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We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.

Fractional differential equations have recently gained much importance and attention. The study of fractional differential equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. Fractional differential equations appear naturally in a number of fields such as physics, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity, Bodes analysis of feedback amplifiers, capacitor theory, electrical circuits, electronanalytical chemistry, biology, control theory, fitting of experimental data, etc. An excellent account in the study of fractional differential equations can be found in [

Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments [

In this paper, we study a Dirichlet boundary value problem of Langevin equation with two different fractional orders. This work is motivated by recent work of Lim et al. [

In Section

A function

Relative to (

The unique solution of the boundary value problem (

As argued in [

Now, we state a known result due to Krasnoselskii (see [

Let

Let

Define

Assume that

Let us fix

Consider the boundary value problem

The existence of solutions for a Dirichlet boundary value problem involving Langevin equation with two different fractional orders has been discussed. We apply the concepts of fractional calculus together with fixed point theorems to establish the existence results. First of all, we find the unique solution for a linear Dirichlet boundary value problem involving Langevin equation with two different fractional orders, which in fact provides the platform to prove the existence of solutions for the associated nonlinear fractional Langevin equation with two different orders. Our approach is simple and is applicable to a variety of real world problems.

The research of J. J. Nieto has been supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.