On the existence of mild solutions to some semilinear fractional integro-differential equations

This paper deals with the existence of a mild solution for some fractional semilinear differential equations with non local conditions. Using a more appropriate definition of a mild solution than the one given in [12], we prove the existence and uniqueness of such solutions, assuming that the linear part is the infinitesimal generator of an analytic semigroup that is compact for $t > 0$ and the nonlinear part is a Lipschitz continuous function with respect to the norm of a certain interpolation space. An example is provided

On some classes of almost automorphic functions and applications to fractional differential equations

AbstractWe study in this paper the properties of C(n)-almost automorphic and asymptoticallyC(n)-almost automorphic functions (a new concept) with values in a Banach space. We then give a new result related to the existence and uniqueness of an asymptotically almost automorphic solution of a semilinear fractional differential equation of the form Dαx(t)=Ax(t)+F(t,x(t),Bx(t)) with t∈R,0<α<1, where A generates a family of α-resolvent family Sα(t) and f satisfies some Lipschitz conditions