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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 A α,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 A α,β (w), α, β ∈ (1, 2), w ∈ R, and fixed point methods. Finally, two examples are presented to illustrate our theoretical results. MSC: 26A33; 34B37

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In this paper, we apply asymptotic behavior on Mittag-Leffler functions E α (z) and E α,α (z) for z > 0 to discuss exp-type Ulam-Hyers stability of c D α t x(t) = λx(t) + f (t, x(t)) for the case λ > 0 on a finite time interval [0, 1] and an unbounded interval (1, ∞).

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