Explicit iteration and unique solution for $ \phi $-Hilfer type fractional Langevin equations

This paper proves that the monotone iterative method is an effective method to find the approximate solution of fractional nonlinear Langevin equation involving $ \phi $-Hilfer fractional derivative with multi-point boundary conditions. First, we apply a approach based on the properties of the Mittag-Leffler function to derive the formula of explicit solutions for the proposed problem. Next, by using the fixed point technique and some properties of Mittag-Leffler functions, we establish the sufficient conditions of existence of a unique solution for the considered problem. Moreover, we discuss the lower and upper explicit monotone iterative sequences that converge to the extremal solution by using the monotone iterative method. Finally, we construct a pertinent example that includes some graphics to show the applicability of our results.