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.