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 $\alpha,\beta\in(1,2)$. By using Laplace transform methods, formula of solutions involving Mittag-Leffler functions $A_{\alpha,\beta}(w)$, $\alpha,\beta\in(1,2)$, $w\in R$, 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 $A_{\alpha,\beta}(w)$, $\alpha,\beta\in(1,2)$, $w\in R$, and fixed point methods. Finally, two examples are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]