Direct and Inverse Fractional Abstract Cauchy Problems

: We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling related inverse problems, extending the treatment given by Alfredo Lorenzi. Some basic assumptions on the involved operators are also introduced allowing application of the real interpolation theory of Lions and Peetre. Our abstract approach improves previous results given by Favini–Yagi by using more general real interpolation spaces with indices θ , p , p ∈ ( 0, ∞ ] instead of the indices θ , ∞ . As a possible application of the abstract theorems, some examples of partial differential equations are given.


Introduction
Consider the abstract equation where B, M, L are closed linear operators on the complex Banach space E, the domain of L is contained in domain of M, i.e., D(L) ⊆ D(M), 0 ∈ ρ(L), the resolvent set of L, f ∈ E and u is the unknown. The first approach to handle existence and uniqueness of the solution u to (1) was given by Favini-Yagi [1], see in particular the monograph [2]. By using real interpolation spaces, see [3,4], suitable assumptions on the operators B, M, L guarantee that (1) has a unique solution. Such a result was improved by Favini, Lorenzi and Tanabe in [5], see also [6][7][8]. In order to describe the results, we list the basic assumptions: (H 1 ) Operator B has a resolvent (z − B) −1 for any z ∈ C, Re z < a, a > 0 satisfying where L(E) denotes the space of all continuous linear operators from E into E.
(H 3 ) Let A be the possibly multivalued linear operator A = LM −1 , D(A) = M(D(L)). Then A and B commute in the resolvent sense: Let (E, D(B)) θ,∞ , 0 < θ < 1, denote the real interpolation space between E and D(B). The main result holds It is straightforward to verify that if B generates a bounded c 0 −group in E, then assumption (H 1 ) holds for B. Analogously, if −B generates a bounded c 0 −semigroup in E, then assumption (H 1 ) holds for B. It was also shown, in a previous paper, that Theorem 1 works well for solving degenerate equations on the real axis, too, see [9].
The first aim of this paper is to extend Theorem 1 to the interpolation spaces (E, D(B)) θ,p , 1 < p < ∞. This affirmation is not immediate. Section 2 is devoted to this proof. In Section 3, we apply the abstract results to solve concrete differential equations. In Section 4, we handle related inverse problems. In Section 5, we study abstract equations generalizing second-order equations in time. In Section 6, we present our conclusions and remarks. For some related results, we refer to Guidetti [10] and Bazhlekova [11].
Our aim is to extend Theorem 1 to 1 < p < ∞. In order to establish the corresponding result, we need the following lemma concerning multiplicative convolution. We recall that L p * (R + ) = L p R + ; t −1 dt and that the multiplicative convolution of two (measurable) functions f , g : R + → C is defined by where the integral exists a.e. for x ∈ R + . Lemma 1. For any f 1 ∈ L p * (R + ) and g ∈ L 1 * (R + ), the multiplicative convolution f 1 * g ∈ L p * (R + ) and satisfies Consider now the chain of estimates

Fractional Derivative
Letα > 0, m = α is the smallest integer greater or equal toα, I = (0, T) for some T > 0. Define The Riemann-Liouville fractional derivative of orderα, or, more precisely, the so-called left handed Riemann-Liouville fractional derivative of orderα, is defined for all Dα t is a left inverse of Jα t , but in general it is not a right inverse. The Riemann-Liouville fractional integral of orderα > 0 is defined as: If X is a complex Banach space,α > 0, then we define the operator Jα as: Define the spaces Rα ,p (I; X) and Rα ,p 0 (I; X) as follows. Ifα ∈ N, set For the Sobolev space W β,p (I; X) of fractional order β > 0, we define where Dα t is the Riemann-Liouville fractional derivative. Notice that ifα ∈ (0, 1), u ∈ D(Lα), then (g 1−α * u)(0) = 0.
We illustrate the previous abstract concepts in the following example Since D(D t ) = W 1,p The solution is i.e., From (4) and (5) it follows that Therefore −D t is the infinitesimal generator of the semigroup U(τ), τ ≥ 0. Letα > 0. Then for f ∈ L p (I, X) Therefore D −α t has an inverse which is denoted by Dα t .We have Statement (e) implies that ifα ∈ (0, 1], However, this reads equivalently (λ − Lα) −1 ≤ C/|λ| provided that λ is in a sector of the complex plane containing Re λ ≤ 0. Therefore, ifα ≤ 1, operator Lα = d dtα = Dα t satisfies assumption (H 1 ) in Theorem 1. Therefore, we can handle abstract equations of the type in a Banach space X with an initial condition g 1−α * u (0) = 0. Then the results follow easily from the abstract model.
for all z in a sector containing Re z ≥ 0.
We refer to to the monograph [2] for many further examples of concrete degenerate partial differential equations to which Theorem 2 applies.

Inverse Problems
Given the problem then corresponding to an initial condition and following the strategy in various previous papers, see in particular Lorenzi [12], we could study existence and regularity of solutions (y, f ) to the above problem such that Φ[My(t)] = g(t), where g is a complex-valued function on [0, T]. This is, of course, an inverse problem. Applying Φ to both sides of Equation (6) we get If Φ[z] = 0, we obtain necessarily Therefore, If L 1 is defined by one can introduce assumptions on the given operators ensuring that the direct problem has a unique strict solution, see [13]. The main step is to verify that assumption (H 2 ) holds for the operators L + L 1 and M.

Application: Generalized Second-Order Abstract Equation
Let us consider the abstract equation, generalizing second-order equation in time, where A, B, C are some closed linear operators in the complex Banach space X, B 1 , B 2 are suitable operators defined on suitable Banach spaces. The change of variables B 1 u = v transforms the given equation to the system which can be written in the matrix form The basic idea is to use a convenient space and a domain of operator matrices. Noting In order to simplify the argument, we take D(B) ⊆ D(A) ∩ D(C). Moreover, we assume that for all z ∈ Σ α , where the involved operators satisfy which guarantees that the problem is of parabolic type. Take Y = D(B) × X with the usual product norm. Then it is shown in Favini-Yagi [2], page 184, that the resolvent estimate holds. Therefore assumption (H 2 ) is satisfied.

Conclusions
It was shown that the degenerate problem including Riemann-Liouville fractional derivative can be handled by means of a general abstract equation. Applications to degenerate fractional differential equations with some related inverse problems were studied. Moreover, generalized second-order abstract equations were well-treated.