On a Langevin equation involving Caputo fractional proportional derivatives with respect to another function

: In this work, we introduce and study a class of Langevin equation with nonlocal boundary conditions governed by a Caputo fractional order proportional derivatives of an unknown function with respect to another function. The qualitative results concerning the given problem are obtained with the aid of the lower regularized incomplete Gamma function and applying the standard ﬁxed point theorems. In order to homologate the theoretical results we obtained, we present two examples.


Introduction
The classical calculus connected to the traditional integrals and derivatives is considered to be the core of modern mathematics. The fractional calculus is the generalization of this calculus as it deals with the integrals and derivatives of any order. There has been a great deal of interest in such type of generalizing calculus because of the findings obtained by some researchers who utilized the fractional integrals and derivatives being at the receiving end of modeling some real world problems that arise in variety of disciplines [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. What makes the fractional calculus distinctive is the fact there are variety of fractional integrals and derivatives and thus a researcher can choose the best fractional operator which suited to the problem under investigation. Moreover, there are two kinds of fractional operators. The first type consist of non-local fractional operators. The second type contains local ones. The local fractional derivatives were initiated first by Khalil et al. [16,17]. The derivatives proposed in these two works were modified by [18,19]. The modified derivative was used by Jarad et al. [20] to generate a new class of fractional operators called fractional proportional operators which contain two parameters and give rise to known fractional operators when one of these parameters tend to certain values. And even more, these operators were generalized in [21,22] and fractional proportional operators with respect to an increasing function were proposed.
The Langevin equation embodying integer order derivative was proposed by Langevin in 1908 [23]. This well known equation delineates the evolution of certain physical phenomena in fluctuating environments [24] and describes anomalous transport [25]. It was extended to the fractional order by Lim et al. [26] who proposed a version of Langevin equations involving two fractional order for the sake of depicturing the viscoelastic anomalous diffusion in the complex liquids. In [27], the authors considered a generalized Langevin equation that lims mechanical random forces. Lozinski et al. [28] considered applications of the mentioned equation in polymer rheology and stochastic simulation. In [29], Laadjal et al. discussed some qualitative properties of solutions to multi-term fractional Langevin equation with boundary conditions.
Recently, Laadjal et al. [30] have studied the existence and uniqueness of solutions to fractional proportional differential equation with the help of incomplete Gamma function.
Motivated and inspired by the aforementioned works, in this article, we deliberate the existence and uniqueness of solutions to the following class of Langevin differential equations: denotes the Caputo fractional proportional derivative (CFPD) with respect to the function v of order i (i = α, β).
Note that from Eq (1.1), we have the following special cases (with the nonlocal boundary conditions (1.2)): Case 2. If ρ = 1, Eq (1.1) reduces to a Langevin equation involving two v-Caputo fractional derivatives Case 3. If ρ = 1 and v(t) = t, Eq (1.1) reduces to a Langevin equation involving the usual Caputo fractional derivatives Moreover, other several special cases can be obtained as well.

Preliminaries
In this section, we present some definitions, propositions, lemmas and theorems needed through the whole article.
Let ρ ∈ (0, 1] and v be strictly increasing continuously differentiable function. The Reimann-Liouville fractional proportional integral (RLFPI) of the function ψ ∈ L 1 [a, b] with respect to the function v is defined by [20] (2.8) Let ρ ∈ (0, 1]. The Caputo fractional proportional derivative (CFPD) of the function ψ ∈ C (n) [a, b] with respect to the function v ∈ C (n) [a, b] is defined by [20] Let ρ ∈ (0, 1]. The Reimann-Liouville fractional proportional derivative (RLFPD) of the function ψ with respect to the function v is defined by [20] (2.12) Remark 6. Note that, for ρ = 1 and v(t) = t, the definitions of the RLFPD and CFPD reduce to the usual definitions of Riemann-Liouville fractional derivative and Caputo fractional derivative, respetively. On other hand note that R Proposition 7 ( [20]). Let ρ ∈ (0, 1], β > 0 and θ > 0 with n − 1 < θ ≤ n, and ψ ∈ L 1 [a, b], we have the following properties: 34,35]). Let θ ∈ C ( (θ) > 0), we have the following definitions: The upper incomplete Gamma function is defined by (2.20) The lower incomplete Gamma function is defined by The upper regularized incomplete Gamma function is defined by The lower regularized incomplete Gamma function is defined by The functions P and Q are also called "Incomplete Gamma functions ratios".

Incomplete Gamma functions vs RLFPIs with respect to another function
In this section, we present new essential lemmas related to the incomplete Gamma functions. These lemmas will be helpful in proving our main results about the existence and uniqueness of solutions for the considered problem.
Remark 12. In all the following results, we assume that v : [a, b] −→ R is a continuous, differentiable and strictly increasing function.
where function P is defined by (2.23). Moreover, Proof. For ρ ∈ (0, 1), from Definition 2.8, we have For ρ = 1 we have Concerning the limit formula (3.2) , we have Finally, formula (3.3) is immediate and hence the proof is completed.
, and δ > 0. Then where the function P is given by (2.23).
Proof. The proof can be accomplished by trailing the same steps as in Lemma 3.3 of [30] and Lemma 13.

9)
Proof. To calculate the above limit, the sign of the term inside the absolute value must be studied. From Remark 12, v (τ) > 0 for all τ ∈ [a, b], and thus for any s 1 , s 2 ∈ [a, b] such that s 2 > s 1 , we have v(s 2 ) > v(s 1 ).
For ρ = 1, we look at the three cases δ = 1, δ < 1 and δ > 1 as follows hence the integral has the value zero as t 2 → t 1 .
Thus, we conclude that The proof is completed.

Equivalence of problem (1.1) and (1.2) to an integral equation
In this section, we prove the equivalence of the considered boundary value problem to an equation involving fractional proportional integral. In all the following results, we assume that: with the nonlocal boundary conditions (1.2) and the solution of the following integral equation are equivalent.
Proof. Applying the operator J α,ρ,v a to both sides of Eq (4.1) and using the first property of Propostion 8, we get Next, applying the operator J β,ρ,v a on both sides of the previous equation yields so, (4.4) From the boundary condition x(a) = 0, we get c 0 = 0. Now, using the boundary condition x(b) = ξx(η), we obtain Substituting the values of c 0 and c 0 in (4.4) we obtain formula (4.2). Now, to prove the other way, it is enough to replace t by a and b to get the boundary conditions (1.2) and to obtain (4.1) it is adequate to apply operators C p D β,ρ,v a and C p D α,ρ,v a consecutively to both sides of (4.2).

Uniqueness result
In this section we hold out the uniqueness of solutions to problem (1.1) and (1.
and we define the constants We should remark that the fixed point of operator T is the solution of the integral Eq (4.4) and consequently the solution of problem (1.1) and (1.2).
Theorem 18. Let ρ ∈ (0, 1) and assume that f : [a, b] × R → R be a continuous function satisfying the assumption: Using (H 1 ) and Lemma 14 we get After simplifications, we reach that where S α+β and S β are given by (5.2). Thus we obtain T B r ⊂ B r .
Next, we prove that the operator T is a contraction mapping. For x, y ∈ X, for all t ∈ [a, b] we have From (H 1 ) and Lemma 14 we get Then, after simplifications, we conclude that which on taking the norm for t ∈ [a, b] produces From the condition (5.3) the operator T is a contraction. Hence, by Banach fixed point theorem the problem (1.1) and (1.2) has a unique solution on [a, b]. The proof is completed.

Existence result
In this section, by using Leray-Schauder alternative fixed point theorem [36], we present the following result about the existence of the solutions for the given problem.
Consider the following hypothesis: (H 2 ) f : [a, b] × R → R are continuous functions and there exist a real positive constants ς 0 and ς 1 such that Theorem 19. Let ρ ∈ (0, 1) and assume that (H 2 ) holds. If Proof. We first show that the operator T is completely continuous. It is clear that the continuity of f implies the continuity of the operator T . Now, let Υ be any nonempty bounded subset of X. Then, there exists N > 0 such that for any x ∈ Υ, x ≤ N. Notice that from condition (H 2 ) for all x ∈ Υ we have Next we prove that T (Υ) is uniformly bounded. Let x ∈ Υ. Then, for any t ∈ [a, b] we have Benefiting from (H 1 ) and Lemma 14 we notch up that Consequently, x < +∞ for any x ∈ Υ. Therefore, T (Υ) is uniformly bounded. Now, we shadow forth the equicontinuity of T on Υ. Let x ∈ Υ.
Taking the advantage of the relation where the function V δ (here δ = β, α + β) is given by (3.9). Thus, from Lemma 15 Then, by making use of Lemma 16, we achieve lim Thus, the operator T is equicontinuous. Hence, by Arzela-Ascoli theorem, we deduce that the operator T is completely continuous.
Finally, we will verify that the set Φ(T ) = {x ∈ X : x = mT x for some 0 < m < 1} is bounded.
For all x ∈ Φ(T ), and for any t ∈ [a, b], we have Then, we obtain the following after simplifications This brings forth to which proves that Φ(T ) is bounded. Thus, by Leray-Schauder alternative theorem, the operator T has at least one fixed point. Hence, the initial value problem (1.1) and (1.2) has at least one solution on [a, b]. The proof is completed.

Special cases
In this section, we elaborate some special cases. From Lemma (13) resprectively.

Applications
In this section, we bring in two examples in order to corroborate our theoretical results.

Conclusions
In this article, we discussed the existence and uniqueness of solutions to a certain type of Langevin equation subject to nonlocal boundary conditions with the assistance of the lower regularized incomplete Gamma function. The derivative involved in this type of Langevin equation is the generalized Caputo propotional fractional derivative which encloses many of the known fractional derivatives. To the best of our knowledge, this article is the first to handle the existence and uniqueness of solutions to differential equations in the frame of such generalized fractional derivatives of a function with respect to another function.