A study of nonlinear Langevin equation involving two fractional orders in different intervals

This paper studies a nonlinear Langevin equation involving two fractional orders α∈(0,1] and β∈(1,2] with three-point boundary conditions. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions for the problem. The existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results. Some illustrative examples are also discussed. © 2011 Elsevier Ltd. All rights reserved.