On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

In this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form 1Dρ=(1−ρ)+ρD,ρ∈[0,1],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D^{\rho }= (1-\rho )+ \rho D, \quad \rho \in [0,1], $$\end{document} where D is the ordinary differential operator.