On mild solutions of the p-Laplacian fractional Langevin equations with anti-periodic type boundary conditions

Abstract

This work aims at investigating the unique existence of mild solutions of the problem for the p-Laplacian fractional Langevin equation involving generalized fractional derivatives, which are defined with respect to an appropriate function, with anti-periodic type boundary conditions. Herein, we assume that the source function of the problem may have a singularity. Under reasonable assumptions on regularity of the problem, we transform it to a non-local integral equation with the two parameters Mittag-Leffler function in kernels. Based on the integral equation, we obtain the existence and uniquesness results using the Schaefer, nonlinear Leray-Schauder alternatives and Banach fixed-point theorems. Furthermore, we also aim to study the continuity of the mild solutions with respect to perturbations in the inputs of fractional derivatives, such as fractional orders, friction constant, appropriate function and associated parameters, from which we deduce that the solution of Langevin's equation with the fractional Hadamard derivative is the ‘limit’ of the one with the fractional Caputo-Katugampola derivative. Lastly, numerical examples are given to confirm our theoretical findings.