DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020187 DYNAMICAL SYSTEMS SERIES S FRACTIONAL CAUCHY PROBLEMS AND APPLICATIONS Mohammed AL Horani Department of Mathematics, The University of Jordan Amman, Jordan Mauro Fabrizio and Angelo Favini Dipartimento di Matematica, Universit` a di Bologna Piazza di Porta S. Donato, 5 – 40126 Bologna, Italy Hiroki Tanabe Takarazuka, Hirai Sanso 12-13, 665-0817, Japan Abstract. We are devoted to fractional abstract Cauchy problems. Required conditions on spaces and operators are given guaranteeing existence and unique- ness of solutions. An inverse problem is also studied. Applications to partial differential equations are given to illustrate the abstract fractional degenerate differential problems. 1. Introduction. Many papers and monographs are devoted to BMu(t) − Lu(t)= f (t) , 0 ≤ t ≤ T (1) where T> 0, B, M , L are closed linear operators on the complex Banach space E with D(L) ⊆ D(M ), 0 ∈ ρ(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 [9], see in particular the monograph [10]. By using the real interpolation space (E,D(B)) θ,∞ ,0 <θ< 1, see [12], [13], suitable assumptions on the operators B, M , L guarantee that (1) has a unique solution. This result was improved by Favini, Lorenzi, Tanabe in [7], see also [5], [6], [8]. 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 ‖(λ − B) −1 ‖ L(E) ≤ c |Re λ| +1 , Re λ < a . (2) (H 2 ) Operators L, M satisfy ‖M (λM − L) −1 ‖ L(E) ≤ c (|λ| + 1) β (3) for any λ ∈ Σ α :=  z ∈ C : Re z ≥−c(1 + |Imz|) α , c> 0, 0 <β ≤ α ≤ 1  . (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: B −1 A −1 = A −1 B −1 . The following result is established, see [7], 2010 Mathematics Subject Classification. Primary: 26A33; Secondary: 34G10. Key words and phrases. Fractional derivative, abstract Cauchy problem, evolution equations, inverse problem. 1