New solvability and stability results for variable-order fractional initial value problem

In this paper, we employ a new methodology to investigate the existence, uniqueness, and stability for initial value problem in the frame of Riemann–Liouville derivative of variable-order. In contrast to the methods used in the literature, which concentrated on using the idea of piecewise constant functions and the generalized intervals concept, our method is straightforward and based on an improved fractional operator that demonstrates the solvability and stability of the main problem under less strict presumptions. The paper is supported by applications which has examples and illustrative figures.