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A Study on -Caputo-Type Hybrid Multifractional Differential Equations with Hybrid Boundary Conditions
In this research paper, we investigate the existence, uniqueness, and Ulam–Hyers stability of hybrid sequential fractional differential equations with multiple fractional derivatives of -Caputo with different orders. Using an advantageous generalization of Krasnoselskii’s fixed point theorem, we establish results of at least one solution, whereas the uniqueness of solution is derived via Banach’s fixed point theorem. Besides, the Ulam–Hyers stability for the proposed problem is investigated by applying the techniques of nonlinear functional analysis. In the end, we provide an example to illustrate the applicability of our results.
Over the last couple of years, the concept of fractional differential equations (FDEs) has been validated as being an effective and powerful gadget to model complex and real world phenomena due to their wide range of applications in several fields of sciences and engineering; see [1–5] and the references therein. Recently, FDEs gained the attention of mathematicians and researchers working in different disciplines of science and technology, which resulted in plenty of research papers that have been carried out on FDEs. That made a valuable contribution ranging from the qualitative theory of the solutions of FDEs, such as existence, uniqueness, stability, and controllability to the numerical analysis. Speaking in this context, the stability analysis of functional and differential equations is important in many applications, such as optimization and numerical analysis, where computing the exact solution is rather hard. There are various kinds of stability; one of those types has recently received considerable attention from many mathematicians, so-called Ulam–Hyers (U-H) stability. The source of U-H stability goes back to 1940 by Ulam , next by Hyers . A variety of works have been done by many authors in regard of the U-H stability of FDEs; for example, the authors in  studied the existence and stability results for implicit FDE. Some recent developments in Ulam’s type stability are discussed by Belluot et al. . Ibrahim, in , obtained the generalized U-H stability for FDEs. Some approximate analytical methods for solving FDEs can be found in [11–14]; also, computational analyses of some fractional dynamical and biological models were investigated recently, see [15, 16].
On the contrary, quadratic perturbation of nonlinear differentials, also known as the hybrid differential equations, had rapid progress over the last years; this is due to its importance, which lies in the fact that they include perturbations that facilitate the study of such equations by using the perturbation techniques. These equations are also considered as a particular case in dynamic systems. The starting point for this field is when Dhage and Lakshemikantham  formulated a hybrid differential equation, where they investigated the existence and uniqueness of the solutions to the following hybrid equation:
Their results were based on the fixed point theorem (FPT) for the product of two operators in Banach algebra.
In 2011, Zhao et al.  extended Dhage’s work  to fractional order and studied the existence of solutions to the following Riemann–Liouville (RL)-type hybrid FDEs:
After several years, Sitho et al.  derived a new existence result for the following hybrid sequential integro-differential equations:
In , Baitiche et al. studied the existence of solutions for the following hybrid sequential FDEs:
They generalized Darbo’s FPT for the product of two operators associated with measures of noncompactness.
Some existence results for -Caputo-type hybrid FDEs are obtained in [21, 22]. For recent analysis techniques in FDEs involving generalized Caputo FD, we refer to [23, 24]. Just recently, Boutiara et al.  discussed some qualitative analyses to the following fractional hybrid system:
The above findings motivated us to study the existence, uniqueness, and U-H stability of solutions for the following -Caputo hybrid fractional sequential integro-differential equation (for short, -Caputo HFSIDE):
endowed with the hybrid fractional integral boundary conditions:withwhere , , and denote the -Caputo FD of order , respectively, and and are the -RL fractional integral of order and , respectively, , , with , and is appropriate positive real constants.
The modernity of our proposed problem in contrast with past problems is that, in this work, we consider a kind of general case of boundary value problems in a setup of a -Caputo HFSIDE delineated by (6) and (7). To be sure, the advantage of this work is that the applied FD has the freedom of choice of the kernel which makes it conceivable to bring together and cover most of the preceding results on hybrid FDEs.
The research paper is organized as follows. Section 2 presents some basic mathematical pieces of knowledge required throughout the paper. The main results for -Caputo HFSIDE (6), (7) are proved in Section 3. In Section 4, we give an illustrative numerical example, and Section 5 is related to a brief conclusion.
Let and be Banach spaces of continuous real-valued functions defined on . Clearly, is a Banach algebra with the normand multiplication
Let an increasing function satisfy , for all . For effortlessness, we set and .
Definition 1 (see ). The -RL fractional integral of order of an integrable function is defined by
Definition 2 (see ). The -Caputo FD of order of a function is defined bywhere and .
In case, if , we have
As a special case, if , then the above definitions reduce to the well-known classical fractional definitions; for more details, see [1,3].
Lemma 1 (see ). Let . The following holds:(i)If , then(ii)If , , thenwhere .
Concerning the applied FPTs, we will suffice with indication to [27, 28].
3. Main Results
Lemma 2. Let and . For any functions , , andwith,, the following linear fractional BVP,supplemented with the conditions,has a unique solution, that is,where , .
proof. Applying the -RL fractional integral of order to both sides of (16) and using Lemma 1, we obtainBy multiplying to both sides of (16), we find thatOn the contrary, we haveFrom (20) and (21), we find thatIntegrating from 0 to and using the fact that , , and from the condition in (17), we havewhere ; by multiplying to both sides, we obtainNext, applying -RL fractional integral of order to both sides of (24) and using Lemma 1, we obtainwhere with the help of conditions , , and , ; we find . Then, we apply the third condition of (17) in (25), and we obtainSome computations give usInserting , and in (25), leads to solution (18).
Conversely, by Lemma 1 and by taking on both sides of (25), we obtainNext, operating on both sides of the above equation, with the help of Lemma 1, we obtainNow, it remains to review the boundary conditions (17) of our problem (16). Substituting in (18) with the fact that , leads to . Next, we apply on (18); then, we substitute ; it follows that . Substituting and , we find that the two resulting equations are equal, and from it, we get thatThis means that satisfies (16) and (17). Therefore, is solution of problem (16) and (17).
Lemma 3. For ,we have(1)(2)(3)
proof. To prove property (1), we haveFrom the above integrals and left side of (1), we obtainThe proofs of properties (2) and (3) are similar to proof of (1).
Now, we consider the following assumptions: (H1) and , and there exist positive functions and , such that for and . (H2) There exist functions such that (H3) There exist and , such thatwhere
3.1. Existence of Solutions
In this section, we prove the existence of a solution for problems (6) and (7) by applying Dhage FPT .
Theorem 1. Suppose (H1)–(H3) hold; then, problems (6) and (7) have at least one solution in.
proof. First, we set , and .
Now, we define asDefine two operators and asThen, using assumptions (H1)-(H2), we have, for and each ,Also,Now, we also consider two operators and defined byWe need to prove that and satisfy all assumptions of Dhage’s theorem . This can be proven in the forthcoming steps:Step 1: is a contraction map. Indeed, let . Then,Using Lemma 3 and hypotheses (H1)-(H2), we obtainMoreover,Hence, by (50), is a contraction map.Step 2: is compact and continuous on . Firstly, we prove that is continuous on .Let be a sequence such that in . It follows from Lebesgue dominant convergence theorem that, for all ,Hence, . Thus, is a continuous on . Besides, we prove that is uniformly bounded on . Indeed, for any , we haveTherefore, , for all , which implies that is uniformly bounded on . Now, we show that is equicontinuous. Let with . Then, for any , we haveAs , . This means that is equicontinuous. Thus, Arzelá–Ascoli theorem shows that is a compact operator on .Step 3: we prove that , for all . For any , we havewhich implies , so . Hence, all assumptions of Dhage’s theorem  are satisfied. So, the equation has at least one solution in . So, there exists a solution of problems (6) and (7) in .
3.2. Uniqueness of Solutions
Here, we prove the uniqueness theorem of (6) and (7) relying on Banach’s FPT .
Theorem 2. Suppose that (H1)-(H2) and the following hypothesis hold: (H4) , and there exist positive function, such thatIfthen problems (6) and (7) have a unique solution.
proof. We set the operator asConsider , and we set . First, we show that . As in the previous proof (Step 3) of Theorem 1, we can obtain the following.
For and ,This shows that . Next, we prove that is a contraction. For ,From (44), (50), and (54), we obtain