MILD SOLUTIONS FOR ABSTRACT FRACTIONAL DIFFERENTIAL EQUATIONS. CARLOS LIZAMA AND GASTON M. N’GU ´ ER ´ EKATA Abstract. We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a, k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C 0 -semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behavior of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley-Torvik equations. 1. Introduction Fractional differential equations are generalizations of ordinary differential equations to an arbitrary (non- integer) order. Fractional differential equations have attracted considerable interest because of their ability to model complex phenomena. These equations capture nonlocal relations in space and time with power-law memory kernels. Due to the extensive applications of FDEs in engineering and science, research in this area has grown significantly in the past years (see, e.g., [4, 7, 11, 13, 18, 19, 23, 25, 35, 38, 41] and the references therein). The main purpose of this paper is to establish a general procedure as to derive mild solutions to a wide class of fractional differential equations. This is a fundamental and complex problem that has been recently discussed in [15], following the publication of several papers therein cited. Our method is based on an extensive use of properties of Laplace transforms and (a,k)-regularized families, a concept introduced by C. Lizama [29]. We anticipate it can be used for more classes of fractional differential equations with various types of fractional derivatives not covered in this paper. Our study of the variation of constant formulas for abstract fractional differential equations or, equiva- lently, the representation of their solution by means of families of bounded and linear operators, has been motivated by the recent paper [15], which treats the problem of the existence of solution for abstract differ- ential equations with fractional derivatives in time. In that paper, the authors observed that the concepts of mild solutions used in several recent literature on the subject are not appropriate, because the used concept of solution is not realistic. The authors then proposed the use of the well developed theory of resolvent operators for integral equations [39]. However, it is well known that not all fractional differential equation can be formulated as an integral equation, so that the method proposed in [15] fails in the general case. The paper is organized as follows. In Section 2, we recall the very recent facts about (a,k)-regularized families since the paper [29]. Then we present in Section 3, how to derive the variation of constants formulas for various classes of fractional differential equations with the Caputo derivative. This is the main part of the paper. We finally present some applications in Section 4, where we study the existence and asymptotic behavior of solutions of some fractional differential equations. First, using the Leray-Shauder alternative theorem, we prove the existence of a solution to the fractional differential equation with nonlocal conditions (1.1) D α t u(t)= Au(t)+ D α−1 t f (t,u(t)),t ∈ [0,T ],u(0) + g(u)= u 0 ,u ′ (0) = 0, Key words and phrases. Linear and semilinear evolution equations; regularized operator families; mild solutions. The first author is partially financed by Fondecyt Grant number 1100485. 2010 Mathematics subject classification. Primary: 45N05; Secondary: 43A60. 1