Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

• The initial value problem is transformed to a fixed point problem.
• The sufficient conditions for existence and uniqueness of solutions are established.
• The relationship between the two sufficient conditions is discussed.

The solvability of initial value problems for nonlinear Langevin equation involving two fractional orders are discussed in this paper. An existence result for the solution is obtained using the Leray–Schauder nonlinear alternative. In addition, sufficient conditions for unique solution are established under the Banach contraction principle. The existence results for the initial value problems of nonlinear classical Langevin equation follow as a special case of our results.