Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions

Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, porous media, and so forth. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent developments on the subject, see 1–8 and the references therein. Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments. However, for systems in complex media, ordinary Langevin equation does not provide the correct description of the dynamics. One of the possible generalizations of Langevin equation is to replace the ordinary derivative by a fractional derivative in it. This gives rise to fractional Langevin equation, see for instance 9–12 and the references therein. In this paper,we consider the following boundary value problem of Langevin equation with two different fractional orders: