Existence and Uniqueness of Solutions for Fractional Order Cauchy-Lipschitz Differential Equations via the \(\alpha \) -Fixed Point Theorem

In this paper,we generalize the notion of the differential equation using the definition of \(\alpha \) -fixed point given by Azennar et al. (J. Interdisciplinary Math. https://doi.org/10.1080/09720502.2021.1887606 ) using the definition of fractional derivative given by Khalil et al. (J. Comput. Appl. Math. 264:65–70, 2014), and as an application, we consider a nonlinear fractional differential equation with an initial condition using this new formulation.