Existence and Uniqueness Results for a nonlinear fractional differential equations of order $\sigma\in(1,2)$

Abstract

The main objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $\sigma\in(1,2),$ when the nonlinear term has a discontinuity at zero. Hereafter, by using some tools of Lebesgue spaces such as H\"older inequality, we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness theorems for the problem.