Abstract
Abstract We study the boundedness and compactness of Riemann–Liouville integral operators on the so-called Morrey spaces which are nonseparable spaces. There are no approximation or contractive skills in this kind of spaces. Moreover, unlike the use of dual or maximal point of view in integrable function spaces, the idea of our proof proceeds from the compactness of the truncated Riemann–Liouville fractional integrals by using a criterion for strongly pre-compact set. Constructing a truncated Marchaud fractional derivative function, we show the characterization of the solution to Abel’s equation in Morrey spaces. With the aid of fixed-point theorem, we establish the existence and uniqueness of solutions to a Cauchy type problem for fractional differential equations. We also give an example to illustrate the sufficiency of the conditions in our main result.
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