Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives ✩

Abstract 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.