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

TL;DR

This paper investigates the local existence and uniqueness of solutions for nonlinear fractional differential equations of order between 1 and 2, addressing discontinuities in the nonlinear term using Lebesgue space techniques.

Contribution

It establishes existence and uniqueness results for a class of nonlinear fractional differential equations with discontinuous nonlinearities, extending previous theories.

Findings

Proved local existence of solutions for the problem.

Established uniqueness theorems including Nagumo-type, Krasnoselskii-Krein-type, and Osgood-type.

Applied Lebesgue space tools to handle discontinuities.

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.