator
as follows
where
For any
, set
Then,
. Define an operator
as follows
where
Obviously,
is a mild solution of (
1) if and only if the operator equation
has a solution
.
In view of (H2), we have
Thus, there exists a constant
such that
i.e.,
Clearly, is a nonempty, convex, and closed subset of , and is a nonempty, convex, and closed subset of .
Lemma 9. Assume that (H0), (H1) and (H2) hold. Then the set V is equicontinuous.
Proof. Step I. We first prove that is equicontinuous.
As
, we find
Hence, for
,
, we obtain
For any
and
, we have
Hence, is equicontinuous.
Step II. We prove that is equicontinuous.
Let , for any , . Then .
For
, in view of (H2), there exists
such that
For
, in virtue of (H2) and (
5), we find
When
,
, we have
For
, we find
where
One can deduce that
, as
. Noting that
then by Lebesgue dominated convergence theorem, we find
so,
as
.
For
be enough small, we have
where
By (H0) and Lemma 7, it is easy to see that as . Similar to the proof that , tend to zero, we obtain and as . Thus, tends to zero as . Clearly, as .
For
, if
, then
and
. Thus, for
Therefore, is equicontinuous. Furthermore, V is equicontinuous. □
Lemma 10. Assume that (H1) and (H2) hold. Then, uniformly for .
Proof. In fact, for any
, by (H2) and Lemma 6, we find
By (H2), we derive
which implies that
uniformly for
. This completes the proof. □
Lemma 11. Assume that (H1) and (H2) hold. Then .
Proof. From Lemmas 9 and 10, we know that
. For
and any
, by (
4) and (
6), we have
Therefore, . □
Lemma 12. Suppose that (H1) and (H2) hold. Then Φ is continuous.
Proof. Indeed, let
be a sequence in
which is convergent to
. Consequently,
Let
,
,
. Then
. In view of (H1), we have
On the one hand, using (H2), we get for each
,
On the other hand, the function
is integrable for
,
. By Lebesgue dominated convergence theorem, we obtain
Thus, for
,
Therefore,
as
. Hence,
is continuous. The proof is completed. □
Theorem 1. Assume that is compact. Furthermore suppose that (H1) and (H2) hold. Then the Cauchy problem (1) has at least one mild solution. Proof. Clearly, the problem (
1) exists a mild solution
if and only if the operator
has a fixed point
, where
. Hence, we only need to prove that the operator
has a fixed point in
. From Lemmas 11 and 12, we know that
and
is continuous. In order to prove that
is a completely continuous operator, we need to prove that
is a relatively compact set. In view of Lemmas 9 and 10, the set
is equicontinuous on
for any
, and
uniformly for
. According to Lemma 8, we only need to prove
is relatively compact in
X for
. Obviously,
is relatively compact in
X. We only consider the case
. For
and
, define
on
as follows:
By Lemma 4, we know that
is compact because
is compact for
. Further,
is compact, then the set
is relatively compact in
X for any
and for any
. Moreover, for every
, we find
Thus,
is also a relatively compact set in
X for
. Therefore, Schauder’s fixed point theorem implies that
has at least a fixed point
. Let
. Thus,
which implies that
is a mild solution of (
1). The proof is completed. □
In the case that is noncompact for , we need the following hypothesis:
- (H3)
there exists a constant
such that for any bounded set
,
where
is the Kuratowski’s measure of noncompactness.
Theorem 2. Assume that (H0), (H1), (H2) and (H3) hold. Then the Cauchy problem (1) has at least one mild solution. Proof. Let for all and , . By Lemma 11, , for , . Consider set , and we will prove set is relatively compact.
In view of Lemmas 9 and 10, the set is equicontinuous and uniformly for . According to Lemma 8, we only need to prove is relatively compact in X for .
Let
,
,
. By the condition (H3) and Lemma 1, we have
On the other hand, by the properties of measure of noncompactness, for any
we have
Thus
where
. From (
7), we know that
or
holds. Therefore, by the inequality in ([
17] p. 188), we obtain that
, then
is relatively compact. Consequently, it follows from Lemma 8 that set
is relatively compact, i.e., there exists a convergent subsequence of
. With no confusion, let
,
.
Thus, by continuity of the operator
, we have
Let
. Thus,
is a mild solution of (
1). The proof is completed. □
By Theorems 1 and 2, we have the following corollaries.
Corollary 1. Assume that is compact for and (H1) holds. Furthermore suppose that (H2) there exists a function and such that andThen the Cauchy problem (1) has at least one mild solution. Corollary 2. Assume that (H0), (H1), (H2) and (H3) hold. Then the Cauchy problem (1) has at least one mild solution. Example 1. Let . Consider the following fractional partial differential equations on infinite interval We define an operator A by with the domain Then A generates a compact, analytic, self-adjoint semigroup . Then problem (8) can be rewritten as followswhere for satisfies (H1), and , . Let , for . Thenwhere . This means that the condition (H2) is satisfied. By Corollary 1, the problem (8) has at least a mild solution.