On a Nonlinear Fractional Langevin Equation of Two Fractional Orders with a Multiplicative Noise

Abstract

We consider a stochastic nonlinear fractional Langevin equation of two fractional orders Dβ(Dα+γ)ψ(t)=λϑ(t,ψ(t))w˙(t),0<t≤1. Given some suitable conditions on the above parameters, we prove the existence and uniqueness of the mild solution to the initial value problem for the stochastic nonlinear fractional Langevin equation using Banach fixed-point theorem (Contraction mapping theorem). The upper bound estimate for the second moment of the mild solution is given, which shows exponential growth in time t at a precise rate of 3c1expc3t2(α+β)−1+c4t2α−1 on the parameters α>1 and α+β>1 for some positive constants c1,c3 and c4.