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

Fractional Langevin type delay equations with two fractional derivatives

“…One of them is so called fractional Langevin type equation, which is obtained from the classical Langevin equation by replacing classical derivative by fractional derivative. The nonlinear Langevin type equations involving two fractional orders was introduced and investigated in [10]- [15].…”

Langevin equation involving one fractional order with three-point boundary conditions

“…where c D α and c D β are the Caputo fractional derivatives and f : [0, 1] × R → R is a given continuously differentiable function, m − 1 < α m and n − 1 < β n, m, n ∈ N [22]. It worth mentioning that mainly, fractional Langevin equations have been studied extensively.…”

“…With different unit intervals of values to two fractional orders α and β, many authors introduced their works. For instance, [3,4,22,24,25] concerned with m − 1 < α m and n − 1 < β n, m, n ∈ N, [1,15,21] concerned with 0 < α, β 1, [2,7,12,27] concerned with 0 < α 1 and 1 < β 2, [5,11] concerned with 1 < α 2 and 0 < β 1, and Gao et al [8] concerned with 1 < α, β 2.…”

Single upper-solution or lower-solution method for Langevin equations with two fractional orders

“…In [26], by using the Leray-Schauder nonlinear alternative, the authors studied the following initial value problem of Langevin equations with two fractional orders:…”