1. Introduction
Consider the differential equation
where
A is a linear closed operator, which has a dense domain
in a Banach space
,
,
is a given function. Let
be the Riemann–Liouville integral for
and the Riemann–Liouville derivative for
. Here
, where
, is the Dzhrbashyan–Nersesyan fractional derivative [
1]. Note that this derivative includes as partial cases the Gerasimov–Caputo (
,
,
) and the Riemann–Liouville (
,
,
) fractional derivatives of an order
from
.
In recent decades, fractional-order equations have been actively used in modeling various complex systems and processes in physics, chemistry, social sciences, and humanities [
2,
3,
4,
5,
6]. We note recent works [
7,
8,
9,
10,
11,
12], combining theoretical studies in various fields of fractional integro-differential calculus and their use in real-world modeling problems, particularly when modeling biological processes in virology, which is especially important at present. Readers should also note the works [
13,
14], which consider some applied problems with the Dzhrbashyan–Nersesyan fractional derivative.
The initial value problem
with
for Equation (
1) in the scalar case (
,
) is studied by M.M. Dzrbashyan, A.B. Nersesyan in [
1]. The unique solvability theorem for such a problem with
and a matrix
A was obtained in [
15]. Various equations with partial derivatives of Dzhrbashyan and Nersesyan were studied in papers [
16,
17,
18,
19,
20,
21]. Problem (
1), (
2) with a linear continuous operator
in an arbitrary Banach space
was researched in [
22] considering the methods used to resolve families of operators; see [
23].
The results obtained in this work generalize the corresponding results of the theory of analytic semigroups of operators solving first-order equations in Banach spaces [
24,
25]. We also note the works in which the theory of analytical resolving families is constructed for evolutionary integral equations [
26], equations with a Gerasimov–Caputo [
27] or Riemann–Liouville [
28] derivative, fractional multi-term linear differential equations in Banach spaces [
29], and equations with various distributed fractional derivatives [
30,
31,
32,
33,
34].
After the Introduction and Preliminaries, in the second section of the present work, the notion of a
k-resolving family for homogeneous Equation (
1), i.e., with
,
, is introduced. In the third section, it is shown that the existence of
k-resolving families,
, follows from the existence of a zero-resolving family. In the fourth section, a criterion of the existence of a zero-resolving family of operators to the homogeneous Equation (
1) is found in terms of conditions for a linear closed operator
A. The class of operators which satisfy these conditions is denoted as
. Various properties of the resolving families are investigated, and a perturbation theorem for operators from
is proved in the fifth section. For problem (
1), (
2) with a function
f, which is continuous in the graph norm of
A or Hölderian, the existence of a unique solution is obtained in the sixth section. In the last section, this result is used to prove the theorem on a unique solution existence for an initial-boundary value problem to a fractional linearized model of the viscoelastic Oldroyd fluid dynamics.
The theoretical significance of the obtained results lies in the fact that they give a correct statement of an initial problem and conditions for its unique solvability for equations with the Dzhrbashyan–Nersesian fractional derivative and with an unbounded linear operator at the unknown function. The unboundedness of the operator in the equation makes it possible to reduce initial-boundary value problems to various equations and systems of partial differential equations in problems of this type.
2. Preliminaries
Let
be a Banach space. For the function
, the Riemann–Liouville fractional integral of an order
has the form
For the function z, the Riemann–Liouville fractional derivative of an order , where is defined as . Further, we use the notation for ; is the identical operator.
Let
be a set of numbers
,
We will use the denotations
,
hence
. Further, we will assume that
. Define the Dzhrbashyan–Nersesyan fractional derivatives, which correspond to the sequence
, by relations
Example 1. Take , , , , then , , , are the Riemann–Liouville fractional derivatives. In particular, .
Example 2. If , , , , then , , is the Gerasimov–Caputo fractional derivative.
Example 3. In [23], it is shown that the compositions of the Gerasimov–Caputo and the Riemann–Liouville fractional derivatives , , , may be presented as Dzhrbashyan–Nersesyan fractional derivatives for some sequences . Let
,
. Then, for a function
, we use
to denote the Laplace transform, and for too-large expressions for
z as
. In [
22], it is proved that
denotes the Banach space of all linear continuous operators on a Banach space ; denotes the set of all linear closed operators, which are densely defined in and act into . For an operator , its domain is endowed by the norm , which is a Banach space due to the closedness of A.
Consider the initial value problem
to the linear homogeneous equation
where
,
is the Dzhrbashyan–Nersesyan fractional derivative, associated with a set of real numbers
,
,
, by (
3), (
4),
.
A solution to problem (
6), (
7) is a function
, such that
,
,
, (
7) holds for all
and conditions (
6) are valid. Hereafter,
.
Denote , , , for and formulate an assertion that is important for further considerations.
Theorem 1 ([
34])
. Let , , , be a Banach space, . Then, the next statements are equivalent.(i) There exists an analytic function . For every , there exists such a that the inequality is satisfied for all for .
(ii)
H is analytically extendable on for every there exists , such that for all 3. -Resolving Families of Operators
Definition 1. A set of linear bounded operators is called k-resolving family, , for Equation (7), if it satisfies the next conditions: (i) is a strongly continuous family at ;
(ii) , for all , ;
(iii)
For every is a solution of initial value problem , , to Equation (7). Let be the resolvent set of operator A.
Proposition 1. Let , , . For there exists a k-resolving family of operators for Equation (7), such that at some , , for all . Then, for , and a k-resolving family of operators for Equation (7) is unique. Proof. Due to identity (
5) and Definition 1 for arbitrary
,
Therefore, the operator
is invertible and equality (
8) holds. Since
for
, we have
. Due to equality (
8) from the uniqueness of the inverse Laplace transform, we see the uniqueness of a
k-resolving family for Equation (
7). □
Proposition 2. Let , , . There exists a 0-resolving family for (7), such that at some , for all . Then, for every , there exists a unique k-resolving family . Moreover, and at some for all , . Proof. Since every
has a nonzero limit
as
, due to ([
29], Lemma 1)
as
. Therefore, for every
,
and there are Riemann–Liouville fractional integrals for this function.
Define for
the families
. By this construction, it satisfies condition (i) in the Definition 1. For
,
since
satisfies condition (ii) in Definition 1 and the operator
A is closed. So, condition (ii) holds for
, where
.
For
, multiply the equality
, which follows from point (iii) of Definition 1 for
after the Laplace transform action, by
and obtain the equality
, i.e.,
, which means that
is a
k-resolving family for Equation (
7) due to the uniqueness of the inverse Laplace transform. Hence equality (
8) is valid and a
k-resolving family of Equation (
7) is unique by Proposition 1. □
Remark 1. The parameter in the formulation of Proposition 2 defines the power singularity of the family at zero. At the beginning of the proof of Proposition 2, it was shown that we have two possibilities only: the singularity at zero has a power of , or a singularity is absent in the case . Due to Proposition 2, the k-resolving family has the singularity of the power , or it is absent at zero, if .
Theorem 2. Let , , , there exist a k-resolving family of operators of (7) for some , such that at some , for all . Then, there exists a limit in the norm of the space , if and only if . Proof. Note that
due to (
5), Definition 1 and Proposition 1. Hence for
,
Since, for large enough
we have
Assume that
is a continuous function on
and
. For arbitrary
, take
, such that for all
; therefore, due to the inequality
for
, we have
as
. Hence, for large enough
. Consequently,
is a continuously invertible operator, so
.
Let
,
,
,
. Due to equality (
9), we obtain for
Take
for small
; then,
as
. □
Remark 2. An analogous result of Theorem 2 is well-known for resolving semigroups of operators for first-order equations (see, e.g., [35]). On resolving families of operators for equations, which are solved with respect to a Gerasimov–Caputo derivative, a similar theorem was obtained in work [27]. 4. Generation of Analytic -Resolving Families
Let . A k-resolving family of operators is called analytic, if at some it has an analytic continuation to . An analytic k-resolving family of operators has a type at some , , , if, for arbitrary , , there exists , such that the inequality is satisfied for all .
Remark 3. From Proposition 2 and Remark 1 it follows that for a k-resolving family of operators , we may have , or .
Definition 2. An operator belongs to the class , , , , , , if:
(i) For all we have
(ii)
For arbitrary , , there exists a constant , such that for every If
, the operators
are defined, where
,
,
,
,
,
.
Theorem 3. Let , , , , .
(i)
If there exists an analytic 0-resolving family of operators of the type for (7), then (ii)
If , then for every there exists a unique analytic k-resolving family of operators of the type for (7). Moreover, for , . Proof. Choose
,
is the positively oriented closed loop,
then
.
If , then following Theorem 1 with , the operators family is analytic of the type , it implies point (i) of Definition 1, and point (ii) of this definition is evidently fulfilled.
For any
,
, we have such a
, that for every
So, for , there exists the Laplace transforms , , therefore, .
If
,
, then
Hence,
since by the Cauchy theorem
as
for
.
At the same time, due equality (
5)
for
, hence
Further, for every
for
thus,
Finally,
Acting on the inverse Laplace transform, we get the equality
, so
is a zero-resolving family of operators for Equation (
7). Then, by Proposition 2 for every
, there exists a
k-resolving family of operators, which coincide with operators
. Every such family is analytic with the type
; see the proof of Proposition 2 and Remark 3.
If there exists a zero-resolving family with the type
, equality (
8) at
and Theorem 1 with
implies that
. □
Remark 4. Note that , if .
Remark 5. An analogous for Theorem 3 result on the first-order equations is called the Solomyak–Yosida theorem on generation of analytic semigroups of operators [24,25]. Previously, similar results were obtained for evolutionary integral equations [26], differential equations with a Gerasimov–Caputo fractional derivative [27], with a Riemann–Liouville derivative [28], for multi-term linear fractional differential equations in Banach spaces [29], and equations with distributed fractional derivatives [30,31,33,34]. Corollary 1. Let , , , , , . Then, for any problem (6), (7) has a unique solution, and it has the form The solution is analytic in .
Proof. After Theorem 3, we need to prove the uniqueness of a solution only. If problem (
6), (
7) has two solutions
,
, then the difference
is a solution of (
7) with the initial conditions
,
. Redefine
y on
for any
as a zero function. The got function
satisfies equality (
7) at
without the point
T. Using the Laplace transform obtained from Equation (
7) and zero initial conditions, the equality
Since
, we have
for
. Therefore,
for arbitrary
, hence
on
and a solution of problem (
6), (
7) is unique. □
Remark 6. For the k-resolving operators of Equation (7) have the form (see [22]) Here, according to the Mittag–Leffler function is denoted. Indeed, decomposing the resolvent in the series for large enough and using the Hankel integral, we obtain these equalities.
Theorem 4. Let , , , , , . Then .
Proof. For some , such that , take , hence , , since . Then, for sufficiently large . Therefore, , since . Here, we use the principal branch of the power function.
So, for
, where
,
and by Lemma 5.2 [
36] the operator
A is bounded. □
Remark 7. For strongly continuous resolving families of the equation with a Gerasimov–Caputo derivative, such a result was proved in [27]. 5. Inhomogeneous Equation
Let
. Consider the equation
A solution of the initial value problem
to Equation (
10) is a function
, such that
,
,
, for all
equality (
10) is fulfilled and conditions (
11) are valid.
Lemma 1. Let , , , , , , . Then,is a unique solution for the initial value problemto (10). Proof. Since , for sufficiently large , hence for , , , for . Analogously, for , .
Further,
hence
. Define
f by zero outside the segment
; then,
,
. Repeating the analogous reasoning sequentially, we get
,
for
,
,
.
Since
, we have
so,
for all
. Thus, the function
satisfies equality (
10). The proof of a solution’s uniqueness is the same as for the homogeneous equation. □
Let
for some
be the set of all functions
, satisfying the Hölder condition:
Lemma 2. Let , , , , , , , . Then, problem (10), (13) has a unique solution; it has form (12). Proof. Since
A is closed,
therefore,
, as
(see the previous proof). Therefore, for all
Note that for any
since for large enough
At the same time, for sufficiently large
; therefore, the family
is bounded uniformly. Since
is dense in
, for every
.
Thus,
as
. Therefore,
,
.
Other arguing is the same as in the proof of the previous lemma. □
Corollary 1, Lemma 1 and Lemma 2 imply the following result.
Theorem 5. Let , , , , , , , . Then problem (10), (11) has a unique solution, it has the form 6. Perturbation Theorem
Theorem 6. Let , , , , , , , for some there exists , such that for all , . Then, for sufficiently large . Proof. Choose
,
for some
,
, then from (
14), it follows that
where
is the constant from Definition 2. Note that the value
is close to one, for a sufficiently large number
l
is close to zero. So, for such a
l, we have
So,
with
, for all
,
□
Remark 8. For every condition, (14) is satisfied with , . Remark 9. Theorem 6 generalizes the similar theorem for generators of analytic semigroups of operators [37]. Note that there are also analogous results for generators of resolving families for equations with distributed fractional derivatives in [30]. 7. Application to a Model of a Viscoelastic Oldroyd Fluid
Let
,
,
,
,
be a bounded region, which has a smooth boundary
. We consider a fractional linearized model of the viscoelastic Oldroyd fluid dynamics with the order
(see [
38])
Here, , , , are Dzhrbashyan–Nersesyan fractional derivatives with respect to time t, are spatial variables, is the fluid velocity vector, is a function of memory for the velocity, which is defined by a Volterra integral with respect to t for v, is the pressure gradient of the fluid, is the Laplace operator with respect to all the spatial variables, , , , . The constants and the functions are given.
Take , , . The closure of in the norm of will be denoted by , and in the norm of the space by . We also denote , is the orthogonal complement for in the Hilbert space , , are the projectors.
The operator
, extended to a closed operator in the space
with the domain
, has a real, negative, discrete spectrum with finite multiplicities of eigenvalues, condensed at
only [
39]. Denote by
eigenvalues of
B, numbered in non-increasing order, taking into account their multiplicities. Then,
will be used to denote the orthonormal system of eigenfunctions, which forms a basis in
[
39].
In order for Equation (
19) to be fulfilled, take
and define in
an operator
Theorem 7. Let , , , , , , the operator A be defined by (20). Then, for some , . Proof. Let
,
,
,
, then for
so,
and instead of estimates of the form
, it will be enough to get inequalities
.
Take
,
, where
is sufficiently large, then for
Since
for
, we have
, for sufficiently large
l, the value
is small enough and
Fix
l,
; then, for
, we have
and
if we take
l, such that
Further, for large
for sufficiently large
l, since
Thus, with , with a chosen sufficiently large . □
Theorem 8. Let , , , , . Then, problem (15)–(19) has a unique solution. Proof. Problem (
15)–(
19) is represented as abstract problem (
10), (
11) due to the above choice of
and
A. Since we find the vector functions
and
with the values in
for every
, instead of Equation (
17), we consider its projection on
In this case, the projection of Equation (
18) on
has the form
hence,
. Theorem 7 and Theorem 5 imply the required statement. □
Remark 10. If we found and , we obtain the pressure gradient using the formula from the projection of Equation (17) on the subspace . 8. Conclusions
On the one hand, the results obtained will become the basis for the study of various classes of semilinear and quasilinear equations with the Dzhrbashyan–Nersesyan derivative. It is supposed to consider cases when the nonlinearity in the equation is continuous in the norm of the graph of the operator A and when it is Hölderian. In addition, there are plans to investigate similar equations with a degenerate linear operator at the Dzhrbashyan–Nersesyan derivative, linear, semi-linear and quasilinear. On the other hand, abstract results will be used to study various initial-boundary value problems for partial differential equations and their systems encountered in applications.