THE GENERALIZED NONLOCAL BOUNDARY CONDITION FOR FRACTIONAL LANGEVIN EQUATION WITH A WEAKLY SINGULAR SOURCE

Abstract

We investigate the nonlinear Langevin equation regarding the fractional derivative of a function depending on another function $\psi$ with generalized nonlocal conditions. We assume that the source function of the problem may have a singularity that appears from the discontinuity of the source function at $t=a$. We construct a new formula solution for the problem via the Mittag-Leffler function with two parameters. Based on the obtained formula solution, we propose some suitable conditions such that the problem has at least one or has a unique mild solution. The desired results are proved by applying some well-known fixed point theorems such as the Schaefer, nonlinear Leray–Schauder alternatives and Banach. Furthermore, we discuss the continuous dependence of mild solutions of the problem on the inputs (fractional orders, friction constant, appropriate function and nonlocal conditions) from which we deduce that the solution of the Langevin equation with Caputo–Katugampola fractional derivative ($\psi (t)=(t^{\rho }-1)∕\rho$) converges to the solution of the Langevin equation with Hadamard fractional derivative ($\psi (t)=\text{ln}t$) as $\rho \to 0^{+}$. Finally, two examples are given to illustrate our theoretical findings.