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

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 Eα,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.

In 1908, Langevin [12] applied Newton's second law to a Brownian particle to give an elaborate description of Brownian motion which is now called the "Langevin equation" [13].
The classical Langevin equation for the apparently random movement of a Brownian particle in a fluid due to collisions with the molecules of the fluid is described by where x denotes the position of the particle, m denotes the particle's mass, and f denotes the force acting on the particle from molecules of the fluid surrounding the Brownian particle.The force f may be written as a sum of two parts.The first one is the viscous force proportional to the particle's velocity with coefficient λ.The second one denoted by η(t) is the random force arising from rapid thermal fluctuation [14].The fractional Langevin equation was introduced by Mainardi et al. in the early 1990s [14,15].Much work since then has been devoted to the study of the fractional Langevin equations in the field physics (e.g., [16][17][18][19][20][21][22]).Moreover, the fractional Langevin equations have been applied to describe various anomalous diffusive process, such as single file diffusion and crossover dynamics between different diffusive regimes (see, e.g., [23][24][25][26]).
In this paper, we study the following anti-periodic boundary value problem of nonlinear fractional Langevin equations: where α, ξ ∈ (0, 1), β, α As mentioned in [7] and the references therein, the existence results of fractional differential equations involving Caputo differential operator of order α, β ∈ (0, 1) are obtained by Mittag-Leffler functions E α (z) and E α,β (z), since E α (z) and E α,β (z) have "good" properties, such as explicit boundedness, monotonicity and nonnegativity.However, for α, β ∈ (1, 2), the above properties do not hold anymore, which leads to difficulties for the theoretical analysis.In this paper, using some techniques, we study the properties of E β (z) and E β,θ (z)(β, θ ∈ (1, 2)) and obtain the existence result of solutions to (1) and ( 2) under the weak assumptions on f (t, u(t)).
The plan of this paper is as follows.In Section 2, we present some basic concepts, notations about fractional calculus.In Section 3, we prove some properties of Mittag-Leffler functions.In Section 4, we present the definition of solution to (1) and (2).In Section 5, we employ Krasnoselskii's fixed point theorem to obtain the existence of solutions to problem (1) and ( 2).An example is given in Section 6 to demonstrate the application of our result.

Preliminaries
In this paper, we denote by C(J, R) the Banach space of all continuous functions from J to R, L p (J, R) the Banach space of all Lebesgue measurable functions l : J → R with the norm l L Definition 1. [3,4] The fractional integral of order γ with the lower limit a for a function x(t) ∈ L 1 ([a, +∞), R) is defined as where Γ(•) is the gamma function.Definition 2. [3,4] If x(t) ∈ AC n ([a, b], R), then the Riemann-Liouville fractional derivative L D γ a + x(t) of order γ exists almost everywhere on [a, b] and can be written as ), then the Caputo derivative c D γ a + x(t) of order γ exists almost everywhere on [a, b] and can be written as Definition 4. [5] The Hilfer fractional derivative of order 0 ≤ γ ≤ 1 and 0 < α < 1 with lower limit a is defined as x)(t).
We introduce the following fixed point theorem.
Theorem 1. (Krasnoselskii's fixed point theorem) Let B be a closed, convex, and nonempty subset of a Banach space U, and let A 1 , A 2 be operators such that: Then there exists z ∈ B such that z = A 1 z + A 2 z.

Properties of Mittag-Leffler Functions
In this section, we prove some properties of the Mittag-Leffler functions.
where C is a positive constant.
Similar to the arguments in [3,4], we can obtain the following results.

Solutions of BVP
In this section, we present the formulas of solutions to problem (1) and ( 2).
Lemma 7. [4] For θ > 0, a general solution of the fractional differential equation c D θ 0 + u(t) = 0 is given by , and [θ] denotes the integer part of the real number θ.
Formally, by Lemma 7, for C i ∈ R(i = 0, 1, 2), we have c D Based on the arguments of (see [4], pp.222-223) and Lemma 2, we obtain Definition 6.A function u : J → R is said to be a solution of (1) and ( 2) Proof.From the Hölder inequality, we have where For t ∈ J, y > p 1 , using the Hölder inequality, we have For convenience, we define Lemma 10.Assume that (H1) holds.For u ∈ C β , t ∈ J, we have Proof.It follows from the definition of derivative for the Lebesgue integration and ( 14) that Next, we show that (F α+β−1 u)(t) ∈ AC(J, R).For every finite collection {(a j , b j )} 1≤j≤n on J with n ∑ j=1 (b j − a j ) → 0, noting Lemmas 1, 5 (ii), 9 and ( 14), we derive exists.Moreover, similar to (15), one has Noting that Lemma 2 and ( 14) we can see From the Definition 2, ( 16) and ( 17), we get and and Noting that Definitions 3 and 4, ( 17) and ( 19) we obtain For convenience, we shall use the following notation: Lemma 11.Assume that (H1) holds.A function u is a solution of the following fractional integral equation if and only if u is a solution of the problem (1) and (2), where Proof.(Sufficiency) Let u be the solution of ( 1) and ( 2), Lemmas 8, 2 and 4 imply where a, b, c are constants.Using the boundary value condition (2), we derive that a = 0 and bE β,2 (−λ) Now we can see that (20) holds.
(Necessity) Let u satisfy (20).Noting that Lemma 10, ( c D α Clearly, the boundary value condition (2) holds and hence the necessity is proved.
For convenience of the following presentation, set

Existence Result
In this section, we deal with the existence of solutions to the problem (1) and (2).To this end, we consider the following assumption.

Proof. We consider an operator
Clearly, F is well defined.Obviously, the fixed point of F is the solution of problem ( 1) and (2).
By ( 11)-( 14) and (H1), the following inequalities hold: Moreover, from Lemma 1, there exists a constant Furthermore It follows from ( 21) and ( 23) that (F u) β ≤ r.Now, we can see that (F u)(t) ∈ B r for any u ∈ B r and t ∈ J. Setting According to (H2), ( 13) and ( 14), we obtain This implies that F 2 is a contraction mapping.Let {u n } be a sequence such that u n → u in C β , then there exists ε > 0 such that u n − u β < ε for n sufficiently large.By (H2), we have Moreover, f satisfies (H1), we get f (t, u n (t)) → f (t, u(t)) as n → ∞ for almost every t ∈ J.
and the Lebesgue dominated convergence Similarly, from Lemma 1 and Moreover, by Lemmas 1 and 5, ( 21) and ( 22), {(F 1 u)(t) : u ∈ B r } is an equicontinuous and uniformly bounded set.Then, F 1 is a completely continuous operator on B r .The proof now can be finished by using Theorem 1.

Conclusions
In this paper, we have presented existence results to the nonlinear Langevin fractional differential equations with the anti-periodic boundary value conditions and some properties of the Mittag-Leffler functions E β (z) and E β,θ (z)(β, θ ∈ (1, 2)).We prove the equivalence of the problem (1) and ( 2) and the integral Equation ( 20) under the weak assumption (H1).Moreover, when β, θ ∈ (1, 2), E β (z) and E β,θ (z) do not possess the monotonicity and nonnegativity, using Lemma 6, we successfully obtain some estimates for the Mittag-Leffler functions.Our results are new and significantly contribute to the existing literature on fractional order differential equation with anti-periodic boundary value conditions.In fact, our approach is simple and can easily be applied to a variety of real world problems.
In this area, our future work will focus on studying the more complex model, such as the boundary value problem for the mixed type fractional differential equations with the Caputo and the Riemann-Liouville fractional derivative.
p = J |l(t)| p dt 1 p < ∞ and by AC([a, b], R) the space of all absolutely continuous functions defined on [a, b].Moreover, for n = 1, 2,