1. Introduction
Recent years have seen a wide spread of fractional analysis as well as the theory of fractional-order differential equations and inclusions in contemporary mathematics. The increasing interest in this subject is explained by its numerous applications in various branches of applied mathematics, physics, engineering, biology, economics, and other sciences (see, e.g., the monographs of [
1,
2,
3]). A number of various approaches to the solving of boundary value problems for differential equations and inclusions in the case of fractional order
have been widely described in the literature (see [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19] and references therein).
It is well known that, beginning with the classical Cauchy–Picard theorem, as the usual assumption providing the existence and uniqueness of the solution to the initial value problem
the Lipschitz condition of the function
f in the space variable
x is considered. However, for a large class of such problems, especially in the case of an infinite-dimensional phase space, this condition looks rather onerous. That is why, in a large number of works (see, for example, Refs. [
20,
21,
22,
23,
24]), the Lipschitz condition is replaced with a certain type of monotonicity condition of
f in
For example, in the case of a Hilbert space
H with the inner product
, this condition can take the form
for some
Notice that the existence and uniqueness results also remain true in the case of differential inclusion, i.e., when the right-hand side
f is a multivalued map.
To the best of our knowledge, for fractional-order differential equations and inclusions (see, e.g., [
1,
2,
3] and references therein), the results of such a type have not been obtained up to the present time. In our opinion, the property of the Caputo fractional derivative
of a function
with the values in a Hilbert space, which was recently proved in the works [
25,
26,
27], opens up some opportunities in this direction.
In the present paper, we obtain results on the existence and uniqueness of a mild solution to the Cauchy problem for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a multivalued nonlinearity satisfying a condition of type (3). The paper is organized in the following way. In the next section, we recall necessary information from fractional calculus, the theory of measures of noncompactness, and the topological degree theory for condensing multivalued maps. In the third section, we formulate the problem and develop some approximation results based on the Yosida approximations of the linear part of the inclusion. This allows us to obtain the result (Theorem 1) about the a priori estimate for a mild solution of the problem. This theorem is used to get the result (Theorem 2) on the existence of a mild solution to the problem on an arbitrary bounded interval. Further, the main result of the fourth section is Theorem 3 on the uniqueness of a mild solution to the problem. Finally, in the fifth section, we present the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type as an example.
2. Preliminaries
2.1. Fractional Derivative
In this section, we will recall some notions and definitions that we will need in the following sections (details can be found, e.g., in [
1,
2,
3,
28]).
Let E be a real Banach space.
Definition 1. The Riemann–Liouville fractional derivative of the order of a continuous function is the function of the following form:provided that the right-hand side of this equality is well defined. Here,
is the Euler gamma-function
Definition 2. The Caputo fractional derivative of the order of a continuous function is the function defined in the following way:provided that the right-hand side of this equality is well defined. Definition 3. A function of the formis called the Mittag–Leffler function. The Mittag–Leffler function has the following asymptotic representation as
(see, e.g., [
28,
29]):
Denote
by
. Notice that the second of the above formula implies that in the case where
and
, we have
Notice that from the relations (see, e.g., [
30]):
and
where
it follows that
Remark 1 (See, e.g., [
31]).
Consider a scalar equation of the form
with the initial condition
where
is a continuous function. By a solution of this problem, we mean a continuous function
satisfying condition (11) whose fractional derivative
is also continuous and satisfies Equation (10). It is known (see [
1], Example 4.9) that the unique solution of this equation has the form
We will need the following auxiliary assertion, which is an analogue of the known Gronwall lemma on integral inequalities.
Lemma 1. Let a bounded measurable function such that is a continuous function andwhere is a bounded measurable function. Then, Proof. Consider a scalar equation
with the initial condition
From inequality (13) and Equation (14), we have
There exists a non-negative function
such that
The solution to the last equation is the following non-negative function
Thus,
because
and
; we finally get the inequality
□
2.2. Measures of Noncompactness and Condensing Multivalued Maps
Let us recall some notions and facts (details can be found, for example, in [
32,
33,
34,
35,
36]).
Let be a Banach space. We introduce the following notation:
.
Definition 4. Let be a partially ordered set. A function is called the measure of noncompactness (MNC) in if, for each , we have:where denotes the closure of the convex hull of Ω.
A measure of noncompactness is called:
- (1)
monotone if for, each , implies ;
- (2)
nonsingular if, for each and each , we have .
If is a cone in a Banach space, the MNC is called:
- (1)
regular if is equivalent to the relative compactness of ;
- (2)
real if is the set of all real numbers with the natural ordering;
- (3)
algebraically semiadditive if for every
It should be mentioned that the Hausdorff MNC obeys all of the above properties. Another example can be presented by the following measures of noncompactness defined on where is the space of continuous functions with the values in a separable Banach space E:
(i) the modulus of fiber noncompactness
where
,
is the Hausdorff MNC in
E and
;
(ii) the modulus of equicontinuity, defined as
Notice that these MNCs satisfy all above-mentioned properties except regularity. To obtain a regular MNC in the space
, we can consider the MNC
with the values in the cone
with the natural partial order.
Definition 5. Let be a closed subset; a multivalued map (multimap) is called upper semicontinuous (u.s.c.) if the pre-image of each open set is open in
Definition 6. A u.s.c. multimap is called condensing with respect to an MNC β (or β-condensing) if, for every bounded set that is not relatively compact, we have More generally, given a metric space
of parameters, we will say that a u.s.c. multimap
is a condensing family with respect to an MNC
(or
-condensing family) if, for every bounded set
that is not relatively compact, we have
Let be a bounded open set, a monotone nonsingular MNC in , and a -condensing multimap such that for all , where and denote the closure and the boundary of the set V.
In such a setting, the topological degree
of the corresponding vector multifield
satisfying the standard properties is defined, where
i is the identity map on
. In particular, the condition
implies that the fixed-point set
is a nonempty subset of
To describe the next property, let us introduce the following notion.
Definition 7. Suppose that β-condensing multimaps have no fixed points on the boundary and there exists a β-condensing family such that:
- (i)
for all
- (ii)
Then, the multifields and are called homotopic: The homotopy invariance property of the topological degree asserts that if , then
Let us also mention the following property of the topological degree, which we will need in later.
The normalization property: If
then
3. Existence of a Solution
Let
H be a separable Hilbert space. We will consider the Cauchy problem for a semi-linear fractional-order differential inclusion in
H:
where
, and the linear operator
A satisfies the following condition:
is a linear closed (not necessarily bounded) operator generating a bounded
-semigroup
of linear operators in
H such that
for some
It will be assumed that the multimap obeys the following conditions:
The multifunction admits a measurable selection for each and i.e., there exists a measurable function such that for a.e.
The multimap is u.s.c. for each and for a.e.
For each
and
, there exists a function
such that
,
implies
For each
, there exists
such that, for every bounded set
, it holds that:
where
denotes the Hausdorff MNC in the space
H.
for each
and
it holds:
From conditions
–
, it follows that for each
, the superposition multioperator
is defined by the formula
(see, for example, [
32,
33]).
Let us recall (see, for example, [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19]) that a mild solution to problems (15) and (16) is a function
of the form
where
, and the function
is defined by Formula (7).
Lemma 2. (See [31], Lemma 3.4.) The operator functions and possess the following properties: For each , the operator functions and are linear bounded operators; more precisely, for each , we have The operator functions and are strongly continuous, i.e., functions and are continuous for each
Remark 2. Notice that if A is a bounded operator, then the solution defined by Formula (18) satisfies the following differential equation (see [3]): Now, suppose that
is any mild solution to problems (15) and (16). Take a selection
satisfying (18). Then, condition
implies that
where
and
Now, taking a piecewise linear function
satisfying the conditions of Lemma 3, consider the function
where
and
as
Now, consider Yosida approximations for the operator
A:
It is known (see, e.g., [
37,
38]) that
are bounded, mutually commuting operators,
converges to
A pointwise on
, and each
generates the uniformly continuous contraction semigroup
.
We introduce the approximations
with the formula
Lemma 3. For each , there exists a set of a Lebesgue measure and a piecewise linear function with a finite number of nodes belonging to such that Proof. Notice that we can assume, without loss of generality, that the selection
f is a continuous function. In fact, consider the functions
defined by the formula
where
Recall that a point
is called a Lebesgue point of the function
f if
If we rewrite
then
for a.e.
as
, since, for a measurable function, the Lebesgue points form a complete measure space (see [
39]). Notice that functions
are continuous and
Hence, each function may be approximated with an arbitrary degree of accuracy in the space by piecewise linear functions with a finite number of nodes belonging to .
Take a sequence
. Applying the Egorov theorem to functions
(see [
37]), for a given
, we may find
such that
, and the sequence
uniformly converges to
f on
. So, we will have, for a sufficiently large
k,
Taking now a piecewise linear function
satisfying
we will get the desired function. □
Lemma 4. The expressiontends to zero as uniformly on Proof. Denoting
we get from (20) that
For a given
, we choose
such that
From the construction of the function
(see relations (26) and (27)), it follows that for a sufficiently small
, we get
Then, for the case
, we have
For
, the following estimate holds:
Since
for
, we obtain the estimate
Now, choosing
so that
we get the desired statement. □
Corollary 1. The expression tends to zero as .
Proof. Since the operator function is strongly continuous and as , the first term in this sum tends to zero uniformly on The second term uniformly tends to zero due to Lemma 4. □
Remark 3. If in Lemma 4, we replace with the Mittag–Leffler function , then, repeating the above reasonings, we get as uniformly on
Lemma 5. For a fixed , the sequence converges to as uniformly on
Proof. Since for each fixed
, we have
uniformly with respect to
(see [
38]), we also have the uniform convergence
which implies the desired convergence for a fixed
. □
Notice that due to the closedness of the operator
A, we have for
:
By the definition of the operator functions
and
, we have for
:
Since for a given piecewise linear function , the set is compact in H, it follows that the range of the function lies in , and therefore, is a compact set.
Lemma 6. For a fixed , we have as uniformly with respect to
Proof. Since
for each fixed
(see [
38]), we have
uniformly with respect to
Since
can be expressed through
by the Formula (24), we get thedesired convergence. □
Now, we can get the following assertion about the a priori estimate of a solution to problems (15) and (16).
Theorem 1. Under the above conditions, there exists a continuous function such thatfor every mild solution x of problems (15) and (16) defined on an interval Proof. Take a sequence of positive numbers
and choose a sequence of approximations
so that
Further, according to Lemma 4, we find
such that for
, the following holds:
Since
lie in
and
, for each fixed
that is uniformly bounded in
n (see Lemma 6), we may indicate
such that for
, we have
Take
Notice that, simultaneously, we construct the corresponding sequences of functions and sets .
By virtue of Remark 2, we have
From [
25,
26], it follows that
Now, let us estimate the right-hand side of inequality (31). To do this, let us mention that condition implies that
for each
and
it holds
where
For sufficiently large
we get the inequality
By virtue of Lemma 1, assuming that
the following inequality holds true:
Notice that the third and fourth terms tend to zero as . In fact, in the third term, the integral is uniformly bounded on and we can apply Remark 3 to the fourth term.
Passing in (39) to the limit as
, we get
Therefore, the right-hand side determines the function of the a priori estimate on the interval □
From Theorem 1, we can obtain the following result on the existence of a solution to problems (15) and (16) on an arbitrary interval
Theorem 2. Under the above conditions, there exists a mild solution to problems (15) and (16) on for each
Proof. Consider the family of multivalued integral operators
defined in the following way:
where
is the superposition multioperator defined by (17).
It is clear that each fixed point
of the multimap
is a mild solution to the problem
Moreover, it is known (see [
5,
8,
10,
11,
12,
13,
14]) that the family (34) has compact convex values and is condensing with respect to the MNC
in
(see
Section 2). Since the multioperators
satisfy conditions
–
independently on
, by applying Theorem 1, we conclude that there exists a constant
such that all solutions to problems (35) and (36) satisfy the a priori estimate
So, the multioperators from family (34) are fixed-point free on the boundary of the ball of the space centered at zero of the radius . Notice that the range of the multioperator consists of the single function as its fixed point.
Now, applying the homotopy and normalization properties of the topological degree, we obtain
which yields, by the existence property of the topological degree, the desired result. □
4. Uniqueness of a Solution
Now, we are in position to present our main result.
Theorem 3. Under the above conditions, problems (15) and (16) have a unique mild solution on for each
Proof. Suppose the contrary, that there are two different mild solutions
on
for problems (15) and (16). Take a sequence of positive numbers
and choose a sequences of approximations
and
so that
Further, according to Lemma 4, we find
such that for
, the following holds:
Since
and
for each fixed
that is uniformly bounded in
n (see Lemma 6), we may indicate
such that for
, we have
Take
Notice that, simultaneously, we construct the corresponding sequences of functions and sets .
By virtue of Remark 2, we have
From [
25,
26], it follows that
Now, let us estimate the right-hand side of inequality (38). Using the properties of Yosida approximation for the first term and adding and subtracting
in the second term, we have
Adding and subtracting
in the last term, we get
Using the properties
and
, we have
Now, using the properties of the norm and the scalar product, we finally obtain
For sufficiently large
we get the inequality
By virtue of the analog of Lemma 1, the following inequality holds true:
Notice that the second, third, and fourth terms tend to zero as . In fact, in the third term, the integral is uniformly bounded on , and we can apply Remark 3 to the second and fourth terms.
Passing in (39) to the limit as
, we get
Therefore, for each , it holds that □
5. An Example
Consider the following Cauchy problem for a system governed by a partial differential inclusion of a fractional diffusion type:
where
is the Caputo partial derivative in
t of order
,
,
,
is the feedback multimap which will be defined below, and
Considering
as
, where
, we will reduce the above problem to abstract problems (15) and (16) in the space
. In so doing, the operator
A is defined by the formula
We will assume that the function f generates the superposition operator
defined as
In order to conclude that this operator is well defined, it is sufficient to assume that the function
f is continuous,
for all
, and
f has a sublinear growth in the third variable:
where
a and
b are some nonnegative constants.
We now describe the "feedback" multimap
For a given concave locally Lipschitz-functional
we denote by
its subdifferential. It is known (see [
40], Propositions 2.1.2, 2.1.5, and 2.1.9) that
is a u.s.c. multimap in
with compact convex values, which is monotone in the following sense:
for all
Now, let
be any fixed orthonormal system of functions from
For a given
, we define a vector
assuming
where
We now define the multioperator
as
From the properties of multivalued maps (see, e.g., [
33]), it follows that
D is u.s.c. and has compact convex values; moreover, from (43)–(45), it follows that
D is monotone, i.e.,
for all
Now, we can substitute problems (41) and (42) with the following problem in the space
H:
where
If we suppose that
is continuously differentiable in
v,
is bounded for all
, and
then
From the properties of u.s.c. compact-valued maps (see [
33], Theorem 1.2.35), it follows that
D transforms bounded subsets of
H into relatively compact ones. However, then,
where
So, all conditions of Theorems 2 and 3 are fulfilled, and we conclude that problems (41) and (42) have a unique mild solution.