# Complex variable approach to analysis of a fractional differential equation in the real line

## TL;DR

This paper develops a complex variable approach to establish existence theorems for fractional differential equations involving complex derivatives, providing insights into solution properties and local existence under Riemann-Liouville fractional derivatives.

## Contribution

It introduces a novel complex variable method for fractional differential equations and proves a Peano type existence theorem for such problems.

## Key findings

- Established a Peano type existence theorem for complex fractional differential equations.
- Analyzed geometric properties of solutions in special cases.
- Provided partial results on local existence with Riemann-Liouville derivatives.

## Abstract

The first aim of this work is to establish a Peano type existence theorem for an initial value problem involving complex fractional derivative and the second is, as a consequence of this theorem, to give a partial answer to the local existence of the continuous solution for the following problem with Riemann-Liouville fractional derivative: \begin{equation*} \begin{cases} &D^{q}u(x) = f\big(x,u(x)\big), \\ &u(0)=b, \ \ \ (b\neq 0). \\ \end{cases} \end{equation*} Moreover, in the special cases of considered problem, we investigate some geometric properties of the solutions.