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

Abstract

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

Anti-Periodic Boundary Value Problems for Nonlinear Langevin Fractional Differential Equations

“…Proof. (i) From Lemma 3.4 of [10], we have E β (−λt β ) ≤ 1. Using Lemma 3, the second and the third inequalities hold.…”

“…In recent years, the subject of fractional differential equations is gaining much importance and attention. For details, see [1][2][3][4][6][7][8][9][10][11] and the references therein.…”

“…Recently, there has been a significant development in solving fractional Langevin equation (see [7][8][9][10] and the references therein). To the best of our knowledge, there are few papers dealing with anti-periodic BVP involving fractional Langevin equation with two fractional orders c D α [10].…”

Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders

“…Recently, several contributions mindful with the uniqueness and existence results for fractional generalized Langevin equations, have been published, see [6][7][8][9][10][11][12][13][14][15] and the references given therein.…”

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

“…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.…”