Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

Salem, Ahmed; Mshary, Noorah; El-Shahed, Moustafa; Alzahrani, Faris

doi:https://doi.org/10.1155/2021/9995969

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9995969 | https://doi.org/10.1155/2021/9995969

Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

Ahmed Salem,¹Noorah Mshary,^1,2Moustafa El-Shahed,³and Faris Alzahrani¹

Academic Editor: Shahram Rezapour

Received03 Jul 2021

Revised21 Jul 2021

Accepted08 Nov 2021

Published23 Nov 2021

Abstract

In this work, through using the Caputo–Hadamard fractional derivative operator with three nonlocal Hadamard fractional integral boundary conditions, a new type of the fractional-order Sturm–Liouville and Langevin problem is introduced. The existence of solutions for this nonlinear boundary value problem is theoretically investigated based on the Krasnoselskii in the compact case and Darbo fixed point theorems in the noncompact case with aiding the Kuratowski’s measure of noncompactness. To demonstrate the applicability and validity of the main gained findings, some numerical examples are included.

1. Introduction

Over the past few years, the fractional calculus (which is basically an expansion of the traditional calculus) has provided its remarkable contribution in addressing lots of physical and biological phenomena as well as describing many engineering dilemmas [1]. It has been confirmed in several literature studies that many real-world problems associated with many applied science fields can be described more convenient using the fractional-order differential equations (FoDEs) rather than that of using the ordinary differential equations (ODEs) [2]. In particular, the FoDEs played and still play a significant role in developing lots of models that outline several engineering problems and physical applications such as electromagnetics, porous media, control, and viscoelasticity [3]. For further facts on the basic principles of fractional calculus and the FoDEs, the reader may return to the references [4–7]. The fractional calculus, new interesting research field, is attracting the interest of mathematicians and researchers. With fractional operators, many real-world problems are being investigated, in particular the presenting of mathematical model for the transmission of COVID-19 and mathematical models of HIV/AIDS and drug addiction in prisons [8].

The so-called fractional-order Langevin equation (FoLE) (which was first established in [9]) is considered as a practical tool employed in describing some physical phenomena’s evolution [10]. For instance, some operations related to the porous media and the reaction-diffusion can be outlined using this type of equations (see [11–13]). From these motivations, we find that exploring the existence of solutions for such equation will further enrich our understanding and knowledge about this subject. Actually, lots of results related to the solution’s existence and uniqueness of the FoLE (in view of some well-known operators such as the Riemann–Liouville and Caputo operators) were extensively deduced through several research papers (see, e.g., [14–17]). For the purpose of studying such kind of problems, several strategies and schemes could be implemented such as the coincidence theory [18], the upper and lower solution method [19, 20], the fixed point theorems [21, 22], and many others. For example, some convenient findings associated with the solution’s existence of the FoLE have been pointed out in [23]. Besides, this aspect together with the solution’s uniqueness aspect of the same equation have been well explored in [24]. However, for further recent outcomes, we refer the reader to papers [25, 26] and references therein.

The Sturm–Liouville operator has several applications in variety fields of science such as engineering and mathematics [27]. The authors in [28] presented an approach to the fractional version of the Sturm–Liouville problem. Kiataramkul et al. [10] combined Sturm–Liouville and Langevin fractional differential equations and they called generalized Sturm–Liouville and Langevin equation (GSLLE). Furthermore, they studied the existence and uniqueness of solutions for GSLLE with antiperiodic boundary conditions.

In 1892, another fractional derivative, which is called later as the Hadamard fractional derivative, was proposed by Hadamard. This derivative is distinguished from the others by its construction that includes a logarithmic function of arbitrary exponent. Latest achievements about this derivative and its corresponding Hadamard FoDEs can be found in [29]. In [30], Jarad et al. proposed the so-called Caputo–Hadamard fractional-order derivative, which is a new modification of the Hadamard fractional derivative operator. This new version has proved its suitability in dealing with some physical interpretable initial conditions [31]. In order to get a full overview about this operator and its properties, we refer the reader to the same reference [30]. From the perspective that asserts, there are limited results in literature that addresses the boundary value problems (BVPs) in view of the Caputo–Hadamard derivative [31].

This paper attempts to present a theoretical study about the existence of solutions for the nonlinear GSLLE,subject to the following three point integral boundary conditions:where is the Caputo–Hadamard derivative operator of order , is the Hadamard integral operator of order , and , , , and . The functions , and are continuous. It is remarkable to point out that when for all , then our problem reduces to the Sturm–Liouville fractional BVP and when and are real constants functions for all , then our problem reduces to the Langevin fractional BVP.

Here, we investigate the existence of solution in two cases: in the compact case by utilizing the well-known fixed point theorem due to Krasnoselskii. In the noncompact case, we apply the Kuratowski’s measure of noncompactness technique through using the Darbo’s fixed point theorems. The underlying idea is there are two measures of noncompactness commonly used; these associate numbers to sets in such a way that compact sets all get the measure 0, and other sets get measures that are bigger according to “how far” they are removed from compactness. There are many contributors who used measure of noncompactness to investigate the existence of solution to FoDE [32–34] and references given therein.

The remaining of this paper is ordered as follows: Section 2 presents briefly some fundamental concepts related to the fractional calculus and fixed point theorems. In Section 3, some mathematical preparations and principles are established to achieve our goals through defining some essential terms and deducing some auxiliary lemmas. Section 4 employs these preparations to attain the main results of this work, which concern with exploring the existence of solution for the GSLLE. Section 5 illustrates two numerical examples followed by Section 6 that outlines the major points and concluded remarks.

2. Preliminaries

This part aims to present some fundamental fixed point theorems and essential preliminaries related to the fractional calculus, especially in regard to the Hadamard fractional-order operators.

Definition 1 (see [29]). The Hadamard integral operator of the fractional-order value for a function is outlined aswhere .

Definition 2 (see [30]). The Caputo-Hadamard derivative operator of fractional-order for at least -times differentiable function is outlined aswhere , , , and .

Lemma 1 (see [30]). Suppose and such that . Then,

Lemma 2 (see [30]). Let or and such that . Then,where .

Lemma 3 (see [30, 29]). Let , then we have(i),(ii),(iii),(iv),(v),where , and .

For simplicity, we will consider in all aforementioned preliminaries, and consequently and will be denoted by and , respectively. However, some essential fixed point theorems are reported below for completeness.

Definition 3. (see [3]). Suppose that is a Banach space and is the collection of subsets of . The Kuratowski’s measure of noncompactness is a map outlined aswhere

Lemma 4 (see [3]). Suppose that is a Banach space, , , and are bounded sets of , and is the Kuratowski’s measure of noncompactness defined above. Then, we have the following assertions:(i) is compact .(ii), where .(iii), where .(iv), for any .(v)If the map is bounded and equicontinuous, it follows

Lemma 5 (Ascoli–Arzela theorem [3]). The family in has a uniformly-convergent subsequence , if it is uniformly bounded and equicontinuous maps on , and is relatively compact in the Banach space , for any .

Theorem 1 (Krasnoselskii’s fixed point theorem [3, 26]). Suppose that is a closed convex bounded set of a Banach space , and are two operators satisfying(1), for each .(2) is a contraction mapping.(3) is continuous and is a compact subset of .Then, has a fixed point.

Theorem 2 (Darbo’s fixed point theorem [3]). Suppose that is a closed convex bounded set of a Banach space , and is a continuous mapping so thatfor all closed subsets , where . Then, has a fixed point in .

3. Auxiliary Results

This section presents some auxiliary results that will provide a basis for introducing some results associated with the existence of the solutions for the GSLLE.

Lemma 6. The linear BVP given in the followingsubjects to the conditions in (2) which is equivalent to the following fractional-order integral equation:where is a continuous function and the other concepts are defined as in the Introduction section, ,

Proof. Applying the Hadamard fractional-order integral to both sides of (11) yieldsThat is,Again, taking to both sides of the equation above yieldsNow, using the first condition ( given in (2)) yields , and then we haveOn the other hand, using the other two boundary conditions given in (2) leads us to the following two assertions:Solving these two equations gives, respectively, the following two expressions of and :Finally, substituting and in (18) yields the desired result.

Lemma 7. Let and . Then, for the function defined in (13), we can deduce the following assertion:where and .

Proof. Since the function is continuous, then it attains its minimum and maximum which leads towhere are defined in (14). In view of Lemma 3, we getWhence,whereDifferentiation giveswhich means that the function is increasing on and decreasing on where . Thus, for all which implies the desired result.

In this work, the Banach space of all continuous real-valued functions will be considered in which it is equipped with the usual maximum norm:

Define the spaceas a subset of , then one can easily check that represents a Banach subspace equipped with the norm:

In light of Lemma 6, the B.V.P. given in (1) and (2) can be converted into a fixed point problem. This means that the function must hold the two assertions: and for all , where and are the two operators defined on . For achieving this objective, we define the operator as follows:where

In a similar manner in Lemma 7, we can deduce as follows.

Lemma 8. Let and . Then, for the function defined in (13), we can deduce the following assertion:where and .

Lemma 9. Let be defined as in (13), , and . Then, for all , there exists such that whenever .

Proof. Let , and whereThen, from (14) and the fact that , we haveSince , then . Hence,Similarly, we obtainTherefore,which ends the proof.

The previous discussion together with the aforesaid proposed results represent a preparation for the next section that presents the main outcomes of the problem under consideration.

4. The Existence of Solution for the GSLLE

This part intends to introduce some theoretical outcomes associated with the existence of solution for the GSLLE given in (1) and (2). From now on, we will utilize the following appropriate conditions: : the function is continuous. : there exists a positive function such that : there exists a constant such thatfor all and .

For convenience, we definewhere .

4.1. Compactness Case

In this subsection, we assume the compactness of solution and investigate the existence of it through employing the Krasnoselskii’s fixed point theorem to deduce the first result about the existence of solution of the GSLLE.

Theorem 3. Suppose and hold. Ifwhere is defined as in (45), then there is at least one solution to the B.V.P (1) and (2) in .

Proof. For the purpose of proving this theorem, the closed ball is defined in the following manner:where is the radius and assumed so thatwhere is defined as in (44).
One can confirm that is a closed convex bounded set. Actually, this paves the way to implement the Krasnoselskii’s fixed point theorem. To achieve this goal, we intend to prove that , . So, to perform this task, we start with the following assertion:It is known that is positive and increasing on which implies that . Consequently, we haveOn the other hand, we haveNow, from (50) and (51), we obtainSimilarly, we obtainAccordingly, the two inequalities above implyBased on (52) and (54), we can obtain the following assertion:That is,Hence, the first condition given in Theorem 1 is satisfied. For the purpose of showing the second condition of the same theorem which requires to prove that is a contraction mapping, we intend to take into account the term in accordance with the following manner:In other words,Similarly, we can attain the following inequality:Again, applying (58) and (59) into the norm defined above yieldsDue to as we previously assumed, then is indeed contraction mapping.
In the meantime, to address the last condition of Theorem 1, we should first note that is a continuous operator on due to that the function so does. Therefore, the rest claim that should be proved is to show that such operator is compact. In accordance with Lemma 5, it is sufficient to prove that is equicontinuous and uniformly bounded. In regard to the boundedness issue, its enough to track the following assertions:Thus, it remains to prove that is an equicontinuous operator. For this purpose, we intend to estimate the term , where with , as follows:Note that with aiding the results in Lemma 9, the right-hand side goes uniformly to zero as . By the same method, we can deduce thatgoes uniformly to zero as . Thus, based on Lemma 5, one can conclude that is an equicontinuous operator, proving the third condition of Krasnoselskii’s Theorem 1, and hence there is at least one solution of the main problems (1) and (2) on , as required.

4.2. Noncompactness Case

The existence of solution for the FoLE in the case of noncompactness of solution can be further explored by utilizing Kuratowski’s measure of noncompactness through applying Darbo’s fixed point Theorem 2. This matter can be addressed by considering the next existence result.

Theorem 4. Assume that the two assumptions and are satisfied. Ifwhere and are defined in (44) and (45), respectively, then the main problems (1) and (2) have at least one solution on .

Proof. Due to the matter of finding a solution of the main problem is just considered as a fixed point problem, we define operator and its fractional derivative as previously mentioned in (31) and (31), respectively. To begin the proof of this result, we first need to show that the operator is continuous where is defined in (47), but with radiusand . Obviously, the set is closed, bounded, and convex nonempty subset of the Banach space . Now, we need to prove that S() and is continuous on .
To show that is continuous, we let to be a sequence of a Banach space such that as . If , thenConsequently, from the assumption , we haveAlso,Hence, we deduce thatSimilarly, we can attain the following assertion:In conclusion, we getAs , then which implies that the operator is continuous on the space .
In order to prove that S(), take . In view of (61), we getIndeed, for , we haveSimilarly, we can deduce that which with the inequality above lead toBased on (72) and (74), we observe that which concludes that S().
The final step of this proof is to show that the operator satisfies the inequality of the Kuratowski measure of noncompactness in Theorem 2. To do this, let be a closed subset of such that there exists and . Then,Also, let there exist and such thatIn view of continuity of the functions and , the sets and are bounded and equicontinuous and so with the last item of Lemma 4, we obtainand with Definition 3, we getwhich implies thatFrom the second and third items of Lemma 4, we getIn the similar way, we haveHence,Similarly, we can follow the same procedure in regard to . In other word, we can obtain the following assertion:Substituting (82) and (83) into (75) concludes thatIn view of the Darbo fixed point theorem, we confirm that the operator has a fixed point which represents a solution of the main problems (1) and (2).

5. Numerical Examples

In this part, the main B.V.P. reported in (1) and (2) is addressed via two numerical examples. These examples are handled for demonstrating the two results formulated in the previous section.

5.1. Example 1

Consider the B.V.P. as follows:subject to the conditions:

Comparing this problem with the main problem yields to outline that , , , , , , and On the other hand, we have and , which give and . Moreover, we havewhich is continuous and satisfies

This implies that the second condition is satisfied with . By carrying out Mathematica software, it is easy to obtain ,which implies that . Therefore, based on Theorem 1, we conclude that the B.V.P. (85) and (86) have at least one solution on .

5.2. Example 2

Consider the following B.V.P.:subject to the conditions:

Comparing (90) and (91) with (1) and (2) leads us to obtain , , , , , , , , and . In the meantime, we obtain and which implies , , and . Furthermore, we havewhich is a continuous function. In order to see that it satisfies the third condition , we havewhich implies that the function satisfies the third condition with . By carrying out Mathematica software, it is easy to obtain ,which implies that

Hence, according to Theorem 4, we deduce that there is at least one solution to the B.V.P. (90) and (91) on the interval .

6. Conclusion

This paper has introduced two novel results associated with the existence of solutions for the fractional-order Langevin problem that has been formulated in the sense of Caputo–Hadamard fractional-order derivative operator. These two results have been shown in light of two well-known fixed point theorems, namely, the Krasnoselskii’s theorem in the case of compactness and Darbo’s theorem in the case of noncompactness. Two numerical examples have been demonstrated for completeness and illustrate the implementations of our idea on scientific problems. Here, we applied this idea on fractional differential equations; we think that applying this idea on the fractional differential inclusion will be useful in the future study.

Data Availability

The qualitative data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia, under grant no. KEP-PhD-60-130-42. The authors, therefore, acknowledge with thanks DSR’s technical and financial support.

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