Analytic Resolving Families for Equations with the Dzhrbashyan–Nersesyan Fractional Derivative

: In this paper, a criterion for generating an analytic family of operators, which resolves a linear equation solved with respect to the Dzhrbashyan–Nersesyan fractional derivative, via a linear closed operator is obtained. The properties of the resolving families are investigated and applied to prove the existence of a unique solution for the corresponding initial value problem of the inhomogeneous equation with the Dzhrbashyan–Nersesyan fractional derivative. A solution is presented explicitly using resolving families of operators. A theorem on perturbations of operators from the found class of generators of resolving families is proved. The obtained results are used for a study of an initial-boundary value problem to a model of the viscoelastic Oldroyd ﬂuid dynamics. Thus, the Dzhrbashyan–Nersesyan initial value problem is investigated in the essentially inﬁnite-dimensional case. The use of the proved abstract results to study initial-boundary value problems for a system of partial differential equations is demonstrated.

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 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 f ≡ 0, k = 0, 1, . . ., n − 1, is introduced.In the third section, it is shown that the existence of k-resolving families, k = 1, 2, . . ., n − 1, 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 A {α k } (θ 0 , a 0 ).Various properties of the resolving families are investigated, and a perturbation theorem for operators from A {α k } (θ 0 , a 0 ) 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.

Preliminaries
Let Z be a Banach space.For the function z : R + → Z, the Riemann-Liouville fractional integral of an order β > 0 has the form For the function z, the Riemann-Liouville fractional derivative of an order α ∈ (m − 1, m], where m ∈ N is defined as . Further, we will assume that σ n > 0. Define the Dzhrbashyan-Nersesyan fractional derivatives, which correspond to the sequence {α k } n 0 , by relations Example 3. In [23], it is shown that the compositions of the Gerasimov-Caputo and the Riemann-Liouville fractional derivatives D α t D t may be presented as Dzhrbashyan-Nersesyan fractional derivatives D σ n for some sequences {σ 0 , σ 1 , . . ., σ n }.
Let α ∈ (m − 1, m], m ∈ N.Then, for a function z : R + → Z, we use z to denote the Laplace transform, and for too-large expressions for z as Lap [z].In [22], it is proved that L(Z ) denotes the Banach space of all linear continuous operators on a Banach space Z; Cl(Z ) denotes the set of all linear closed operators, which are densely defined in Z and act into Z.For an operator A ∈ Cl(Z ), its domain D A 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 A ∈ Cl(Z ), D σ n is the Dzhrbashyan-Nersesyan fractional derivative, associated with a set of real numbers {α k } n 0 , 0 (7) holds for all t ∈ R + and conditions (6) are valid.
and formulate an assertion that is important for further considerations.

k-Resolving Families of Operators
Definition 1.A set of linear bounded operators {S l (t) ∈ L(Z ) : t > 0} is called k-resolving family, k ∈ {0, 1, . . ., n − 1}, for Equation (7), if it satisfies the next conditions: (i) S k (t) is a strongly continuous family at t > 0; and a k-resolving family of operators for Equation ( 7) is unique.
Proof.Due to identity (5) and Definition 1 for arbitrary Due to equality (8) from the uniqueness of the inverse Laplace transform, we see the uniqueness of a k-resolving family for Equation (7).
Remark 1.The parameter σ 0 in the formulation of Proposition 2 defines the power singularity of the family {S 0 (t) ∈ L(Z ) : t > 0} 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 σ 0 := α 0 − 1 < 0, or a singularity is absent in the case α 0 = 1.Due to Proposition 2, the k-resolving family {S k (t) ∈ L(Z ) : t > 0} has the singularity of the power (7) for some k ∈ {0, 1, . . ., n − 1}, such that S k (t) L(Z ) ≤ Ke at t σ k at some K > 0, a ∈ R for all t > 0.Then, there exists a limit lim Since, for large enough |λ| is a continuous function on [0, 1] and η(0) = 0.For arbitrary ε > 0, take δ > 0, such that for all t ∈ [0, δ] η(t) ≤ ε; therefore, due to the inequality η(t) ≤ K 1 e bt + 1 for t ≥ 0, we have Due to equality (9), we obtain for t > 0 A l e λt dλ λ lσ n .
Take R = 1/t for small t > 0; then, 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].
Proof.After Theorem 3, we need to prove the uniqueness of a solution only.If problem ( 6), (7) has two solutions y 1 , y 2 , then the difference y = y 1 − y 2 is a solution of (7) with the initial conditions D σ k y(0) = 0, k = 0, 1, . . ., n − 1. Redefine y on (T, ∞) for any T > 0 as a zero function.The got function y T satisfies equality (7) at t > 0 without the point T. Using the Laplace transform obtained from Equation ( 7) and zero initial conditions, the equality λ σ n y T (λ) = A y T (λ).Since A ∈ A {α k } (θ 0 , a 0 ), we have y T (λ) ≡ 0 for λ ∈ S θ 0 ,a 0 .Therefore, y T ≡ 0 for arbitrary T > 0, hence y ≡ 0 on R + and a solution of problem ( 6), ( 7) is unique.Remark 6.For A ∈ L(Z ) the k-resolving operators of Equation ( 7) have the form (see [22]) Here, according to E β,γ the Mittag-Leffler function is denoted.Indeed, decomposing the resolvent R σ n (A) in the series for large enough |λ| and using the Hankel integral, we obtain these equalities.

Theorem 4. Let
Here, we use the principal branch of the power function.
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].

Inhomogeneous Equation
Let f ∈ C([0, T]; Z ).Consider the equation A solution of the initial value problem to Equation ( 10) is a function z ∈ C((0, T]; D A ), such that , for all t ∈ (0, T] equality ( 10) is fulfilled and conditions (11) are valid.Denote is a unique solution for the initial value problem to (10). Further, Repeating the analogous reasoning sequentially, we get for all t > 0. Thus, the function z f satisfies equality (10).The proof of a solution's uniqueness is the same as for the homogeneous equation.
Proof.Since A is closed, Therefore, for all t, s ∈ (0, T] Other arguing is the same as in the proof of the previous lemma.
Corollary 1, Lemma 1 and Lemma 2 imply the following result.
Proof.Choose l > sin −1 θ 0 , λ ∈ S θ,la ⊂ S θ,a for some θ ∈ (π/2, θ 0 ), a > a 0 , then from ( 14), it follows that where K A (θ, a) 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 Therefore, 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].

Application to a Model of a Viscoelastic Oldroyd Fluid
Let α k ∈ (0, 1], k = 0, 1, . . ., n, α 0 + α n > 1, σ n ∈ (0, 2), Ω ⊂ R d be a bounded region, which has a smooth boundary ∂Ω.We consider a fractional linearized model of the viscoelastic Oldroyd fluid dynamics with the order N = 1 (see [38]) Here, T > 0, D σ k , k = 0, 1, . . ., n, are Dzhrbashyan-Nersesyan fractional derivatives with respect to time t, x = (x 1 , x 2 , . . ., x d ) are spatial variables, v = (v 1 , v 2 , . . ., v d ) is the fluid velocity vector, w = (w 1 , w 2 , . . ., w d ) is a function of memory for the velocity, which is defined by a Volterra integral with respect to t for v, ∇p = (p x 1 , p x 2 , . . ., p x d ) is the pressure gradient of the fluid, ∆ is the Laplace operator with respect to all the spatial variables, ∆v = (∆v 1 , ∆v 2 , . . ., ∆v d ), ∆w = (∆w 1 , ∆w 2 , . . ., ∆w d ), ∇ the norm of L 2 will be denoted by H σ , and in the norm of the space H 1 by H 1 σ .We also denote H 2 The operator B = Σ∆, extended to a closed operator in the space H σ with the domain H 2 σ , has a real, negative, discrete spectrum with finite multiplicities of eigenvalues, condensed at −∞ only [39].Denote by {λ k } eigenvalues of B, numbered in non-increasing order, taking into account their multiplicities.Then, {ϕ k } will be used to denote the orthonormal system of eigenfunctions, which forms a basis in H σ [39].

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.