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

Abstract

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.