Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

: Some new necessary and sufﬁcient conditions for the existence of analytic resolving families of operators to the linear equation with a distributed Riemann–Liouville derivative in a Banach space are established. We study the unique solvability of a natural initial value problem with distributed fractional derivatives in the initial conditions to corresponding inhomogeneous equations. These abstract results are applied to a class of initial boundary value problems for equations with distributed derivatives in time and polynomials with respect to a self-adjoint elliptic differential operator in spatial variables. natural initial value problem for this equation is a problem with given values of special form distributed derivatives of a solution at initial time. A theorem on the generation of analytics in a sector resolving families of operators for such equations is proved. It gives necessary and sufﬁcient conditions on the closed operator in the equation for the existence of the resolving family. This result allows us to study the unique solvability of the mentioned initial problem to the corresponding inhomogeneous equation. The abstract results of the work are applied to the research of the unique solvability for initial boundary value problems for a class of partial differential equations with a distributed Riemann–Liouville derivative in time.


Introduction
The main goal of this work is the study of the unique solvability issues for a special initial value problem to a class of equations with a distributed Riemann-Liouville derivative. The concept of distributed derivative is firstly encountered, apparently, in the works of A.M. Nakhushev [1,2]. Equations with distributed fractional derivatives appear in various fields of investigations applied to the mathematical modelling of some real processes, when an order of a fractional derivative in a model continuously depends on the process parameters: in the kinetic theory [3], in the theory of viscoelasticity [4] and so on [5][6][7]. Numerical methods of solving such equations were developed in the last decades; see [8,9] and the references therein. The qualitative properties of equations with distributed fractional derivatives are investigated in the works of A.M. Nakhushev [1,2], A.V. Pskhu [10,11], S. Umarov and R. Gorenflo [12], T.M. Atanacković, Lj. Oparnica and S. Pilipović [13], A.N. Kochubei [14] and others.
Consider the distributed order equation c b ω(α)D α t z(t)dα = Az(t) + g(t), t ∈ (0, T], with the Riemann-Liouville derivative D α t and with a closed linear operator A in a Banach where Γ(·) is the Euler gamma function. Let m − 1 < α ≤ m ∈ N, D m t h(t) be the usual derivative of the m-th order of h, D α t h(t) := D m t J m−α t h(t) be the Riemann-Liouville fractional derivative.
The Laplace transform of a function h : R + → Z will be denoted by h or Lap [h], if an expression h is too long. By Z denote the set of functions h : R + → Z, such that the Laplace transform h is defined. The Laplace transform of the Riemann-Liouville fractional derivative of an order α > 0 satisfies the equality (see [21]): Here and further D β t h(0) := lim t→0+ D β t h(t). Denote by L(Z ) the Banach space of all linear continuous operators from Z to Z; Cl(Z ) stands for the set of all linear closed operators, densely defined in Z, acting to the space Z. We supply the domain D A of an operator A ∈ Cl(Z ) by the norm of its graph. Thus, we have the Banach space D A .
Assume that assertion (ii) holds. Take θ ∈ (π/2, θ 0 ), δ > 0 and an oriented contour Therefore, H(λ)e λt X ≤ K(θ)r β−1 e aRe t e −r|t| sin ε and the integral is absolutely convergent, uniformly over compact subsets of Σ θ−π/2 and, consequently, defines an analytic function in the sector By the Fubini theorem and the Cauchy residue theorem we have for λ > a (4), if the next conditions are satisfied:
Denote by ρ(A) the resolvent set of an operator A. Let an operator A ∈ Cl(Z ) satisfy the following conditions: Then we will say that the operator A belongs to the class A R c,ε (θ 0 , a 0 ). Here, as before, m − 1 < c ≤ m ∈ N, c is the upper limit of the integration in the definition of W.

Remark 2.
If we consider problem (4), (6) on a segment [0, T], then we can continue the function y = y 1 − y 2 on [T, ∞) by a continuous bounded way. Reasoning in the same way, we get the uniqueness of a solution on a segment not only in the space Z.
Proof. Due to the proof of Theorem 2, if there exists limit in (8), then it equals the identical operator, since it is so on D A . Let the function is continuous on the segment [0, 1] and η(0) = 0. Therefore, the function η is bounded on [0, 1]. Due to the proof of Theorem 2 and Lemma 3 for all t > 1 r ε−1 e tr cos θ dr + C 2 e at θ −θ e tδ cos ϕ dϕ + 1 ≤ C 3 e at t −ε + C 4 e (a+δ)t + 1 ≤ C 5 e (a+δ)t .
Lemma 5 and Theorem 4 implies the next unique solvability theorem.

Conclusions
Linear differential equations in a Banach space with a distributed Riemann-Liouville derivative and with a closed operator in the right-hand side are studied. It is shown that a natural initial value problem for this equation is a problem with given values of special form distributed derivatives of a solution at initial time. A theorem on the generation of analytics in a sector resolving families of operators for such equations is proved. It gives necessary and sufficient conditions on the closed operator in the equation for the existence of the resolving family. This result allows us to study the unique solvability of the mentioned initial problem to the corresponding inhomogeneous equation. The abstract results of the work are applied to the research of the unique solvability for initial boundary value problems for a class of partial differential equations with a distributed Riemann-Liouville derivative in time.