Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions. (English) Zbl 1207.34007

Summary: We study a Dirichlet boundary value problem for the Langevin equation involving two fractional orders. The Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, the 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 the 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.


34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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