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

The first aim of this work is to establish a Peano-type existence theorem for an initial value problem involving a complex fractional derivative, and then, as a consequence of this theorem, to give a partial answer for the local existence of the continuous solution to the initial value problem:

$\{\begin{array}{cc}\hfill & D_x^{q}u(x)=f(x,u(x)),\hfill \\ \hfill & u(0)=b,(b\ne 0).\hfill \end{array}$

Moreover, for some special cases of the problem, we investigate the corresponding geometric properties of the solutions.