In this paper, we investigate a class of nonlinear Langevin equations involving two fractional orders with nonlocal integral and three-point boundary conditions. Using the Banach contraction principle, Krasnoselskii’s and the nonlinear alternative Leray Schauder theorems, the existence and uniqueness results of solutions are proven. The paper was appended examples which illustrate the applicability of the results.

In the present paper, we discuss a new boundary value problem for the nonlinear Langevin equation involving two distinct fractional derivative orders with multi-point and multi-nonlocal integral conditions. The fixed point theorems for Schauder and Krasnoselskii–Zabreiko are applied to study the existence results. The uniqueness of the solution is given by implementing the Banach fixed point theorem. Some examples showing our basic results are provided.

The paper is devoted to the study of the functionWα,βγ,δ(z), which is an extension of the classical Wright function and Kummer confluent hypergeometric function. The properties ofWα,βγ,δ(z)including its auxiliary functions and the integral representations are proven.

Recently, Khalili and Yadollahzadeh [ 9 ] have investigated the uniqueness and existence of solution $ u(t),\; t\in[0,1] $ for a class of nonlocal boundary conditions to fractional Langevin equation. The authors used the boundary condition $ u'(0) = 0 $ by incorrect method. In the current contribution, we show the correct method for using this condition and study the existence and uniqueness of solution for the same class of equation in slightly different form with anti-periodic and nonlocal integral boundary conditions as well as the boundary condition $ u'(0) = 0 $. An exemplar is provided to illustrate our results.