1. Introduction
At the present time, the theory of differential equations and inclusions of fractional order is the subject of an active study for a large number of researchers. One of the main reasons for this interest is caused by important applications of this theory in physics, engineering, biology, economics, and other sciences (see, e.g., [
1,
2,
3,
4] and the references therein). It should be mentioned in this connection that fractional order models provide an effective and convenient machinery for the description of systems with memory and hereditary properties.
In the present paper, for a semilinear fractional–order functional differential inclusion in a separable Banach space
E of the form
we consider the problem of existence of a mild solution to this inclusion satisfying the periodic boundary value condition (PBVP). Here,
denotes the Caputo fractional derivative of the order
,
is the infinitesimal generator of a bounded
-semigroup,
is a multivalued nonlinearity and the function
describes the prehistory of the solution at the moment
i.e.,
It is worth noting that the introducing into consideration of the delay of the trajectory allows, additionally to the application of the fractional derivative, to reflect the dependence of the current state of a system on its previous characteristics.
Among a large amount of papers dedicated to fractional-order equations and inclusions in Banach spaces, let us mention works [
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15] where existence results of various types were obtained. In particular, in the authors’ paper [
6], the periodic boundary value problem for fractional-order semilinear differential inclusions in Banach spaces was studied by the method of translation multioperator along the trajectories of the inclusion. However, this method can not be extended directly to the case of functional differential inclusions. For this reason, in the present paper, we apply for the solving of the PBVP the method of integral multioperators, combined with the theory of condensing multimaps and the theory of Mittag–Leffler functions.
The paper is organized in the following way. In the next section, we present necessary notions and facts from the fractional and multivalued analysis as well as from the theory of condensing maps. In
Section 3, we study the PBVP for a semilinear fractional-order functional differential inclusion with delay in a Banach space. We introduce and study a multivalued integral operator whose fixed points coincide with mild solutions of our problem. On that base, we prove the main existence result (Theorem 4). In the last section, we consider an example dealing with existence of a trajectory for a time-fractional diffusion type feedback control system with delay satisfying the periodic boundary.
3. Existence Result
For a semilinear fractional–order differential inclusion in a separable Banach space
E of the form
we consider the problem (PBVP) of existence of a mild solution satisfying the following periodic boundary value condition
under the next basic assumptions.
As earlier, the symbol denotes the Caputo fractional derivative of order We suppose that the linear operator A satisfies condition
is a linear closed (not necessarily bounded) operator generating a bounded –semigroup of linear operators in E.
In the sequel, we will use the notation
We will assume that the multivalued nonlinearity obeys the following conditions:
for each the multifunction admits a measurable selection;
for a.e. the multimap is u.s.c.;
there exist functions
such that, for each
, we have
there exists a function
such that for each bounded set
we have:
for a.e.
where
is the Hausdorff MNC in
For a given
consider the multifunction
From above conditions
–
, it follows (see, e.g., [
18] Theorem 1.3.5) that the multifunction
is
–integrable and, therefore, the superposition multioperator
can be defined in the following way:
Now, we consider the operator functions defined in the following way:
where the function
is given by (
2) and (
3).
Remark 2. In a scalar case and with : Lemma 2 (See [
4,
14])
. The operator functions and possess the following properties:- (1)
for each , and are linear bounded operators and moreover, if the semigroup satisfies the estimatewith , then - (2)
the operator functions and are strongly continuous, i.e., functions and are continuous for each
Definition 9. A mild solution of inclusion (7) is a function satisfyingwhere In what follows, we will assume that the next condition holds true:
Consider the multioperator
, given in the following way:
for all functions
y defined as
where
The well posedness of the operator G follows from the next assertion.
Lemma 3. If for some , then and hence
Proof. Equality then follows from the definition of □
Theorem 2. Fixed points of the multioperator G coinside with mild solutions to problems (7) and (8). Proof.
If
x is a solution of problems (
7) and (
8), then, for
, it has the form
where
From condition (
8), it follows that
implying
that yields, for
,
that is
Conversely, let
then, it satisfies, for
Equation (
15) with
whence it is a mild solution of inclusion (
7). The validity of condition (
8) follows from Lemma 3. □
Let us consider some topological properties of the multioperator
Lemma 4. The multioperator G is u.s.c. and has compact values.
Proof. It is clear that it is sufficient to prove the assertion for the multioperator G whose values are naturally restricted to the space . Let us denote this restriction as
The multioperator
may be represented in the form of the following composition:
where
and
From the results of the work [
5], it is known that the multioperator
is u.s.c. with compact values and now the assertion follows from the fact that each of
, and
is a bounded linear operator. □
To prove that the multioperator
G is condensing, consider the cone
endowed with the natural ordering and introduce in the space
the vector measure of noncompactness
defined as
where
is the module of fiber noncompactness
and the second component is the module of equicontinuity:
Theorem 3. Under assumptions , (F1)–(F4), suppose, additionally, that
the semigroup U satisfies estimate (9) for some
Ifwhere is the function from condition then the multioperator G is ν-condensing. Proof. Let
be a nonempty bounded set such that
Let us show that is relatively compact.
From (
19), it follows that
Let
By using estimates (
10) and (
11) and property
and denoting, for
,
we get
For further estimation of
,
let us evaluate the integrals in the last expression by means of Formula (
6):
Now, notice that, if we will take
in Formula (
5), we have
Thus, we get the following equalities
Thus, for
, we obtain
From the last estimate, we get
At the same time, notice that, from the definition of the multioperator
G, it follows that
Taking into account estimates (
21) and (
22), we get
or that is the same
Conditions (
29) and (
20) obviously imply
In the paper [
5], it was shown that, on the interval
therefore, by representation (
16), we also have
From the definition of
G, it follows that
implying, by (
20),
This means, by the Arzela–Ascoli theorem, that is a relatively compact set, concluding the proof. □
Now, we are in position to prove the main result of this paper.
Theorem 4. Under conditions (A), (A1), (F1)–(F4), ifwhere functions γ and μ are from conditions and respectively, η is the constant from condition , then problems (7) and (8) have a solution. Proof. Take arbitrary
and
, then, for some
, we will have, for
the following estimate:
Notice that, by definition of G, the last estimate is also valid for .
Now, if we will take
then the inequality
implies
Therefore, the multioperator
G transforms the closed ball
into itself. By Theorem 1,
G has a fixed point, which is, by virtue of Theorem 2, a solution of problems (
7) and (
8). □
4. Example: A Periodic Problem for a Time-Fractional Diffusion System
At the present time, the research of many authors (see [
3] and the references therein) are devoted to the study of equations of the form
Since the order
of the derivative with respect to time in Equation (
24) can be of arbitrary real order, including
it is called the fractional diffusion-wave equation. This name has been suggested by Mainardi (see [
21]). For
, Equation (
24) becomes the classical diffusion equation; for,
, we have so-called ultraslow diffiision. It is important that the fractional diffusion equation has been related to a dynamical process in fractal media: the order of the resulted equation depends on the fractal, which serves as a model of a porous material (see, e.g., [
22]).
In our example, we consider such a fractional diffusion process subject to control effects.
Let
be the two-dimensional plane of points
. As earlier, by
, we denote the first quadrant of the plane defined by Formula (
17). Denote by
the Hilbert spaces of functions square summable on
We will consider a time-fractional control system whose state will be described by the function , Similarly to the foregoing, let us denote by
The control is characterized by
k sources of external influence whose properties at the moment
are dependent on the prehistory of the system. Their densities are described by the functions
,
and the intensities of sources can be regulated by the controls
measurable functions satisfying the feedback condition with delay of the form
where
is defined as the function
and
W is a u.s.c. multimap from
to Euclidean space
with convex closed values which is globally bounded:
for all
, where
We will consider a time-fractional diffusion type feedback control system with delay governed by the following equation of the order
:
where
is the Laplace operator,
For the above system, we will study the existence of a solution
satisfying the periodic boundary value condition of the form
Consider the differential operator of the form
with the domain
where
denotes the Sobolev space of functions whose normal derivatives
are vanishing on the boundary. Then, as it is shown in [
23], the boundary value problem
for
is solvable and the operator associating to the function
f the solution
u is bounded. Moreover, since the set of functions with a finite support vanishing on a neighborhood of the boundary
is densely embedded in
, by applying the Green formula to the expression
where
, we get, for
, the following estimate:
where the norm and the scalar product are taken in
. Hence, for the resolvent
, the following estimate is true:
This means (see, e.g., [
24] Corollary II.3.5) that the operator
A generates a strongly continuous semigroup
satisfying the estimate
Notice that, since the resolvent
is noncompact, the semigroup
is also noncompact.
Furthermore, we assume that functions satisfy the following conditions:
is measurable for all ;
for a.e. and all where
for all
Then, it is easy to see that the map
where
, defined as
is
-Lipschitz in
with
and compact in
, i.e., the set
is relatively compact in
for each
.
However, then, one can verify (see [
18] Proposition 2.2.2) that the multimap
satisfies conditions
–
of
Section 3 (with
,
in condition
and
in condition
).
Now, we can rewrite our system in the form of the following fractional-order functional differential inclusion in the Hilbert space
To verify condition (
13), let us estimate the norm of the operator
. By using estimate (
29), we have
Now, in accordance with Theorem 4, we conclude that, under condition
system (
26) has a trajectory
satisfying (
28). The corresponding control may be taken as a measurable selection of the multifunction