Existence and uniqueness of solutions of nonlinear Cauchy‐type problems for first‐order fractional differential equations

Abstract

New properties of the first‐order Riemann–Liouville fractional integrals are proved and first‐order fractional derivatives for the equivalence class of functions in are analyzed. Solutions, normal solutions, and generalized normal solutions of the initial value problems (IVPs) for nonlinear first‐order fractional differential equations (FDEs) with fraction are introduced. The equivalences in between the first‐order FDEs and the corresponding (Volterra) integral equations are established based on normal solutions and generalized normal solutions. New results on the existence and uniqueness of generalized normal solutions or nonnegative generalized normal solutions in of the IVPs are obtained for . These results are applied to study the existence and uniqueness of generalized normal nonnegative solutions in of the IVP for the nonlinear first‐order FDE with nonlinearities related to the population models with growth rates of Ricker type.