Existence Results for a New Class of Nonlinear Langevin Equations of Fractional Orders

Abstract

In this paper, we propose a new nonlinear Langevin equation of fractional orders with nonlocal boundary value conditions which is a generalization of previous Langevin equations. By applying some fixed point theorems such as Schauder and contraction mapping principle, the existence and uniqueness of solutions are studied. An example to illustrate our results is given.

Investigating the existence, uniqueness, and stability of solutions in boundary value problem of fractional differential equations

“…Recently, Anti-periodic boundary value problems (BVP) have occurred in the mathematical modeling of various physical processes and have recently received considerable attention. Accordingly, in [15] a study on the existence and uniqueness of a solution u(t), where t ä [0, 1], for a certain class of nonlocal boundary conditions applied to the fractional Langevin equation was recently done by Khalili and Yadollahzadeh. Interestingly, the author applied the boundary condition ( ) ¢ = u 0 0in an improper way.…”