Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann– Liouville fractional derivatives in infinite-dimensional Banach spaces.We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem. © 2012 Elsevier Inc. All rights reserved.