On the Existence and Uniqueness of Solution to Fractional-Order Langevin Equation

In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional-order Langevin equation involving two distinct fractional orders with unprecedented conditions (three-point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional-order Langevin equation. As a result of employing the so-called Krasnoselskii and Leray-Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.


Introduction
Unquestionably, the fractional calculus, which requires the order of the functional operator of calculus to be turned into the fractional case, is a forceful mathematical argument for providing much and more flexibility in dealing with many real-world applications [1,2]. More particularly, the fractional-order differential equations (FODEs), which are considered a cornerstone of this field, have been extensively employed in modeling several phenomena and numerous scientific applications such as diffusion modeling [3], robot manipulators [4], economics [5], and many more [6][7][8][9]. In view of its outstanding importance, lots of mathematicians have paid their attentions to deeply explore the initial and/or boundary value problems (BVPs) established based on such equations (see [10][11][12][13]). Actually, these problems, which may have different boundary conditions, describe several physical real-world issues like thermoelasticity, underground water flow, and heat conduction. One of the most significant of those problems is known as the Langevin equation which was formulated for the first time in 1908 by Langevin, who still much earlier described the progress of some physical matters in fluctuating settings [14,16]. It came out, later on, that such equation had assisted physicians successfully to identify some techniques related to the abnormal diffusion. It also, afterward, succeeded in describing some other techniques in economic field that contain the cost index operating [17]. Besides, its generalization played, and still plays, a significant duty in describing the noise causes in the field of critical dynamics [18]. In 1996, Mainardi and Pironi fractionalized such equation and hence turned it into the fractional-order case to be named later the fractional-order Langevin equation (FoLE) [19]. Several articles then made this equation up at the top of their works (see, e.g., [20][21][22]). It has been recently shown that such equation, for example, but not limited to, plays notable role in outlining the reaction-diffusion models [23,24] and describes some processes associated with porous media [25,26].
In the same framework, many efforts have been underway to explore the two most main aspects associated with the solutions of the FoLEs: the existence and uniqueness [27]. In view of studying such two aspects, several recent articles have been published. In particular, through using some special boundary conditions, some applicable results related to the existence of solutions for the FoLE have been reported in [28]. Both aspects have been investigated when the coupled incommensurate fractional-order systems were handled by Salem and Alnegga and Salem et al. in [15,29]. However, in more recent times, numerous outcomes associated with those aspects have, moreover, been informed widely (see [1,14,21,[30][31][32][33][34]). In more concrete terms, some useful theoretical tools, like the so-called Banach contraction principle and Krasnoselskii's fixed point theorem, have been employed to explore the existence and uniqueness of solutions for several nonlinear FODEs such as the implicit FODEs [35], the fractional-order Cauchy problems [36].
As per the best of our knowledge, there are few published works that address the FoLE with nonlocal integral boundary conditions as informed in [14]. In view of this motivation, this work examines the existence and uniqueness of solution for the nonlinear FoLE with three-point boundary conditions involving two nonlocal integrals. Actually, this boundary value problem, with two different values of fractional order, has the following form: where c D r is the Caputo derivative operator of order r in which k − 1 < r ≤ k and k ∈ ℕ. In this work, we assume that the function f : ½0, T × ℝ × ℝ → ℝ is a continuous function, and 1 < r < 2, 0 < ρ ≤ s ≤ 1, and ℓ ∈ ℝ are the dissipative parameters. We take here the function f , in general form, constitutes from position (displacement) zðtÞ and fractional velocity c D ρ zðtÞof the particle at time t ∈ ½0, T, T > 0. This function perhaps contains external force field, positiondependent phenomenological fluid friction coefficient, intensity of the stochastic force, or zero-mean Gaussian white noise term. The equation (1) subjects to the following boundary conditions: where α, β ∈ ℝ and 0 < ζ < T. It is remarkable to see that the first condition indicates the particle begins its motion from the origin point. Also, all conditions can be interpreted informally as "the displacement of the particle at a certain point proportionates the signed area of the region in the tz-plane that is bounded by the graph of z and the t-axis from the origin to this certain point". One more observation should be recorded; all hypotheses and conditions in this work are assumed on the basis of the various views inspired by the aforementioned cited articles. Anyhow, here is how the rest of this paper is arranged. Section 2 introduces some primary preliminaries associated with fractional calculus and some fixed point theorems, while Section 3 deduces some auxiliary lemmas that pave the way for establishing novel theorems related to the existence and uniqueness of the solution for our main problem. Such theorems are illustrated and shown in Section 4, while Section 5 involves one illustrative example to verify all findings, followed by the last section that summarizes the achievements of the whole work.

Preliminaries
In the whole of this work, the Caputo derivative operator and its inverse operator, the Riemann-Liouville (R-L) integral, are adopted since they suite most physical systems. These two operators as well as some basic concepts and essential preliminaries about fractional calculus are briefly presented here together with some theoretical results associated with the fixed point theorem. Definition 1. [37]. The R-L integral operator of order r for a continuous function z : ½0,∞Þ → ℝ is outlined as where r > 0.
Definition 2. [37]. The R-L derivative operator of order r for a k -times absolutely continuous function z : ½0,∞Þ → ℝ is outlined as where k − 1 ≤ r < k and k ∈ ℕ.
Definition 3. [37]. The Caputo derivative operator of order r for a k-times absolutely continuous function z : ½0,∞Þ → ℝ is outlined as where k − 1 < r ≤ k and k ∈ ℕ.
The following remark states some important relations chosen from the famous books [37]. Actually, such relations will be used during the whole of this work.
Remark 4. Suppose that z : ½0,∞Þ → ℝ is a continuous function, r is a positive real. Then, (i) I r I s zðtÞ = I r+s zðtÞ, s > 0 (ii) c D r I s zðtÞ = I s−r zðtÞ, for r ∈ ½0, s, while c D r I r zðtÞ = z ðtÞ (iii) Lemma 5. Suppose c D r z is a continuous function, then

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Next, some primary keys and essential theorems related to the fixed point theorem are stated for the purpose of getting novel results. Definition 6. [38,39]. A contraction of a normed space X is a mapping S : X → X that meets for some δ < 1.

Theorem 7. (Banach fixed point theorem, Granas and
Dugundji [38,39]). There is a unique fixed point for every contraction mapping on a complete metric space.
Theorem 8. (Krasnoselskii fixed point theorem [38]). For a convex set B ε of a Banach space X and a mapping Sz = S 1 z + S 2 z such that (2) S 2 represents a contraction mapping

(3) S 1 is continuous and compact
Then, S has a fixed point. (2) There exists γ ∈ ð0, 1Þ and z * ∈ ∂O such that γSðzÞ = z, where ∂O is the boundary of O itself

Some Deduced Auxiliary Lemmas
In this part, some auxiliary lemmas are introduced in order to pave the way for exploring the existence and uniqueness of the solution for our main problem. In other words, such lemmas will play a major role in transforming the main issue to a fixed point problem. For this purpose, let us introduce the following two results, with noting that the function f ðt, zðtÞ, c D ρ zðtÞÞ given in (1) will be indicated by f ðtÞ for simplicity.
Lemma 10. Let f ∈ Cð½0, T, ℝÞ. Then, the unique root of the problem given in (1)-(2) can be expressed as where Proof. Applying the integral operator I r to both sides of (1) and then using Lemma 5. yield Again, applying I r to both sides of (12) gives where c i ' s are arbitrary constants, and i = 1, 2, 3. Applying the first condition of (2) yields c 0 = 0, and hence By using the two other conditions given in (2), we obtain the following: Solving the two equations (15) and (16) gives the following: Substituting (17) and (18) into (14) yields the result. Conversely, it is easy by aiding with Remark 4. and Lemma 5. to see that the solution (8) satisfies the fractional differential equation (1) and the boundary conditions in (2). Lemma 11. Let χ 1 and χ 2 be defined as given in (9) and (10), respectively. Then, Proof. Let t, ζ ∈ ½0, T. Define the function uðtÞ as follows:

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One observes that when 0 ≤ t ≤ ζ, we have u ′ ðtÞ = t s−1 ðs ζ − ðs + 1ÞtÞ, which means that uðtÞ is increasing on ð0, t 0 Þ and it is decreasing on ðt 0 , ζÞ, where t 0 = sζ/ðs + 1Þ. Hence, we can deduce that uðtÞ has a maximum value at t = t 0 . On the other hand, when ζ < t ≤ T, we obtain u ′ ðtÞ = t s−1 ððs + 1 Þt − sζÞ, which implies that uðtÞ is increasing on ðζ, TÞ, and so uðtÞ has a maximum value at t = T. Therefore, the result holds due to the following: In similar manner, one can deduce the second statement of this lemma.
To follow up on the progress made in exploring the existence and uniqueness of solutions for the FoLEs, some further results are established below, which will surely serve us in the next sections. In this regard, let C = Cð½0, T, ℝÞ be Banach space of continuous real-valued functions equipped with the usual supremum norm Define the following set One can easily verify that P is a Banach subspace equipped with the following norm: In view of Lemma 10., the main problem given in (1)-(2) will be transformed into its equivalent fixed point problem. In other words, the function zðtÞ should satisfy the two statements; SzðtÞ = zðtÞ and c D ρ SzðtÞ= c D ρ zðtÞ, for all t ∈ ½0, T, where S : C → C is an operator defined on the Banach space C. In order to achieve this goal, consider the operator S as defined a little while ago. That is, where Λ α ð·Þ is defined in (11) and Next, we introduce the following auxiliary result based on all provided aforementioned principles.

Existence and Uniqueness Results
This section proposes some theoretical results related to the existence and uniqueness of the solution for the main target problem, the FoLE given in (1)- (2). First of all, we establish some mathematical assumptions that will be used from now on. Such assumptions are given below. where Advances in Mathematical Physics and χ i , i = 1, 2, 3, 4 are given in (19), (20), (30), and (31), respectively.
In the same vein, we also introduce some further conditions below to establish our proposed results. In fact, these conditions have been recently introduced in [40] for the purpose of investigating the same issue of ours, but they were adopted for handling another problem.
H 1 The function f ðt, zðtÞ, c D ρ zðtÞÞ is continuous on ½0, T H 2 There exist a positive function ψ : H 4 There exist a positive function h : ½0, T → ℝ + and other nondecreasing function Ψ : On the other side, by going back to (26) and (27), one can rewrite each of SzðtÞ and c D ρ SzðtÞ, as a sum of two operators S 1 and S 2 . In other words, SzðtÞ = ðS 1 z + S 2 zÞðtÞ and c D ρ Szð tÞ = ð c D ρ S 1 z+ c D ρ S 2 zÞðtÞ, where After this preparation, we find ourselves ready to present the results that are at the heart of the matter. Here is the first one that deduced on the basis of the Krasnoselskii theorem.
Proof. To prove this theorem, define the closed ball, B ε , as follows: where P and kzk P are defined in Section 3, with radius where Δð·Þ and Δ ′ ð·Þ are defined in (38) and (39), respectively. One can easily verify that B ε is a convex closed set. However, for the purpose of implementing on the Krasnoselskii theorem given in 2.2, we need firstly to show that S 1 z 1 + S 2 z 2 ∈ B ε , for each z 1 ðtÞ, z 2 ðtÞ ∈ B ε . To carry out this step, we begin with Due to z 2 ðtÞ ∈ B ε , then |z 2 ðtÞ | ≤kz 2 ðtÞk ≤ kz 2 ðtÞk P ≤ ε. This matter implies to Thus, the last inequality implies kS 1 z 1 + S 2 z 2 k ≤ ε. In a similar manner, we can show that which implies that

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These conclude that which means that S 1 z 1 + S 2 z 2 ∈ B ε for each z 1 ðtÞ, z 2 ðtÞ ∈ B ε . Thus, the first part of Krasnoselskii Theorem 8 has been shown. The next step is to show that the operator S 2 is contraction. For this purpose, we estimate |S 2 z 1 − S 2 z 2 | as follows: Similarly, we can prove that These lead to which implies that S 2 is contraction if max fΔ, Δ ′ g < 1. At the same time, as the function f is assumed to be continuous according to the assumption H 1 , the operator S 1 is also continuous on B ε . To complete the proof of this theorem, it remains to show that S 1 is compact. In regard to Arzela-Ascoli theorem, it is enough to show that S 1 B ε is uniformly bounded and equicontinuous. For the boundedness, one can easily verify that S 1 z satisfies the following perspective.
In the meantime, to show that S 1 is an equicontinuous operator, let t 1 , t 2 ∈ ½0, T such that t 1 < t 2 . Then, we have Again, using condition H 2 to turn the inequality above to be in the form Consequently, the above inequality leads us to another one of the form Observe that as t 2 → t 1 , the right-hand side goes to zero uniformly. This means that such side does not depend never on z. Therefore, as per Arzela-Ascoli theorem, the operator S 1 is equicontinuous. Hence, in view of Krasnoselskii theorem, we conclude that (1)-(2) has at least one solution.
In the next argument, we will investigate the uniqueness of solution for the problem under consideration. Of course, Banach fixed point theorem, reported in Theorem 7, will be utilized in this investigation. However, the following result is established for achieving that purpose.
Then, the problem given in (1)-(2) has a unique solution on ½0, T. 6 Advances in Mathematical Physics Proof. Consider the set B ε as given in (45) with where W = sup 0t T | f ðt, 0, 0Þ | . For each t ∈ ½0, T and z ∈ B ε , one might estimate the term |f ðt, zðtÞ, c D ρ zðtÞÞ | as follows: The next step is now to show that SB ε ⊆ B ε , ∀z ∈ B ε . Based on the arguments given in the previous theorem, one can obtain the following assertions.
Applying the norm over t gives Therefore, Now, in accordance to Theorem 7, we need to show that the mapping S is contraction. In order to achieve this goal, we should notice that the operator S 2 is contraction as shown in the previous theorem. Hence, it remains to verify that S 1 is also contraction. For this purpose, let z 1 , z 2 ∈ B ε , t ∈ ½0, T and consider |S 1 z 1 ðtÞ − S 1 z 2 ðtÞ | as follows: