Existence Theorems for Hybrid Fractional Differential Equations with ψ -Weighted Caputo–Fabrizio Derivatives

In this study, two classes of hybrid boundary value problems involving ψ -weighted Caputo–Fabrizio fractional derivatives are considered. Based on the properties of the given operator, we construct the hybrid fractional integral equations corresponding to the hybrid fractional diferential equations. Ten, we establish and extend the existence theory for given problems in the class of continuous functions by Dhage’s fxed point theory. Furthermore, as special cases, we ofer further analogous and comparable conclusions. Finally, we give two examples as applications to illustrate and validate the results.


Introduction
Te theory of fractional calculus has attracted the attention of a remarkable number of researchers from various felds in recent years.Te physical meaning of fractional orders is that the dynamical systems of fractional order can be represented by a fractional diferential equation (FDE) with a noninteger derivative.Tese systems are referred described as having fractional dynamics.Undoubtedly, it has been demonstrated that the use of fractional derivatives (FDs) is very benefcial for modeling a wide range of problems and natural phenomena in engineering and applied sciences; for example, see renowned monographs by Osler [1], Samko et al. [2], Kilbas et al. [3], and Diethelm and Ford [4].Te literature contains a variety of FD concepts, including those presented by Riemann-Liouville and Caputo [3], which include the singular kernel k(t, s) � (t − s) − ] /Γ(1 − ]), 0 < ] < 1.
Tese FDs play an important role in modeling numerous physical and biological phenomena.In any case, as was referenced in Caputo and Fabrizio [5,6], certain peculiarities connected with material heterogeneities cannot be well modeled utilizing Riemann-Liouville or Caputo FDs.Because of this reality, the authors in [5] proposed another FD involving the nonsingular kernel k(t, s) � exp(−] (t − s)/1 − ]), 0 < ] < 1; then, Losada and Nieto [7] studied some of its properties.Te existence and uniqueness of solutions are essential properties of mathematical models [8,9] and are among the advantages of applied theory.Te model must possess these properties in order to be reliable and useful.Existence refers to the fact that the model must describe a well-defned problem, which can be solved within a certain mathematical framework.In other words, the model should not have any ambiguity or inconsistency that would make it impossible to solve.In [10][11][12][13][14][15][16][17][18], the authors studied the existence of solutions for various types of FDEs involving the Caputo-Fabrizio FD and other fractional operators.For instance, Abbas et al. [11] handled some existence results for Caputo-Fabrizio type implicit FDEs in b-metric spaces.Te existence and uniqueness of solutions for the following problem CF were demonstrated by Shaikh et al. [14].It is possible to think of hybrid diferential equations as quadratic perturbations of nonlinear diferential equations.Tey are of great interest to scholars because they are particular instances of dynamical systems.Dhage and Lakshmikantham [19] provide details on various perturbations for nonlinear diferential and integral equations.For additional updates on the availability of hybrid FDEs theory, we refer to [19][20][21].For instance, the following hybrid classical has been studied by Dhage and Lakshmikantham [19].
Taking on the analogous approach of [19], Zhao et al. in [20] extended the investigation of hybrid (2) to the following Riemann-Liouville type hybrid FDE: Herzallah and Baleanu [21] discussed the existence results of hybrid FDE (3) using the Caputo FD.Furthermore, various classes of hybrid FDEs subject to diferent conditions have additionally been concentrated on by many specialists, see [22][23][24][25][26][27]; e.g., Ali et al. [26] developed an existence analysis for nonlinear hybrid FDEs with ψ-Hilfer FD and hybrid boundary conditions.For a nonlocal hybrid of Caputo fractional integrodiferential equations, Ahmad et al. [22] presented the existence results.
Almeida [28] proposed a general operator so called ψ-Caputo FD when the kernel is k(t, s) � ψ(t) − ψ(s) and the derivative is ((1/ψ′(t))(d/dt)).Ten, the authors in [29] expanded some of the properties of this operator to include the Laplace transform of it.Regarding this, Jarad et al. [30] developed the idea of weighted FDs with another function.Abdo et al. [31] proved the positive solutions of the following ψ-weighted Caputo problem: Recently, Al-Rafai and Jarrah [32] extended the concept of weighted FD to the ψ-weighted Caputo-Fabrizio FD, where ψ and w are monotone function and weight function, respectively.
Motivated by the abovementioned studies, we discuss the existence of solutions of the following weighted hybrid FDE: and the following ψ-weighted hybrid FDE
Remark 1. Tis work can be a generalization of some of the studied problems in the literature, for example, (i) In problem (5), if we choose w � 1, U(s, x(s)) ≡ 0, and Z(s, x(s)) ≡ 1, then we obtain the following problem: which has been studied by Salim et al. [33].(ii) If we choose a � 1, b � 0, U(s, x(s)) ≡ 0, and Z(s, x(s)) ≡ 1, then our problem (6) reduces to problem (4), which was considered by Abdo et al. [31].
Remark 2 (1) If ψ(s) � s, then problem (6) reduces to problem (5) (2) If ψ(s) � s, w � 1, U ≡ 0, and Z ≡ 1, then problem (6) reduces to problem (7), see [33] (3) If ψ(s) � s, w � 1, a � 1, b � 0, U ≡ 0, and Z ≡ 1, then problem (6) reduces to problem (1), see [14] (4) Many problems with less general operators with various values of w and ψ, such as the one proposed by Caputo and Fabrizio in [5] are part of our current problems 2 Journal of Mathematics Te rest of this work is arranged as follows.Section 2 gives some basic results about generalized Caputo-Fabrizio FD and functional spaces.Our main results of problems ( 5) and ( 6) are discussed in Section 3. Two examples that confrm the validity of the main results are provided in Section 4. Finally, we include the conclusions in Section 5.
Ten, the following FDE has the unique solution Theorem 7 (Dhage's fxed point theorem [34]).Let D be a nonempty, convex, closed subset of a Banach algebra X.

Main Results
Here, we provide some qualitative analyses of two types of Caputo-Fabrizio hybrid problems that are ( 5) and ( 6).
Proof.Applying the operator CF I ];ψ 0;w of the frst equation of ( 14), we have the following equation: Using Lemma 5, we have the following equation: Comparing ( 16) and ( 17), we obtain the following equation: which implies Taking s ⟶ T to both sides of ( 19), we have the following equation: Applying the boundary condition of ( 14) and using (20), we obtain the following equation: 4 Journal of Mathematics Hence, Substituting ( 22) into (19), we obtain (15).
Due to Lemma 8, we can infer the following result: □ Ten, the solution of ( 6) satisfes the following equation: where η ] , μ ] , and β w as in Lemma 8. Now, we need the following assumptions on U, Z, and f.
Proof.Defne the set D � x ∈ X: ‖x‖ X ≤ r  , where r satisfes (As4).Certainly, D is a convex, closed, and bounded subset of X.By Corollary 9, we defne three operators So, we can write the formula (23) in the operator form as follows: Now, we show that O 1 , O 2 , and O 3 fulfll all the assumptions of Teorem 7, through the following claims: □ Step 11.O 1 and O 3 are Lipschitzian on X.
For s ∈℧ and x, x∈ X, we have from (As2) that which implies Tus, O 1 , O 3 : X → X are Lipschitzian on X with Lipschitz constants ‖ϑ Z ‖ and ‖ϑ U ‖, respectively.
Step 12. O 2 : D ⟶ X is a completely continuous.
In the beginning, we show that O 2 is continuous on D.

Let x n
n≥1 be a sequence in D with x n ⟶ x ∈ D. Ten, from Lebesgue's convergence theorem [35], we obtain the following equation: Hence, lim n ⟶ ∞ O 2 x n (s) � O 2 x(s), for all s ∈ ℧.Tus, O 2 is a continuous on D.
Next, let x ∈ D. Ten, by (As3), we have the following equation: Since ψ′, w > 0 and applying the mean value theorem for integral, we obtain the following equation: Hence, (33) becomes Terefore, ‖O 2 x‖ ≤ ϖ, for all x ∈ D, where ϖ given by (As4).Tis consequence proves that O 2 (D) is uniformly bounded set on D. Finally, we show that the set O 2 (D) is an equicontinuous in X.
Let s 1 , s 2 ∈ ℧ with s 1 ≤ s 2 , and x ∈ D. Ten, As s 2 ⟶ s 1 , the continuity of f, ψ and w imply that Tus, O 2 is equicontinuous on D. As a result of the Ascoli-Arzelà theorem [4], O 2 : D ⟶ X is a completely continuous.
Let x ∈ X and y Step 14. Assumption (iv) of Teorem 7 is satisfed, i.e., we have the following equation: Tus, all the assumptions of Teorem 7 are satisfed, and hence, the equation x � O 1 xO 2 x + O 3 x has a solution in D. As a result, ψ-weighted hybrid problem (6) has a solution on ℧.

Special Results.
In this subsection, we discuss some special cases of problem (6).
Consider ψ(s) � s in problem (6), we obtain the following weighted hybrid FDE: where all constants and symbols correspond to those in problem (6).Since the next Lemma is a duplication of Lemma 8 with ψ(s) � s, we shall omit its proof.

Lemma 15.
Let g be continuous function on ℧ with g(0) � 0 and assume that x ⟶ x − U(s, x)/Z(s, x) is increasing in R, a.e., for each s ∈ ℧.Ten, the solution to the following weighted hybrid FDE satisfes the equation where η ] , μ ] , and β w are as in Lemma 8.

Journal of Mathematics
According to Lemma 15, we can defne three operators O 1 , O 3 X ⟶ X, and O 2 : D ⟶ X by In addition, we must provide some constants as follows: and Te following existence theorem can be stated without proof.
Remark 17.Our results for problem (6) are applied for the following special cases: Case 1: if a � 1 and b � 0, then, we have the initial value problem of hybrid FDE: Case 2: if a � 0 and b � 1, then, we have the terminal value problem of hybrid FDE: Case 3: if a � b � 1 and c � 0, then, we have the antiperiodic the problem of hybrid FDE: Case 4: if we choose U(s, x(s)) ≡ 0, and Z(s, x(s)) ≡ 1, then our problems (5) and ( 6) reduce to the following problems:

Conclusions
Te existence and uniqueness of solutions are among the qualitative properties of mathematical models.Tese properties are important because they help ensure that the model provides reliable and accurate results and that the results are applicable to a wide range of situations.Without these qualitative properties, a model may not accurately refect the real-world phenomena it is meant to describe, which can lead to incorrect conclusions and unreliable predictions.In this work, we have successfully analyzed the nonlinear hybrid diferential equations by the application of fractional calculus.Specifcally, problems ( 5) and ( 6) have been considered using the ψ-weighted Caputo-Fabrizio FDs, which incorporate a nonsingular kernel.First, we have provided several special results and various observations for our proposed problems in the frame of ψ-weighted Caputo-Fabrizio FDs, which made our results more generalizable and studyable to a wide range of previously studied and research-worthy problems.Ten, through the utilization of Dhage's fxed point theory for sums of three operators, we have established the existence of solutions to the proposed hybrid problems.Finally, in order to support the theoretical results, we have ofered two practical examples.In the future, it will be interesting if the current systems are studied in the frame of ψ-weighted Atangana-Baleanu-Caputo, recently introduced in [36,37].