Explicit iteration and unique solution for φ-Hilfer type fractional Langevin equations

This paper proves that the monotone iterative method is an effective method to find the approximate solution of fractional nonlinear Langevin equation involving φ-Hilfer fractional derivative with multi-point boundary conditions. First, we apply a approach based on the properties of the MittagLeffler function to derive the formula of explicit solutions for the proposed problem. Next, by using the fixed point technique and some properties of Mittag-Leffler functions, we establish the sufficient conditions of existence of a unique solution for the considered problem. Moreover, we discuss the lower and upper explicit monotone iterative sequences that converge to the extremal solution by using the monotone iterative method. Finally, we construct a pertinent example that includes some graphics to show the applicability of our results.


Introduction
Fractional differential equations (FDEs) have a profound physical background and rich theoretical connotations and have been particularly eye-catching in recent years. Fractional order differential equations refer to equations that contain fractional derivatives or integrals. Currently, fractional derivatives and integrals have a wide range of applications in many disciplines such as physics, biology, and chemistry, etc. For more information see [1][2][3][4][5].
Langevin equation is an important tool of many areas such as mathematical physics, protein dynamics [6], deuteron-cluster dynamics, and described anomalous diffusion [7]. In 1908, Langevin established first the Langevin equation with a view to describe the advancement of physical phenomena in fluctuating conditions [8]. Some evolution processes are characterized by the fact that they change of state abruptly at certain moments of time. These perturbations are short-term in comparison with the duration of the processes. So, the Langevin equations are a suitable tool to describe such problems. Besides the intensive improvement of fractional derivatives, the Langevin (FDEs) have been presented in 1990 by Mainardi and Pironi [9], which was trailed by numerous works interested in some properties of solutions like existence and uniqueness for Langevin FDEs [10][11][12][13][14][15][16][17][18][19]. We also refer here to some recent works that deal with a qualitative analysis of such problems, including the generalized Hilfer operator, see [20][21][22][23][24]. Recent works related to our work were done by [25][26][27][28][29][30]. The monotone iterative technique is one of the important techniques used to obtain explicit solutions for some differential equations. For more details about the monotone iterative technique, we refer the reader to the classical monographs [31,32].
• We use the monotone iterative method to study the extremal of solutions of φ-Hilfer-FLE (1.1).
• We investigate the lower and upper explicit monotone iterative sequences that converge to the extremal solution.
• The results obtained in this work includes the results of Fazli et al. [26], Wang et al. [27] and cover many problems which do not study yet.
The structure of our paper is as follows: In the second section, we present some notations, auxiliary lemmas and some basic definitions which are used throughout the paper. Moreover, we derive the formula of the explicit solution for FLE (1.1) in the term of Mittag-Leffler with two parameters. In the third section, we discuss the existence of extremal solutions to our FLE (1.1) and prove lower and upper explicit monotone iterative sequences which converge to the extremal solution. In the fourth section, we provide a numerical example to illustrate the validity of our results. The concluding remarks will be given in the last section.

Auxiliary notions
To achieve our main purpose, we present here some definitions and basic auxiliary results that are required throughout our paper. Let J := [0, b] , and C (J) be the Banach space of continuous functions υ : J → R equipped with the norm υ = sup{|υ(κ)| : κ ∈ J}.
[2] Let f be an integrable function and µ > 0. Also, let φ be an increasing and positive monotone function on (0, b), having a continuous derivative φ on (0, b) such that φ (κ) 0, for all κ ∈ J. Then the φ-Riemann-Liouville fractional integral of f of order µ is defined by Definition 2.2.
For some analysis techniques, we will suffice with indication to the classical Banach contraction principle (see [35]).
To transform the φ-Hilfer type FLE (1.1) into a fixed point problem, we will present the following Lemma.
Lemma 2.8. Let γ j = µ j + jβ j − µ j β j , ( j = 1, 2) such that µ 1 ∈ (0, 1] , µ 2 ∈ (1, 2] , β j ∈ [0, 1] , λ 1 , λ 2 ≥ 0 and is a function in the space C (J). Then, υ is a solution of the φ-Hilfer linear FLE of the form if and only if υ satisfies the following equation Then, the problem (2.1) is equivalent to the following problem Applying the operator I µ 1 ,φ 0 + to both sides of the first equation of (2.4) and using Lemma 2.4, we obtain where c 0 is an arbitrary constant. For explicit solutions of Eq (2.4), we use the method of successive approximations, that is and By Definition 2.1 and Lemma 2.3 along with Eq (2.6), we obtain Similarly, by using Eqs (2.6)-(2.8), we get Repeating this process, we get P k (κ) as Taking the limit k → ∞, we obtain the expression for P k (κ), that is Changing the summation index in the last expression, i → i + 1, we have From the definition of Mittag-Leffler function, we get By the condition P(0) = 0, we get c 0 = 0 and hence Equation (2.9) reduces to Similarly, the following equation By the condition υ(0) = 0, we obtain c 2 = 0 and hence Eq (2.11) reduces to Put c 0 in Eq (2.12), we obtain (2.14) Substituting Eq (2.10a) into Eq (2.14), we can get Eq (2.2).
On the other hand, we assume that the solution υ satisfies Eq (2.2). Then, one can get υ(0) = 0. Applying H D µ 2 ,β 2 ;φ 0 + on both sides of Eq (2.2), we get By using some properties of Mittag-Leffler function and taking κ = 0, we obtain Thus, the derivative condition is satisfied. The proof of Lemma 2.8 is completed.
As a result of Lemma 2.8, we have the following Lemma.
Step (1): Setting υ 0 = υ and υ 0 = υ, then given υ j ∞ j=0 and υ j ∞ j=0 inductively define υ j+1 and υ j+1 to be the unique solutions of the following problem (3.2) By Theorem 3.3, we know that the above problems have a unique solutions in C (J).
Corollary 3.5. Assume that f : J × R + → R + is continuous, and there exist ℵ 1 , ℵ 2 > 0 such that Then the problem (1.1) has at least one solution υ(κ) ∈ C (J) . Moreover and Proof. From Eq (3.6) and definition of control functions, we get Now, we consider the following problem (3.10) In view of Lemma 2.8, the problem (3.10) has a solution Taking into account Eq (3.9), we obtain It is obvious that υ(κ) is the upper solution of problem (1.1). Also, we consider the following problem (3.11) In view of Lemma 2.8, the problem (3.11) has a solution Taking into account Eq (3.9), we obtain Thus, υ(κ) is the lower solution of problem (1.1). The application of Theorem 3.4 results that problem (1.1) has at least one solution υ(κ) ∈ C (J) that satisfies the inequalities (3.7) and (3.8).

An example
Example 4.1. Let us consider the following problem

Conclusions
In this work, we have proved successfully the monotone iterative method is an effective method to study FLEs in the frame of φ-Hilfer fractional derivative with multi-point boundary conditions. Firstly, the formula of explicit solution of φ-Hilfer type FLE (1.1) in the term of Mittag-Leffler function has been derived. Next, we have investigated the lower and upper explicit monotone iterative sequences and proved that converge to the extremal solution of boundary value problems with multi-point boundary conditions. Finally, a numerical example has been given in order to illustrate the validity of our results.
Furthermore, it will be very important to study the present problem in this article regarding the Mittag-Leffler power low [36], the generalized Mittag-Leffler power low with another function [37,38], and the fractal-fractional operators [39].