Generalizations of Riemann–Liouville fractional integrals and applications

Abstract

The notion of a generalized Riemann–Liouville fractional integral is introduced, and its domain, range, and properties are studied. The new notion and properties provide new insight and understanding into the classical Riemann–Liouville fractional integral and its properties. Based on the generalized Riemann–Liouville fractional integral, equivalences between linear first ‐order fractional differential equations (FDEs) and integral equations are established. These equivalence results are applied to obtain solutions of Abel type integral equations and to study the existence and uniqueness of generalized normal solutions of initial value problems for nonlinear first‐order FDEs.