A Study on ψ -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 ψ -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.


Introduction
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][2][3][4][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. at 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. ere are various kinds of stability; one of those types has recently received considerable attention from many mathematicians, so-called Ulam-Hyers (U-H) stability. e source of U-H stability goes back to 1940 by Ulam [6], next by Hyers [7]. A variety of works have been done by many authors in regard of the U-H stability of FDEs; for example, the authors in [8] studied the existence and stability results for implicit FDE. Some recent developments in Ulam's type stability are discussed by Belluot et al. [9]. Ibrahim, in [10], obtained the generalized U-H stability for FDEs. Some approximate analytical methods for solving FDEs can be found in [11][12][13][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. ese equations are also considered as a particular case in dynamic systems. e starting point for this field is when Dhage and Lakshemikantham [17] formulated a hybrid differential equation, where they investigated the existence and uniqueness of the solutions to the following hybrid equation: eir results were based on the fixed point theorem (FPT) for the product of two operators in Banach algebra.
In 2011, Zhao et al. [18] extended Dhage's work [17] to fractional order and studied the existence of solutions to the following Riemann-Liouville (RL)-type hybrid FDEs: After several years, Sitho et al. [19] derived a new existence result for the following hybrid sequential integrodifferential equations: In [20], Baitiche et al. studied the existence of solutions for the following hybrid sequential FDEs: ey generalized Darbo's FPT for the product of two operators associated with measures of noncompactness.
Some existence results for ψ-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. [25] discussed some qualitative analyses to the following fractional hybrid system: e above findings motivated us to study the existence, uniqueness, and U-H stability of solutions for the following ψ-Caputo hybrid fractional sequential integro-differential equation (for short, ψ-Caputo HFSIDE): endowed with the hybrid fractional integral boundary conditions: where 1 < p ≤ 2, 0 < q ≤ 1, c D p;ψ and c D q;ψ denote the ψ-Caputo FD of order p, q, respectively, and I η i ;ψ and I c;ψ are the ψ-RL fractional integral of order η i > 0, (i � 1, . . . , m) and c > 0, respectively, h ∈ C(I × R, R∖0), f ∈ C (I × R 2 , R), g i ∈ C(I × R, R) with g i (0, 0) � 0, and λ is appropriate positive real constants. e 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 ψ-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 ψ which makes it conceivable to bring together and cover most of the preceding results on hybrid FDEs. e research paper is organized as follows. Section 2 presents some basic mathematical pieces of knowledge required throughout the paper. e main results for ψ-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.
where n � [p] + 1 and D n As a special case, if ψ(θ) � θ, then the above definitions reduce to the well-known classical fractional definitions; for more details, see [1,3].

Main Results
supplemented with the conditions, has a unique solution, that is, proof. Applying the ψ-RL fractional integral of order p − 1 to both sides of (16) and using Lemma 1, we obtain By multiplying ψ ′ (θ)e λΨ 0 (θ) to both sides of (16), we find that On the contrary, we have d dθ From (20) and (21), we find that d dθ .e λΨ 0 (θ) Integrating from 0 to θ and using the fact that G i (0) � 0, i � 1, . . . , m, and from the condition c D q;ψ ϰ(0) � 0 in (17), we have where c 1 ∈ R; by multiplying e − λΨ 0 (θ) to both sides, we obtain Next, applying ψ-RL fractional integral of order q to both sides of (24) and using Lemma 1, we obtain where en, we apply the third condition of (17) in (25), and we obtain Some computations give us Inserting c 0 , c 1 , and c 2 in (25), leads to solution (18). Conversely, by Lemma 1 and by taking c D q;ψ on both sides of (25), we obtain Next, operating c D p;ψ + λ c D p− 1;ψ on both sides of the above equation, with the help of Lemma 1, we obtain Journal of Mathematics Now, it remains to review the boundary conditions (17) of our problem (16). Substituting θ � 0 in (18) with the fact that g i (0) � 0, i � 1, . . . , m, leads to u(0) � 0. Next, we apply c D q;ψ on (18); then, we substitute θ � 0; it follows that c D q;ψ ϰ(0) � 0. Substituting θ � [ and θ � ξ, we find that the two resulting equations are equal, and from it, we get that is means that ϰ(θ) satisfies (16) and (17). erefore, ϰ(θ) is solution of problem (16) and (17).

Existence of Solutions.
In this section, we prove the existence of a solution for problems (6) and (7) by applying Dhage FPT [27].

Journal of Mathematics
‖ϑ‖. (39) Also, Now, we also consider two operators A: E ⟶ E and B: B r ⟶ E defined by I η i +q;ψ g i (s, ϰ(s))(θ).

(41)
We need to prove that A and B satisfy all assumptions of Dhage's theorem [27].
Step 2 : B is compact and continuous on B r . Firstly, we prove that B is continuous on B r .
Let ϰ n (θ) be a sequence such that ϰ n (θ) ⟶ B r in B r . It follows from Lebesgue dominant convergence theorem that, for all θ ∈ I,
us, Arzelá-Ascoli theorem shows that B is a compact operator on B r .
proof. We set the operator G: E ⟶ E as Consider and we set sup θ∈I |K i (θ)| � ‖K‖, i � 1, 2, . . . , m. First, we show that G(B R ) ∈ B R . As in the previous proof (Step 3) of eorem 1, we can obtain the following.
For ϰ ∈ B R and θ ∈ I, is shows that G(B R )⊆B R . Next, we prove that G is a contraction. For ϰ, ϰ∈ B R , From (44), (50), and (54), we obtain As Ξ < 1, G is contractive map. Consequently, by Banach's FPT [28], we conclude that G has a unique fixed point, which is a solution of (6) and (7).

Stability Analysis.
In this portion, we discuss the U-H and generalized U-H stabilities of the solution of the proposed problem. We adopt the following definitions from [29].