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On Solutions of Hybrid–Sturm-Liouville–Langevin Equations with Generalized Versions of Caputo Fractional Derivatives
The main intention of this research article is to introduce a new class of generalized fractional differential equations that fall into the categories of Sturm-Liouville’s, Langevin’s, and hybrid’s problems involving -Caputo fractional derivatives. The existence of the solutions of the proposed equations is discussed by using the technique of the measure of noncompactness related to the fixed point theorem, which is a generalization of Darbo’s fixed point theorem. Additionally, pertinent examples are provided along with the different values of the function to confirm the validity of the reported results.
Fractional differential equations (FDEs) with their various branches such as Hybrid Equation (HE), Langevin Equation (LE), and Sturm-Liouville Equation (SLE) are currently well established, due to the number of papers and books edited worldwide. These types of equations have been applied in many applications in different fields, such as engineering and science. Since in recent years, it has achieved a great deal of development and interest by many researchers, for some of these developments in the theory of fractional differential equations, one can look at the monographs of Kilbas et al.  and Podlubny , where they presented some properties and applications appropriate for various types of fractional operators. Dhage and Lakshmikantham  and Dhage et al.  made excellent results on hybrid problems, as did Zhao et al.  and Ahmad and Ntouyas . The LE  is formulated to be a powerful tool for describing the evolution of physical phenomena in volatile environments. Some of recent Langevin’s problem is studied through [8–10]. However, SLE has many applications in distinct areas of technical knowledge and engineering [11, 12]. The mix of both fractional SLE and fractional LE might give an adequate description of the dynamic processes described in a fractal medium where fractal and memory properties are inserted with a scattered memory kernel. Recently, the authors in  suggested an approach to the fractional model of the SLE and LE. Indeed, they discussed the existence of solutions to the considered systems through fixed point techniques and mathematical inequalities. Muensawat et al.  studied antiperiodic BVPs for fractional systems of generalized SL and LE. Boutiara et al.  considered fractional LE under Caputo function-dependent kernel fractional derivatives. Existence theorem for -fractional HEs has been proven by Suwan et al. . Some qualitative analyses for multiterm LEs with generalized Caputo FDs and diffusion FDE with ABC operators can be found in [17, 18]. The authors in  considered a hybrid LE involving Caputo FD and Riemann-Liouville (RL) fractional integral (FI) as follows:
Motivated by the above works aforesaid and inspired by [19, 21], in this paper, we deal with the existence of solutions for the following BVP to the nonlinear fractional hybrid–Sturm-Liouville–Langevin differential equation: where denotes the -Caputo FD of order , . Here, , are given functions, , and . As in Banach spaces, a closed and bounded set is not generally a compact set; just continuity of the function does not ensure the existence of a solution to differential equations. Our arguments are principally founded on Darbo’s fixed point technique mixed with the technique of measures of noncompactness to set up the existence of solutions for (2). In particular, problem (2) is formed as an overarching structure comprising both fractional SLE, LE, and HE, subjected to boundary conditions involving -Caputo FDs. In fact, choosing on the one hand and , , and on the other hand, reduces the problem (2) into the fractional Sturm-Liouville problem and the fractional Langevin problem, respectively. Besides, if we set and , the problem (2) reduces to the fractional sequential hybrid problem.
Observe also that the current results are consistent with some of the literature results when , and they are new even for the special case: and
Here is a brief outline of the paper. In Section 2, we provide some preliminary facts. Sections 3 and 4 handle the formulation of solutions and the existence of solutions for (2) by using the generalized Darbo’s fixed point theorem (D’sFPT) along with the approach of measures of noncompactness in the Banach algebras. Lastly, we give pertinent examples.
Let us start this section with some auxiliary results used in the forthcoming analysis.
Definition 1 (see ). The -RL FI of order for an integrable function is given by where is the gamma function. One can deduce that where
Definition 2 (see ). For and , the -Caputo FD of a function of order is given by
where for and for .
Also, we can express -Caputo FD by
Lemma 3 (see [1, 20]). Let and . Then, In particular, if , then ,
Lemma 4 (see ). Let . Then, the following holds:
If , then If . Then,
Lemma 5 (see [1, 20]). Let ,, and Then, (i)(ii)(iii), for Let be the closed ball in the Banach space ; if , then Let such that and Conv are a closure and a convex closure of , respectively. And let be the family of the nonempty and bounded subsets of , while denotes the subfamily of all relatively compact subsets of .
Definition 6 (see ). We say that is a noncompactness measure in if all the assumptions below hold: (i) is nonempty and (ii), then (iii)(iv)(v)In the case of being a sequence of closed subsets of with and , then
Definition 7 (see ). Let be a nonempty bounded set and be a Banach space. We say that is a modulus of continuous function, denoted by ; if and , we have Moreover,
Definition 8 (see ). A noncompactness measure in satisfies the condition if for all , where is the Banach algebra.
Lemma 9 (see ). The condition () may be grasped by the noncompactness measure on
Set Now, we present D’sFPT and generalized D’sFPT to prove that there exists at least one fixed point.
Theorem 10 (see [25, 26]). Let be a Banach space and be a nonempty, bounded, convex, and closed set. Let be continuous. Assume that there is with as a noncompactness measure in meeting the following requirements: Then, has a fixed point in
Theorem 11 (see ). Let be a Banach space and be a nonempty, bounded, convex, and closed set, and let be continuous. Assume there exist and such that for each nonempty subset of with where is a noncompactness measure in . Then, has a fixed point in .
3. Solution Formulation
This section presents a formulation of the solution to problem (2) along with the assumptions required in the forthcoming analysis. Foremost, we denote by the space of real valued continuous functions defined on a unit interval . It is clearly the Banach space with the norm:
Multiplication is defined as the usual product of real functions.
To prove the existence of solutions to (2), we need the following lemma:
Lemma 12. The problem (2) is equivalent to the following fractional integral equation:
Proof. Applying the --RL integral on (2), we obtain where . From the BCs of (2), we get Taking the --RL integral of (18), one has where The BCs of (2) give . In this regard, if we apply the --Caputo FD and --Caputo FD to both sides of (17) and use Lemma 5, then the problem (2) immediately is established.
Before giving the essential result, we shall investigate formula (17) under the following assumptions: (i)(AS1) Both functions are continuous(ii)(AS2) and (iii)(AS3) There exists a real number with (iv)(AS4) There exists a continuous nondecreasing function with such that (v)(AS5) There exists such that where
4. Existence Result
The aim of this section is to discuss the existence of solutions to the problem (2). For this end, we apply Theorems 10 and 11.
Theorem 13 Under hypotheses (AS1)–(AS5). Then, the problem (2) has a least one solution in the Banach algebra .
Proof. Consider the operator on the Banach algebra as where From (AS4), we have For the sake of simplicity, we put
Now, we divide the proof into several steps.
Step 1. transforms into itself.
At first, we show that implies that , i.e., for all . Certainly, (AS1) and (AS2) guarantee that if , then . It remains to prove if , then . Let and with . By hypothesis (AS4), we get which tends to be zero uniformly once . It is clear that for all .
Step 2. An estimate of for .
Let and . Then, by using our hypothesis, we have Therefore,
Step 3. The operator is continuous on . Here, is a subset of defined by
with a fixed radius , which satisfies the inequality (AS5).
We shall need to show the continuity of and on , separately. For any and , there exists ; it follows for that Therefore, is continuous on . The continuity of the operator is obtained by Lebesgue dominated convergence (LDC) theorem. Indeed, let be a sequence such that in with as As is continuous, we obtain Since is continuous on , it is uniformly continuous on Now, we set Applying the LDC theorem, we get Thus, is continuous in .
Due to the continuity of and , the operator is continuous in .
Step 4. We estimate and for .
At first, we estimate . Since is uniformly continuous, we obtain for any with with , which implies . Taking and with , under hypothesis (AS5), we get Considering then we can write (38) as Obviously, is uniformly continuous on , and once . Hence, (40) becomes as follows: Next, since is uniformly continuous, we have , with , with , which implies Take into account equations (32), (35), and (36) for each Set Choosing and with yields For simplicity’s sake, we set The factors , , and can be estimated as in the following cases:
Case 1. If , then Case 2. If , then Case 3. If , then Accordingly, we obtain , which implies that
Let . Then,
Step 5. We estimate for .
By Lemma 9 and equations (32), (41), and (48), we obtain Since , the assumption (AS5) gives Thanks to Theorem 10, the contractive condition is fulfilled with , where . By applying Theorem 11, has at least fixed point in . Hence, the problem (2) has at least one solution in .
Here, we provide two examples to illustrate previous results.
Example 14. Consider the problem (2) with following specific data:
Then, the problem (2) reduces to where
,, , and . Thus, (AS1) and (AS2) hold. For (AS3), we obtain . Furthermore, let Then, , and it is a concave function. Since is concave. As a result, the subadditive property of the concave function allows us to conclude
Thus, (AS3) holds, with . Moreover, for every and , we obtain
Hence, (AS4) holds with . Finally, (AS5) permitted to provide us the range of which is obviously
Accordingly, (AS5) confirms that the illustrated example (52) has a solution in due to
Example 15. Depending on the previous example, we present some special cases of with different values for some parameters as in Table 1.
In this work, we have successfully studied some qualitative properties of the solution to a fractional problem that integrates three different types of BVP; more precisely, we have investigated the existence of the solutions of the Sturm-Liouville–Langevin–hybrid-type FDEs. Our analysis has been based on the technique of the measure of noncompactness along with the generalized Darbo’s fixed point theorem. The results were consistent with some of the literature results when , and they are new even for the special case: and
The problem studied can be extended to a more general problem containing -Hilfer FD, and this is what we are considering in future research.
No real data were used to support this study.
Conflicts of Interest
No conflicts of interest are related to this work.
The authors thank the Research Center for Advanced Materials Science (RCAMS) at King Khalid University, Saudi Arabia, for funding this work under the grant number RCAMS/KKU/013/20. Emad E. Mahmoud acknowledges Taif University Researchers Supporting Project number TURSP-2020/20, Taif University, Taif, Saudi Arabia.
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