Existence and uniqueness of solutions for initial value problems of first order Riemann-Liouville type fractional differential equations

Existence and uniqueness of local solutions, nonnegative solutions and solutions for initial value problems (IVPs) of first order nonlinear Riemann-Liouville type fractional differential equations (FDEs) are studied. The nonlinearities are L p L^{p} -Carathéodory functions, so are not necessarily continuous. The results on the existence and uniqueness of local solutions generalize and unify the previous results where the nonlinearities are continuous and some of the results on the existence of nonnegative solutions or solutions are new even when the nonlinearities are continuous. As applications, the new results are used to study the existence of nonnegative local solutions and solutions of the IVPs of the first order FDEs with nonlinearities arising from the logistic type population models.