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Langevin Equations with Generalized Proportional Hadamard–Caputo Fractional Derivative
We look at fractional Langevin equations (FLEs) with generalized proportional Hadamard–Caputo derivative of different orders. Moreover, nonlocal integrals and nonperiodic boundary conditions are considered in this paper. For the proposed equations, the Hyres–Ulam (HU) stability, existence, and uniqueness (EU) of the solution are defined and investigated. In implementing our results, we rely on two important theories that are Krasnoselskii fixed point theorem and Banach contraction principle. Also, an application example is given to bolster the accuracy of the acquired results.
In recent years, fractional calculus has gained great importance by numerous renowned mathematicians. The most essential feature of this topic is that it allows us to execute integrations and differentiations in any order, not necessarily integer ones Such a benefit has been encouraged by the applications in various areas, conceivably including fractal phenomena which appear in many sciences such as physics and engineering. Also, discovering the fractional derivatives and their generalizations was done by well-known mathematicians such as Riemann, Caputo, Hadamard, Euler, Liouville, Laplace, Laurent, and Fourier and was constantly the major path of research in the fractional calculus area. Moreover, these derivatives will provide us new chances to get generalized solutions of fractional differential equations [1–6]. It is worth to mention that the most important comprehensive treatments of differential equations with fractional order were initiated by Caputo  in 1969, and it was later called Caputo’s derivative. There are many generalizations and modifications of this derivative [8–12], for example, the authors in  used Caputo’s derivative to modify the Hadamard derivatives to a more beneficial concept that called Hadamard–Caputo derivatives (HCD). In 2019, Rahman et al.  introduced an integral form of Hadamard fractional derivatives which give generalized forms than the HCD and have shorted as GHCD. Many authors used HCD and their generalizations to study the existence, uniqueness, and stability of fractional differential equations [1, 6, 13, 14]. The researchers in  utilized the HCDs to present some results on the existence and stability for solutions of fractional Langevin equations with some conditions related to nonperiodic type boundary and nonlocal integral. Recently, Devi et al.  used two fixed point theorems due to Krasnoselskii and Banach as well as the HCDs of distinctive orders connected with nonlocal integral to establish the existence, uniqueness, and HU stability of solutions for fractional Langevin equations with nonperiodic boundary conditions.
According to varied literature, like [15–17], it has verified that the FLEs are perfectly mathematical models for single-file prevalence and the conduct of unshackled particles driven by internal noises. This motivates us to examine the solutions of the FLEs and their features. So, in the present work, we use the generalized proportional Hadamard–Caputo derivative to investigate the FLEs with nonlocal integrals and nonperiodic boundary conditions. For the proposed equations, the HU stability and EU of the solution are defined and analyzed. In implementing our results, we relied on two main fixed point theorems, namely, Krasnoselskii’s theorem and Banach contraction operator. Furthermore, to reinforce the accuracy of the gained results, an application example is presented with adequate values for the parameters.
Our paper arranged as follows. In Section 2, we present all basic concepts related to the HCDs and their generalizations with some previous results which serve our results. Section 3 devoted to present our main results which in turn was divided into two subsections, namely, 3.1, introduced to the existence and uniqueness of the solutions of the FLEs, and 3.2, prepared to study the stability results for the solutions of the FLEs.
We use the set to refer to all absolutely continuous functions such that its derivative of order is absolutely continuous on [1, e], where and is the renowned Euler’s number.
Definition 1 (see ). If is both integrable and continuous, the one side fractional integral of Hadamard of -order is given by
Definition 2 (see ). The -order Hadamard fractional derivative is given bywhere and .
Definition 3 (see ). The -order fractional HCD is given by
Rahman et al.  recently defined a generalized proportional Hadamard fractional integral of order q.
Definition 4 (see ). The -order generalized proportional Hadamard fractional integral is given bywhere and .
Finally, Jarad et al.  presented a broader range of fractional proportional integrals and derivatives.
Definition 5 (see ). The -order of a broader range of the one side fractional proportional integrals is given bywhere , , and .
Definition 6 (see ). The -order of the one side of a broader range of fractional proportional derivative is given bywhere , , and .
Definition 7 (see ). The -order of the one side of Caputo-broader range of fractional proportional derivative is given bywhere , , and .
Remark 1. Obviously, if we put in Definitions 5 and 7, we obtain the one side generalized proportional Caputo–Hadamard fractional integral and derivative, respectively.
By using Proposition 3.1, Proposition 4.1, and Remark 1.1 in , we obtain the following lemma.
Lemma 1 (see ). Let such that and . Then, for any and , we have(i)(ii)
The following theorem is essential in implementing our results.
Theorem 1 (see ). Let be a nonempty convex, bounded, and closed subset of a Banach space . Consider are two operators from to such that(i) for each (ii) is continuous and compact(iii) satisfies a contraction conditionThen, such that .
3. Main Results
This section comprises our new results for the fractional Langevin equations with generalized proportional Hadamard–Caputo derivative. By using Krasnoselskii and Banach fixed point theorems, we investigate the existence, uniqueness, and HU stability for these FLEs. Before we can show our results, we have to prove the following helpful lemmas.
Lemma 2. Let such that , , , and . Then,
Also, the fractional differential equationhas a solutionwhere .
Proof. Also,By applying equation (11) in (12), we havewhere the replacing of the integer by has been modified. Hence, the solution of the FDE (9) is given by
Lemma 3. The solution of the following fractional Langevin equation in the integral formis given by
Proof. Applying the generalized proportional Hadamard fractional integral to equation (15), we getRepeating integration by using generalized proportional Hadamard fractional integral operator of order , we obtainSinceThen, by substituting in (18) from (19) and (20), we getwhere , and are real constants.
By using the boundary condition, we get the following.
From , , and .
Therefore,From the conditions,By putting in (17), we getAlso, by putting in (17) and (22) and from equation (24), we haveSubstituting in (22) from equations (24) and (25), we obtain
Suppose that the space is a Bananch space and its norm is defined by such that, for each , it implies that is continuous.
Let us start by defining an operator in order for
It should be noted that if the operator has a fixed point, the solution of equation (15) exists.
3.1. Existence Result
We now introduce the following conditions to confirm our existence result with the support of fixed point procedure.
Theorem 2. Let us imagine that the continuous function satisfying the following inferences:(I)(II)Then, the FLEs (15) has a minimum of one solution on ifwhere
Proof. Choose , whereDefine the closed ball .
By combining the operators which are described as follows:Keep this in mind on .
First, we will show that , for each .Taking advantage of the , for , and from , we getHence, .
Second, we will demonstrate that is a contraction operator. Let be two elements in Banach space such thatTaking advantage of the for , and from , we getHence, is a contraction map, as implied by assumption that .
Finally, we shall demonstrate that is continuous and compact.
To begin with, since is a continuous function on , operator is also continuous.
must be uniformly bounded and equicontinuous on in order to be seen to be compact.This proves that is uniformly bounded.
The compactness of operator is then demonstrated. For each , we get