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

Abstract

In this paper, we focus on the existence of solutions of the nonlinear Langevin fractional differential equations involving anti-periodic boundary value conditions. By using some techniques, formulas of solutions for the above problem and some properties of the Mittag-Leffler functions E α , β ( z ) , α , β ∈ ( 1 , 2 ) , z ∈ R are presented. Moreover, we utilize the fixed point theorem under the weak assumptions for nonlinear terms to obtain the existence result of solutions and give an example to illustrate the result.

Upper-solution or lower-solution method for Langevin equations with n fractional orders

“…With different unit intervals of values to two fractional orders α and β, many authors introduced their works. For instance, [5,8,21,23,24] 16,20] concerned with 0 < β, α ≤ 1, [3,10,14,25] concerned with 0 < α ≤ 1 and 1 < β ≤ 2, [8,11,13] concerned with 1 < α, β ≤ 2. Motivated by work, we study the existence and uniqueness of solutions to the initial value problem of the Langevin equation involving n Fractures of different periods as follows:…”

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.…”