Inverse Problems for Degenerate Fractional Integro-Differential Equations

This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of non-fractional equations. Our method is based on transforming the inverse problem to a direct problem and identifying the conditions under which this direct problem has a unique solution. The conditions under which the unique strict solution can be compared with the case of a mild solution, obtained in previous studies under quite restrictive requirements, are on the underlying functions. Applications from partial differential equations are given to illustrate our abstract results.

In order to obtain an existence result, the authors of Reference [23] were compelled to require that ( see Reference [23], Theorem 3.2, p. 3409) u 0 ∈ D(B −1 A), f : J × X 2 → X B −1 A is completely continuous and there exists a positive constant L 1 such that Moreover, it is assumed, in order to apply fixed point arguments, that a certain obtained constant is less than 1. Then problem (1) and (2) admits a mild solution on J. This result shows how many restrictive assumptions must be done to obtain only a mild solution to a weakly degenerate equation (recall that it is assumed Our problem consists in studying existence, uniqueness and regularity of a pair (y, f ) ∈ C([0, τ]; D(L)) × C([0, τ]; C) solving, in a strict sense, the integro-differential problem where L, L 1 , M are closed linear operators acting on the complex Banach space X, 0 <α < 1, being the necessary compatibility relation to be satisfied in advance. Analogous problems withα = 1 have been considered by many authors, above all for M = I, the identity operator, see in particular [15,25]. The case forα = 1 without the integral sign has been considered recently in Reference [6], see also Al Horani et al. [3][4][5]. Also one can find some related results in Reference [7] where the authors extended, on the grounds of Reference [8] and the previous results of Favini and Lorenzi [26], see also Favini and Yagi [27], pp. 157-162. The plan of this paper is as follows. In Section 2 we recall previous results on possibly degenerate differential and integro-differential equations. Section 3 is devoted to the preliminaries for the general caseα ∈ (0, 1). In Section 4 we consider the special caseα = 1. Section 5 is related to the main casẽ α ∈ (0, 1). Section 6 contains some examples and applications.
It must be noted that the conditions on f and k in Reference [23] are very restrictive and one expects that such conditions can imply strict solutions. At this purpose, we recall that our required strict solution y(t) is defined on the whole interval [0, τ] and Ly, Dα t My have convenient Holder regularity in time.
More general problems like Dα t (My(t)) = Ly(t) + . . , n 2 could be of interest in the future.

Previous Results and Preliminaries
This section is devoted to recall previous results that shall be used in the sequel. We begin with the following lemma from [15].
where y 0 ∈ D(L) ⊆ D(M) ∩ D(L 1 ), 0 ∈ ρ(L), zM − L has a bounded inverse for any z in the region The following result is important, see Reference [6].
, Xθ A , withθ > 1 − β, M, L, L 1 being closed linear operators on X, and for all ϕ ∈ (0, 1) , Then for all θ ∈ (0, 1) admits a unique strict solution, that is, Notice that when we apply Φ to both sides of Equation (8) we get so that necessarily Therefore, (8) takes the form where L 2 is defined by . These conditions are strongly restrictive. More precise and better results canceling, in particular, f (0) + Ly 0 have been obtained by Favaron-Favini, see Reference [25], Theorem 48.

Lemma 4.
Assume that L has a bounded inverse, y 0 ∈ D(L), 5α + 2β > 6, Here λ 0 is a fixed constant such that λ 0 M + L has a bounded inverse.
Assume (H 1 ) to hold together with the hypothesis that the closed linear operator B has a resolvent (z − B) −1 for all z ∈ C : Re z < a, a > 0 such that Then under the hypotheses above, equation BMu = Lu + f admits a unique strict solution u such that Lu, BMu ∈ (E, D(B)) ω,p , If X is a complex Banach space, introduce operator B X by It is well known that ρ(B X ) = C and B X is a positive operator in C([0, τ]; X) of type π/2. Powers for B X are defined as follows for all δ > 0, f ∈ C([0, τ]; X) and any t ∈ (0, τ]. Since B −δ X is injective, one defines for δ > 0 It is known that if δ ∈ (0, 2), B δ X is positive of type δπ 2 . Moreover, the following interpolation result holds.
It follows that since operator Bα X satisfies the spectral property described above, for anyα ∈ (0, 1]

The Integro-Differential Problem forα = 1
Of concern is the inverse problem . The unknown is the pair (y, f ), f ∈ C([0, τ]; C). In order to avoid problems for the sum of closed operators, we assume that z 0 M + L has a bounded inverse and introduce the new variable x = (z 0 M + L)y. Then (11)-(13) takes the form One may note that all involved operators are bounded. Observe also and that for all λ ∈ Σ α . In this case A = (z 0 M + L)M −1 , as expected. Applying Φ to both sides of Equation (14), we get If Φ[z] = 0, then Therefor, we get a direct problem, precisely, One applies Lemma 4 and notice that zM z has the same spectral properties of zM(z 0 M + L) −1 − I, provided that z ∈ Xθ A for someθ > 1 − β, see Favini and Tanabe [16]. Thus our assumptions reduce to 5α Therefore, we can establish the result as follows.

Applications
In this section we introduce two concrete cases of partial differential equations in which all our hypotheses run well and Theorem 2 can be applied. Of course, by using Favini and Yagi [27], many other concrete applications could be described. We begin with the following example. Example 1. Consider the inverse problem to find (y, f ) satisfying Ω being a bounded set in R n with a smooth boundary, k is continuous on [0, τ], m(x) ≥ 0, m ∈ C(Ω), (0), h sufficiently smooth. Of course the ambient space is L p (Ω). The resolvent estimates hold with α = 1, β = 1/p, p > 1. Similar situation is found in Favini and Yagi [27], pp. 79-80.
One obtains a differential system to which the quoted results from Favini and Yagi [27] apply.

Conclusions
Some well known results for the case of non-fractional equations have been extended. Existence, uniqueness and regularity of solutions to the inverse problem related to degenerate fractional integro-differential equations have been studied. Some conditions on the underlying functions are imposed to guarantee the existence of a unique strict solution under less restrictive requirements than those presented in Reference [23,24], for example. This holds for Fedorov and Ivanova [7,13]. Applications from partial differential equations are given to illustrate our abstract results.