On fractional Langevin equation involving two fractional orders

• We introduce a new norm that is convenient for the fractional and singular differential equations.
• The existence and uniqueness of initial value problems for nonlinear Langevin equation with two different fractional orders are discussed.
• By an example we show that we can not speak about the existence and uniqueness of solutions of nonlinear Langevin equation with two diffrent fractional orders just by using the previous methods.

In numerical analysis, it is frequently needed to examine how far a numerical solution is from the exact one. To investigate this issue quantitatively, we need a tool to measure the difference between them and obviously this task is accomplished by the aid of an appropriate norm on a certain space of functions. For example, Sobolev spaces are indispensable part of theoretical analysis of partial differential equations and boundary integral equations, as well as are necessary for the analysis of some numerical methods for the solving of such equations. But most of articles that appear in this field usually use ‖.‖∞ in the space of C[a, b] which is very restrictive. In this paper, we introduce a new norm that is convenient for the fractional and singular differential equations. Using this norm, the existence and uniqueness of initial value problems for nonlinear Langevin equation with two different fractional orders are studied. In fact, the obtained results could be used for the classical cases. Finally, by two examples we show that we cannot always speak about the existence and uniqueness of solutions just by using the previous methods.