Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

: Among the many different deﬁnitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan– Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Lefﬂer functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan– Nersesyan derivative in the case of relative p -boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables. Abstract results for non-degenerate and degenerate equations in Banach spaces are applied to the investigation of a class of initial boundary value problems for partial differential equations with a time-fractional derivative and with polynomials in a self-adjoint elliptical differential operator with respect to spatial variables.


Introduction
One of the rapidly developing areas of modern mathematics is the theory of fractional differential equations and their applications [1][2][3][4][5][6][7] (also see the references therein). Among the many different definitions of the fractional derivative, the Riemann-Liouville [8] and Gerasimov-Caputo [8][9][10] derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan-Nersesyan fractional derivative [11], which generalizes the Riemann-Liouville and Gerasimov-Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. In this sense, the concepts of the Riemann-Liouville and Gerasimov-Caputo derivatives are symmetric. We investigate initial value problems with the Dzhrbashyan-Nersesyan fractional derivative, and the results obtained in these symmetric cases will be valid for the initial problems of equations with the Riemann-Liouville and the Gerasimov-Caputo derivatives, respectively. To begin , let us give the following definition.
In [11], M.M. Dzhrbashyan and A.B. Nersesyan proved the existence of a unique continuous solution lying in L p (0, l; R) for the initial value problem D σ k z(0) = z k , k = 0, 1, . . . , n − 1 (3) for the equation D σ n z(t) + p 0 (t)D σ n−1 z(t) + · · · + p n−1 (t)D σ 0 z(t) + p n (t)z(t) = f (t) with some functions p k : (0, T) → R, k = 0, 1, . . . n − 1, f : (0, T) → R. In the partial case, p 0 ≡ p 1 ≡ · · · ≡ p n−1 ≡ f (t) = 0, p n ≡ a ∈ R, the solution is presented in the form of a linear combination of the Mittag-Leffeler functions. Various differential equations with the Dzhrbashyan-Nersesyan derivative were considered in the works of A.V. Pskhu. For example, in [12], the fundamental solution of a diffusion-wave equation with the Dzhrbashyan-Nersesyan time-fractional derivative was obtained, and the unique solvability of the initial value problem D σ k z(x, 0) = z k (x), k = 0, 1, . . . , n − 1, x ∈ R n for the equation in R n × (0, T] was studied. In [13], similar issues were researched for the case of the discretely distributed Dzhrbashyan-Nersesyan time-fractional derivative. In this paper, we study the unique solvability issues (in the classical sense) for some classes of linear equations with operator coefficients in Banach spaces. In Section 2, the formula of the Laplace transform for the fractional Dzhrbashyan-Nersesyan derivative is obtained, and the initial value problem (3) with z k from a Banach space Z, k = 0, 1, . . . , n − 1, for the class of homogeneous equations D σ n z(t) = Az(t) with a linear bounded operator in Z is studied; z : R + → Z. Using the Laplace transform, we obtain the resolving operators' families for this equation, which are presented in the form of the Mittag-Leffler functions with an operator argument. In Section 3, the same initial value problem for the inhomogeneous equation with a function f ∈ C([0, T]; Z ) is investigated. These results are used for the proof of the unique solvability of the problem Here, X , Y are Banach spaces, L ∈ L(X ; Y ) (linear and continuous operator from X into Y), and M ∈ Cl(X ; Y ) (linear closed operator with a dense domain D M in the space X and with an image in Y ). We consider the case ker L = {0}; hence, Equation (5) is called a degenerate evolution equation. For this equation, we will use the condition of (L, p)-boundedness of the operator M. It allows us to reduce this equation to a system of two equations on two mutual subspaces. One of them has the form (4), and the other has a nilpotent operator at the fractional derivative. It is shown that the initial value problem is more natural for the degenerate Equation (6). Here, P is a projector on one of the abovementioned subspaces along the other subspace. A theorem of the existence and uniqueness of a classical solution of the problem in (6) and (7) is also obtained. Abstract results for non-degenerate and degenerate equations in Banach spaces are applied to the investigation of a class of initial boundary value problems for partial differential equations with a time-fractional derivative and with polynomials in a self-adjoint elliptical differential operator with respect to spatial variables. This article is a continuation of the previous work of the authors, who investigated equations in Banach spaces with other fractional derivatives [14][15][16][17] with applications to initial boundary value problems for partial differential equations and systems of equations.

Homogeneous Equation with the Dzhrbashyan-Nersesyan Fractional Derivative
Consider the fractional Dzhrbashyan-Nersesyan derivative, which is a generalization of two well-known fractional derivatives: the Riemann-Liouville and Gerasimov-Caputo [11] derivatives. Let us present their definitions.
Let α > 0, z : [0, T] → Z, for some T > 0 and Banach space Z. The Riemann-Liouville fractional integral of an order α > 0 of a function z has the form The Riemann-Liouville fractional derivative of an order α > 0 for a function z is defined as where m − 1 < α ≤ m ∈ N, and D m t := d m dt m is the integer-order derivative. Further, we use the notations R D α Let {α k } n 0 = {α 0 , α 1 , . . . , α n } be the set of real numbers that satisfy the condition 0 < α k ≤ 1, k = 0, 1, . . . , n ∈ N. We denote Further, it is assumed that the condition σ n > 0 is met everywhere. We define the Dzhrbashyan-Nersesyan fractional derivatives, which are associated with a sequence {α k } n 0 , with the relations Let function z : R + → Z, α > 0, m = α ; then, the Laplace transform, which we will denote as z-or when the expressions are too large for z, we denote it as Lap[z]-has the form Let L(Z ) be the Banach space of all linear bounded operators on Z, A ∈ L(Z ), and let D σ n be the Dzhrbashyan-Nersesyan fractional derivative, which is defined by a set of numbers {α k } n 0 = {α 0 , α 1 , . . . , α n }, 0 < α k ≤ 1, k = 0, 1, . . . , n ∈ N using Formulas (8) and (9). It is required that the inequality σ n > 0 is satisfied. Consider the equation with the initial conditions (11) is fulfilled for all t ∈ R + , and conditions (12) are true. Here, Let a solution of (11) have the Laplace transform; then, Equation (11) implies that For a fixed value l ∈ {0, 1, . . . , n − 1}, consider the problem for Equation (11). If its solution has the Laplace transform, then the equality (13) for it has the form λ σ n z(λ) − λ σ n −σ l −1 z l = A z(λ).
From here, we have . So, we define the operators for k = 0, 1, . . . , n − 1: Note that due to the boundedness of the operator A, The Mittag-Leffler function is used here: , Z ∈ L(Z ).
is the unique solution to the problem in (11) and (14).
Proof. For any l ∈ {0, 1, . . . , n − 1}, we have Therefore, Lemma 1 is valid for all l. From the linearity of the problem in (11) and (12), we get what we need.

Remark 1.
The result for Z = R was obtained in [11].
Due to the boundedness of the operator A, for all t > 0. Thus, equality (15) is satisfied for the function z f . The uniqueness of the solution can be proved in the same way as for the homogeneous equation above.
From Theorem 1 and Lemma 2, we immediately get the following result.
is a unique solution of the problem (12) in (15).

Degenerate Equation
are projections.

Lemma 4.
Let H ∈ L(X ) be a nilpotent operator with a power p ∈ N 0 , h : [0, T] → X , such that (D σ n H) l h ∈ C((0, T]; X ) at l = 0, 1, . . . , p, D σ k (D σ n H) l h ∈ C([0, T]; X ) for k = 0, 1, . . . , n − 1, l = 0, 1, . . . , p. Then, there exists a unique solution to the equation It has the form Proof. Let z = z(t) be a solution of Equation (20). We act with the operator H on both parts of (20) and get the equality HD σ n Hz(t) = Hz(t) + Hh(t). Due to the theorem's conditions, there exists a fractional derivative D σ n for the the right-hand side of this equality, as well as for its left-hand side. Acting with the operator D σ n on both parts of this equality, we will have (D σ n H) 2 z = D σ n Hz + D σ n t Hh = z + h + D σ n Hh. At the p-th step, sequentially continuing this reasoning, we obtain the equality By virtue of the continuity and nilpotency of the operator H, we have (D σ n H) p+1 z = (D σ n ) p+1 H p+1 z ≡ 0.
Hence, equality (21) for is true the function z. This equality implies the existence of a solution to Equation (20) (it is checked by substituting this function into the equation) and its uniqueness. Indeed, the difference of two solutions corresponds to a solution of Equation (20) with the function h ≡ 0. According to Formula (21), its solution is identically equal to zero. The lemma has been proved.
If we use the operator M −1 0 (I − Q) ∈ L(Y 0 ; X 0 ) in the same way, then we get the equation By Theorem 2 and with Z = X 1 , (24) and (26) has a unique solution, and it has the form By virtue of Lemma 4, if conditions (22) are fulfilled, the problem in (25) and (27) has a unique solution: In this case, the following conditions are used: D σ k (D σ n G) l M −1 0 (I − Q)g ∈ C([0, T]; X ) for k = 0, . . . , n − 1, l = 0, 1, . . . , p.
The existence and uniqueness theorem for the problem in (19) and (28) is proved similarly with help of a reduction to the system in (24) and (25) with initial conditions (26) and without conditions (27).