Nonlocal problems for Langevin-type differential equations with two fractional-order derivatives

In this paper, we intend to study nonlocal problems for Langevin-type differential equations with two fractional derivatives of orders α,β∈(1,2) . By using Laplace transform methods, formula of solutions involving Mittag-Leffler functions Aα,β(w) , α,β∈(1,2) , w∈ R , and nonlocal terms of such equations are presented by studying the corresponding linear Langevin-type equations with two fractional derivatives. Meanwhile, existence results of solutions are established by utilizing boundedness, continuity, monotonicity, nonnegative of Mittag-Leffler function Aα,β(w) , α,β∈(1,2) , w∈ R , and fixed point methods. Finally, two examples are presented to illustrate our theoretical results.