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

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. (e 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.


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][5][6][7]. e 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].
e 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][12][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][15][16][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.
e Sturm-Liouville operator has several applications in variety fields of science such as engineering and mathematics [27]. e 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. is 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. is 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].
is 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 CH D μ is the Caputo-Hadamard derivative operator of order μ, H J c is the Hadamard integral operator of order c, 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. e 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. ere are many contributors who used measure of noncompactness to investigate the existence of solution to FoDE [32][33][34] and references given therein. e 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.

Preliminaries
is part aims to present some fundamental fixed point theorems and essential preliminaries related to the fractional calculus, especially in regard to the Hadamard fractionalorder operators.
e Hadamard integral operator of the fractional-order value μ ∈ R + for a function f ∈ L p [a, b] is outlined as where 0 ≤ a ≤ t ≤ b < ∞.
For simplicity, we will consider a � 1 in all aforementioned preliminaries, and consequently H J μ a and H D μ a will be denoted by J μ and D μ , respectively. However, some essential fixed point theorems are reported below for completeness.
Definition 3 (see [3]). Suppose that B is a Banach space and Υ B is the collection of subsets of B.

if it is uniformly bounded and equicontinuous maps on [0, T], and f(t 0 ) is relatively compact in the Banach space B, for any
Theorem 1 (Krasnoselskii's fixed point theorem [3,26]). Suppose that C ε is a closed convex bounded set of a Banach space B, and S 1 , S 2 : B ⟶ B are two operators satisfying Theorem 2 (Darbo's fixed point theorem [3]). Suppose that C ε is a closed convex bounded set of a Banach space B, and S: C ε ⟶ C ε is a continuous mapping so that for all closed subsets W ∈ C ε , where 0 ≤ k < 1. en, S has a fixed point in C ε .

Auxiliary Results
is 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.
e linear BVP given in the following subjects to the conditions in (2) which is equivalent to the following fractional-order integral equation: where g: Proof. Applying the Hadamard fractional-order integral H J μ to both sides of (11) yields at is, Again, taking H J c to both sides of the equation above yields Now, using the first condition (x(1) � 0 given in (2)) yields c 2 � 0, and then we have On 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 c 0 and c 1 : Finally, substituting c 0 and c 1 in (18) yields the desired result. en, for the function χ(t, s) defined in (13), we can deduce the following assertion: where α 1 � min t∈ [1,T] α(t) and α 2 � max t∈ [1,T] α(t).
Proof. Since the function α: [1, T] ⟶ R + is continuous, then it attains its minimum and maximum which leads to where b i , i � 1, 2 are defined in (14). In view of Lemma 3, we get Whence, where Differentiation gives which means that the function f is increasing on (1, t 0 ) and

□
In this work, the Banach space of all continuous realvalued functions B � C( [1, T], R) will be considered in which it is equipped with the usual maximum norm: Define the space as a subset of B, then one can easily check that Y 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. is means that the function x(t) must hold the two assertions: For achieving this objective, we define the operator S as follows: where

Mathematical Problems in Engineering
In a similar manner in Lemma 7, we can deduce as follows.
□ e 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.

The Existence of Solution for the GSLLE
is 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:
Proof. For the purpose of proving this theorem, the closed ball C ϵ is defined in the following manner: where ε is the radius and assumed so that where ψ ℓ (·) is defined as in (44). One can confirm that C ε 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 S 1 x 1 + S 2 x 2 ∈ C ε , ∀x 1 (t), x 2 (t) ∈ C ε . So, to perform this task, we start with the following assertion: It is known that log t is positive and increasing on [1, T] which implies that log c+μ t ≤ log c+μ T. Consequently, we have On the other hand, we have Mathematical Problems in Engineering Now, from (50) and (51), we obtain Similarly, we obtain Accordingly, the two inequalities above imply Based on (52) and (54), we can obtain the following assertion: (55) at is, Hence, the first condition given in eorem 1 is satisfied. For the purpose of showing the second condition of the same theorem which requires to prove that S 2 is a contraction mapping, we intend to take into account the term |S 2 x 1 − S 2 x 2 | in accordance with the following manner: In other words, Similarly, we can attain the following inequality: Again, applying (58) and (59) into the norm ‖ · ‖ Y defined above yields Due to max Ψ 0 , Ψ ℓ < 1 as we previously assumed, then S 2 is indeed contraction mapping.
In the meantime, to address the last condition of eorem 1, we should first note that S 1 is a continuous operator on C ε due to that the function h so does. erefore, 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 S 1 C ε is equicontinuous and uniformly bounded. In regard to the boundedness issue, its enough to track the following assertions: us, it remains to prove that S 1 is an equicontinuous operator. For this purpose, we intend to estimate the term |S 1 x(t 2 ) − S 1 x(t 1 )|, where t 1 , t 2 ∈ [1, T] with t 2 > t 1 , as follows: 8 Mathematical Problems in Engineering Note that with aiding the results in Lemma 9, the righthand side goes uniformly to zero as t 2 ⟶ t 1 . By the same method, we can deduce that goes uniformly to zero as t 2 ⟶ t 1 . us, based on Lemma 5, one can conclude that S 1 is an equicontinuous operator, proving the third condition of Krasnoselskii's eorem 1, and hence there is at least one solution of the main problems (1) and (2) where ψ ℓ (·) and Ψ ℓ are defined in (44) and (45), respectively, then the main problems (1) and (2) have at least one solution on [1, T].
Proof. Due to the matter of finding a solution of the main problem is just considered as a fixed point problem, we define operator S: B ⟶ B and its fractional derivative D ℓ S: B ⟶ B as previously mentioned in (31) and (31), respectively. To begin the proof of this result, we first need to show that the operator S: C ε ⟶ C ε is continuous where C ε is defined in (47), but with radius and N � sup t∈ [1,T] |h(t, 0, 0)| < ∞. Obviously, the set C ε is closed, bounded, and convex nonempty subset of the Banach space Y. Now, we need to prove that S(C ε ) ⊂C ε and S is continuous on C ε . To show that S: C ε ⟶ C ε is continuous, we let x n n∈N to be a sequence of a Banach space Y such that x n ⟶ x as n ⟶ ∞. If t ∈ [1, T], then Mathematical Problems in Engineering 9 Consequently, from the assumption H 3 , we have Also, Hence, we deduce that Similarly, we can attain the following assertion: In conclusion, we get As n ⟶ ∞, then ‖Sx n − Sx‖ Y ⟶ 0 which implies that the operator S is continuous on the space Y.
In order to prove that S(C ε )⊂C ε , take x ∈ C ε . In view of (61), we get Indeed, for t ∈ [1, T], we have Similarly, we can deduce that | CH D ℓ S 2 x(t)| ≤ εΨ ℓ which with the inequality above lead to Based on (72) and (74), we observe that ‖Sx‖ Y ≤ ε which concludes that S(C ε )⊂C ε . e final step of this proof is to show that the operator S: C ε ⟶ C ε satisfies the inequality of the Kuratowski measure of noncompactness in eorem 2. To do this, let W be a closed subset of C ε such that there exists W i , i � 1, 2, . . . , n and W⊆ ∪ n i�1 W i . en, Also, let there exist U⊆ ∪ n i�1 U i and V⊆ ∪ n i�1 V i such that In view of continuity of the functions h and α, the sets U and V/α are bounded and equicontinuous and so with the last item of Lemma 4, we obtain and with Definition 3, we get which implies that From the second and third items of Lemma 4, we get ≤ L α 1 Γ(c + μ + 1) log c+μ T + χ(T)log c+μ ζ + χ(zeta)log c+μ T |χ(ζ, T)| σ(W) � Lψ 0 (c + μ)σ(W). (80)

(84)
In view of the Darbo fixed point theorem, we confirm that the operator S has a fixed point which represents a solution of the main problems (1) and (2).