Integrated Resolving Functions for Equations with Gerasimov–Caputo Derivatives

: The concept of a β -integrated resolving function for a linear equation with a Gerasimov– Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability of the Cauchy problem to linear homogeneous and inhomogeneous equations are found in the case of the existence of a β -integrated resolving function. The necessary and sufﬁcient conditions for the existence of such a function in terms of estimates on the resolvent of its generator are obtained. The example of a β -integrated resolving function for the Schrödinger equation is given. Thus, the paper discusses some aspects of the symmetry of the concepts of integrability and differentiability. Namely, it is shown that, in the absence of a sufﬁciently differentiable resolving function for a fractional differential equation, the problem of the existence of a solution can be solved by an integrated resolving function of the equation.


Introduction
One of the traditional approaches for solving broad classes of initial boundary value problems involving partial differential equations is using, from the theory of differential equations in Banach spaces, in particular, the theory of semigroups of operators [1][2][3][4].As it is known, the resolving function S : R + → L(Z ) that defines the solution z(t) = S(t)z 0 of the Cauchy problem z(0) = z 0 to a first-order equation (D 1 is the first order differentiation operator) in a Banach space Z with, generally speaking, an unbounded linear operator A form a semigroup, i.e., S(s)S(t) = S(s + t) for all s, t ≥ 0, since the formal solution of such an equation is given by the exponential function.For fractional order equations D α z(t) = Az(t), (D α z is the Gerasimov-Caputo derivative of the order α > 0) the semigroup property of resolving function is absent, and there is no generalization of this property in the general case, since the class of Mittag-Leffler functions that are the formal solutions of such equations is too wide.However, in other respects, the advantages of these functionalanalytical approaches remain valid in the study of fractional differential equations.Such advantages include the possibility of studying wide classes of initial boundary value problems for partial differential equations and systems of equations within one class of initial problems for an equation in a Banach space: proving the existence and uniqueness of the solution, obtaining a representation of a solution in the linear case and an approximate solution in the nonlinear one, etc.The technique of resolving families of operators (resolving functions) is used successfully when studying integro-differential equations [5,6], integral evolution equations [7,8], various fractional differential equations [9][10][11][12][13][14].
An important generalization of the notion of the operator semigroup is the k-times integrated operator semigroup [15,16].The theory of the k-times integrated operator semigroups (see [15][16][17][18][19]) enables one to investigate the solvability of the Cauchy problem z(0) = z 0 for a first-order Equation (1) in the case where the operator A does not generate an operator semigroup S, but generates a k-times integrated semigroup of operators S k .In the case of the existence of the semigroup S, S k is the k-th order primitive of S.
The Cauchy problem for a linear homogeneous equation was studied in terms of solution operators (resolving functions) in [9].Here, D k are derivatives of k-th order, k = 0, 1, . . ., m − 1, D α is the Gerasimov-Caputo derivative of the order α ∈ (m − 1, m], m ∈ N (see the definition in the next section).In [9], the necessary and sufficient conditions in terms of the operator A resolvent are obtained for the existence of the solution operator (resolving function) of Equation ( 3).The properties of strongly continuous and analytic, exponentially bounded resolving functions were studied.The aim of the present work is to extend the concept of a k-times integrated semigroup to the resolving functions of the fractional differential Equation (3).In this case, β-integrated resolving functions will be considered not only for β ∈ N 0 := N ∪ {0}, as for operator semigroups, but also for a fractional order β ∈ R + of integration.This will allow us to assert the existence of a solution to the Cauchy problem (2) for Equation (3) or for the corresponding inhomogeneous equation in the case where there is no (sufficiently differentiable) resolving function, but there is a β-integrated resolving function (its existence conditions are less stringent).In partial cases, we obtain k-times integrated semigroups [15,16] (α = 1, β = k ∈ N) and the resolving functions of the fractional order equation [9] (α > 0 is arbitrary, β = 0).Thus, the symmetry properties (in some sense) of the concepts of integrability and differentiability are studied.
In the second section, some necessary definitions are given, including the new notion of a β-integrated resolving function.Some properties of this function are proved.Necessary and (separately) sufficient conditions for the existence of a β-integrated resolving function are obtained in terms of estimates for the resolvent of the generator A in some right complex half-plane.Third section contains the issues of the existence of a unique solution for the Cauchy problem to Equation (3), the corresponding general inhomogeneous equation and some special inhomogeneous equations.Mild solutions and classical solutions are considered.In the fourth section, two theorems on the necessary and sufficient conditions for the existence of a β-integrated resolving function are obtained.These conditions are formulated in terms of the resolvent of generator A on the semi-axis.The last section concerns the β-integrated resolving function for the linear time-fractional Schrödinger equation with the Dirichlet boundary condition.

β-Integrated Resolving Functions and Some of Their Properties
Introduce the denotation R + := R + ∪ {0}, for h : R + → Z, where Z is a Banach space, the Riemann-Liouville integral of the order β > 0 is J 0 h(t) := h(t), D m is the derivative of the m-th order, m ∈ N. The Gerasimov-Caputo derivative of the order α ∈ (m − 1, m] for h : R + → Z is defined as [9] (p.11, Formula (1.20)).
Remark 1. A.N.Gerasimov [20] and M. Caputo [21] introduced the concept of a fractional derivative, named here by their names, independently of each other.A discussion of these issues can be found in [22].
The Laplace transform of a function h : R + → Z will be denoted by h.The Laplace transform of the Riemann-Liouville integral and the Gerasimov-Caputo derivative satisfies the equalities (see, e. g., [9,23]).
Denote by L(Z ) the Banach algebra of all linear bounded operators from Z to Z, the set of all linear closed operators, densely defined in Z, acting to the space Z will be denoted by Cld(Z ).The denotation for the set of all linear closed operators, defined in Z, acting on the space Z is Cl(Z ).Endow the domain D A of an operator A ∈ Cl(Z ) by the norm of its graph Consider the Cauchy problem for a linear homogeneous equation where m − 1 < α ≤ m ∈ N, A ∈ Cl(Z ).As a solution to problem (4), ( 5) is a function Remark 2. Often, the family of operators {S(t) ∈ L(Z ) : t ≥ 0} from Definition 1 is called a solution operator [9] (p.20, Definition 2.3) or a resolving family of operators [11][12][13][14].The second option seems more convenient to us.In addition, in this paper, it is more natural to use the corresponding mapping S : R + → L(Z ) not a family of operators, so we use the shorter term "resolving function".
Let X be a Banach space.For F : R + → X , define the exponential growth bound as Let S be a resolving function of Equation (5) with ω(S) < ∞.It is known that [9] (p.21, Formula (2.6)).For β ≥ 0, consider the function S β : R + → L(Z ), where for z 0 ∈ Z Then, for Reλ > ω(S), there exists the Laplace transform We call A a generator of a β-integrated resolving function, if ω ≥ 0 exists and a strongly continuous function In this case, S β is called the β-integrated resolving the function generated by A.

Remark 5.
From Definition 2, it follows the one-to-one correspondence of generators and βintegrated resolving functions for a fixed β ≥ 0.
, where E a,b (z) is the Mittag-Leffler function.Indeed, for we have where Γ t := tΓ.
A be a generator of a β 1 -integrated resolving function.Then, A is the generator of a β 2 -integrated resolving function.
(vi) Let x, y ∈ Z such that, for all, t ≥ 0 Then, x ∈ D A and Ax = y.
Remark 6.It is known that the generators of β-integrated resolving functions may be not densely defined (see Remark 3.2.3 in [16] for Apparently, there is no generalization of the semigroup property for the resolving functions in the case of α = 1, β = 0. Hence, there is no analogue of functional relation (3.9) ( [16] p. 124) characterizing k-times integrated semigroups (the case of α = 1, β = k ∈ N), and there is no generalization of Proposition 3.2.4 for α = 1.

Cauchy Problem for Equations with a Generator of β-Integrated Function
Consider the Cauchy problem where T > 0, z k ∈ Z, k = 0, 1, . . ., m − 1, f ∈ L 1 (0, T; Z ) and an operator A generates a β-integrated resolving function for β ≥ 0. By a mild solution of problem ( 10) and ( 11), we mean a function z ∈ C([0, T]; Z ) such that for all t ∈ [0, T], J α z(t) ∈ D A and By a classical solution of ( 10) and ( 11), we understand a function z (10) hold and equality (11) is valid for all t ∈ [0, T].
Let p be the minimal integer, which is equal to or greater than p ∈ R.
A generate a β-integrated resolving function S β such that ω(S β ) < ∞, then the following assertions are valid.
Hence, for λ > ω y(λ) = (λ α − A) −1 0 ≡ 0 and x ≡ z.For z k ∈ D A l+1 , we similarly have that ) is a classical solution to (10) and ( 11) with f ≡ 0. If two classical solutions exist, then there are two mild solutions.The previous assertion implies the uniqueness of a solution.
Thus, due to Lemma 2 (iv) is a mild solution to problem (10) and (11).Its uniqueness can be proven analogously to the homogeneous case.
consequently, taking into account (12), we have is a unique classical solution of problem (10) and (11).
(ii) For all z 0 , z 1 , . . ., z m−1 ∈ Z, there exists a unique classical solution of problem and it is exponentially bounded.
Proof.If A generates a β-integrated resolving function S β , take We have K > 0, ω > max{ω(S β ), 0}, such that for all t ≥ 0 S β (t) L(Z ) ≤ Ke ωt , hence, obviously hold; hence, y is a classical solution of problem ( 13) and (14).Its uniqueness on every segment [0, T] can be shown as in the proof of assertion (i) in Theorem 3; hence, the solution is unique on R + .Consequently, statement (i) implies assertion (ii).
Hence, for some c > 0 and for all t ≥ 0, λ > ω Here, we use asymptotic formula [9, p. 12, (1.27)] for the Mittag-Leffler function.The last inequality is possible, if z 0 = 0 only.Hence, due to (15) exists for all t > 0 and all z 0 ∈ Z and so S β is a β-integrated resolving function generated by A. Consequently, assertion (ii) implies (i).

Criterion of Existence for β-Integrated Resolving Function
Herein, we will use the next two statements.

Define
Lip ω (R + ; Z ) := G : R + → Z : Then, the following assertions are equivalent: Proof.Assume that statement (ii) holds.Then, A also generates a (β + 1)-integrated resolving function S β+1 = J 1 S β on Z which satisfies the assertion (ii) of Theorem 7. Hence, statement (i) follows from that theorem.
Remark 8.The implication from (ii) to (i) is valid for arbitrary α > 0 and for A ∈ Cl(Z ).
It is known that σ(A) = {iλ k : k ∈ N}, where λ 1 ≥ λ 2 ≥ • • • ≥ λ k ≥ . . .are the real non-positive eigenvalues of the corresponding Laplace operator numbered in ascending order, taking into account their multiplicities.Let {ϕ k } be the corresponding eigenfunctions of A, which form an orthonormal basis in L 2 (Ω).

Conclusions
The case is investigated when an unbounded operator at the unknown function in a linear equation resolved with respect to the Gerasimov-Caputo derivative does not generate a resolving function (a resolving family of operators), but satisfies weaker conditions sufficient to generate a β-integrated resolving function.Conditions for the unique solvability of the equation in the sense of classical and mild solutions are obtained.Necessary and sufficient conditions for the existence of a β-integrated function is found in terms of the resolvent of an operator A. These results are planned to be extended to multi-term equations with Gerasimov-Caputo derivatives, equations with Riemann-Liouville, Hilfer, and Dzhrbashyan-Nersesyan fractional derivatives and distributed derivatives.
then there exists a unique classical solution to problem(10) and(11).