In this article, we find the necessary conditions for the existence and uniqueness of solutions to a system of hybrid equations that contain mixed fractional derivatives (Caputo and Riemann-Liouville). We also verify the stability of these solutions using the Ulam-Hyers (U-H) technique. Finally, this study ends with applied examples that show how to proceed and verify the conditions of our theoretical results.

1. Introduction

Although the concept of fractional calculus was established 300 years ago, interest in this type of derivative appeared for a short period. So that it is no secret to anyone that the most important use of fractional derivatives is to find analytical solutions to differential equations if possible, or by using numerical analysis methods to find an approximation to these solutions. In this study, we will focus on the idea of studying theories that investigate the existence of a solution to a system of hybrid fractional equations that contain mixed fractional derivatives with boundary conditions attached to them.

As mentioned before, fractional calculus as a concept is not very recent. It is worth mentioning here the great names who have given a lot to this science, such as A.V. Letnikov, J. Hadamard, J. Liouville, B. Riemann M., and Caputo L. worked in this field. These names must be mentioned by way of example. To get acquainted with some of the names of scientists who have made great contributions to fractional calculus in the modern world, we ask the reader to look at [1].

Fractional derivatives have played a very important role in mathematical modeling in many diverse applied sciences, see [2, 3]. For example, the authors in [4] employed the fractional derivative of the Psi-Caputo type in modeling the logistic population equation, through which they were able to show that the model with the fractional derivative led to a better approximation of the variables than the classical model. In addition, the authors in [5] employed the fractional derivative of the Psi-Caputo type and used the kernel Rayleigh, to improve the model again in modeling the logistic population equation.

As a final example, the authors in [6] employed the fractional derivatives of the Caputo and Caputo-Fabrizio type by modeling the equation that gives the relationship between atmospheric pressure and altitude, and they were also able to show that the fractional equation gave less error in estimating atmospheric pressure at a certain altitude. There are many scientific papers in the literature that prove the superiority of fractional derivatives over classical ones.

There are a large number of manuscripts published in the literature that investigate the issue of the existence of a solution to fractional differential equations, whether they are sequential equations of type or nonsequential equations [7–14].

In 2012, the authors in [7] studied a nonlinear three-point boundary value problem of sequential fractional differential equations. Green’s function of the associated problem involving the classical gamma function is obtained. Existence results are obtained using Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem.

(1)

Here, D is the ordinary derivative, ψ : [0, 1] × ℝ⟶ℝ, λ ∈ ℝ + , δ is a real number such that δ ≠ ((λ + e^−λ − 1)/(λη + e^−λη − 1)).

In 2019, Ahmad et al. [15] developed the existence theory for a new kind of nonlocal three-point boundary value problems for differential equations involving both Caputo and Riemann–Liouville fractional derivatives. The existence of solutions for the multivalued problem concerning the upper semicontinuous and Lipschitz cases is proved by applying nonlinear alternative for Kakutani maps and Covitz and Nadler fixed point theorem.

(2)

where φ : [0, 1] × ℝ⟶ℝ, δ ∈ ℝ, ζ ∈ (0, 1).

It is known that fractional calculus and FDEs are used in different fields such as physics, signal and image processing, control theory, robotics, economics, biology, and metallurgy, see for example [16, 17] and references therein. On the other hand, recently, many researchers have paid much attention to hybrid differential equations of fractional order. This is because of the development and new advanced applications of fractional calculus. The fractional hybrid modeling is of great significance in different engineering fields, and it can be a unique idea for future combined research between various applied sciences, for example, see [18] in which fractional hybrid modeling of a thermostat is simulated, for some recent results on hybrid.

For FDEs, we refer to [19, 20]. Freshly, some authors have studied different characteristics of hybrid FDEs including the existence of solutions, see for some detail [21–29], and some go further and studied Hyers-Ulam stability for FDEs by different mathematical theories, see for some detail [26].

Zhao et al. [29] investigated the existence result for the fractional hybrid differential equations with Riemann–Liouville fractional derivatives given by

(3)

where ^RLD^r is Riemann–Liouville fractional derivative, f : [0, T] × ℝ⟶ℝ/{0}, g : [0, 1] × ℝ⟶ℝ are assumed to be continuous.

Hilal and Kajouni [30] studied the Caputo hybrid BVP of the form

(4)

in which f : [0, T] × ℝ⟶ℝ/{0}, g : [0, 1] × ℝ⟶ℝ are assumed to be continuous and a₁ + a₂ ≠ 0.

In [31], the authors have considered the following coupled hybrid system. A new generalization of Darbo’s theorem associated with measures of noncompactness is the main tool in their approach:

(5)

supplemented with nonlocal hybrid boundary conditions.

Inspired by the aforementioned studies, the following sequential hybrid BVP is considered for investigating the existence of the solution and for the stability of its solution via the U-H sense

(6)

After this introductory section of this work, the manuscript is organized as the following hierarchical structure: Section 2 delivers the basic elements of fractional calculus definitions, Section 3 introduces the main results of the work, Section 4 introduces the (U-H) stability result for our problem, and the last section is arranged for a numerical example to support the theoretical results.

2. Preliminaries

In this part, we present some basic elements and definitions needed to find solutions to the main mathematical problem presented in this study.

Definition 1 (see [3].)The Riemann-Liouville (RL) fractional integral is defined by

(7)

Definition 2 (see [3].)The Caputo fractional derivative of order ν of a function ϑ : ℝ⁺⟶ℝ is given by

(8)

Theorem 3 (see [3], Banach’s contraction mapping principle.)Let (S, d) be a complete metric space; H : S⟶S is a contraction then

(i)
H has a unique fixed point s ∈ S; that, is H(s) = s
(ii)
∀s₀ ∈ S, we have

Theorem 4 (see [3], nonlinear alternative of Leray-Schauder type.)Assume that V is an open subset of a Banach space U, 0 ∈ V, and be a contraction such that is bounded then either

(i)
F has a fixed point in , or
(ii)
∃μ ∈ (0, 1) and v ∈ ∂V such that v = μF(v) holds

Theorem 5 (see [2], Arzela-Ascoli theorem.)F ⊂ C(U, ℝ) is compact if and only if it is closed, bounded, and equicontinuous.

3. Main Results

Lemma 6. If h ∈ C([0, 1], ℝ), and

(9)

then the solution to the problem mentioned above is given by

(10)

Proof. Taking to , then take to the resulting equation, we get

(11)

Substitution of χ(0) = 0 and χ^′(0) = 0 in Equation (11) gives a₂ = 0 and a₀ = 0, respectively, and consequently, Equation (6) becomes

(12)

Use of the condition χ(1) = δχ(ζ) in Equation (12) yields

(13)

Inserting a₁ in Equation (12) gives

(14)

Alternatively, we have

(15)

Equation (15) is equivalent to Equation (10), which makes the proof done.

Denote the Banach space by C = C[0, 1] with the norm

. Then, the product space (C × C, ‖(χ, ϑ)‖) with the norm ‖(χ, ϑ)‖ = ‖χ‖ + ‖ϑ‖, ∀(x, y) ∈ C × C is indeed a Banach space too. We define an operator ϒ : C × C⟶C × C as

(16)

where

(17)

To construct the necessary conditions for the results of uniqueness and existence of the problem (6), let us consider the following hypotheses.

(C1) Let the functions f and g are assumed to be continuous and bounded; that is, ∃λ_f, λ_g > 0 such that

(18)

(C2) Let the functions ψ and φ are assumed to be continuous, and ∃υ_i, ℓ_i > 0, (i = 1, 2) such that

(19)

(C3) There is positive constants ω₀, θ₀, and ω_i, θ_i ≥ 0 (i = 1, 2) such that

(20)

(21)

(C4) Let S ⊂ C × C be a bounded set, then ∃κ_i > 0, (i = 1, 2) such that |ψ(τ, χ(τ), ϑ(τ))| ≤ κ₁, and

(22)

Observe that

(23)

To facilitate the calculations below, let us say

(24)

Theorem 7. If both (C1) and (C2) are satisfied, and assume that [λ_ℏΛ₁(υ₁ + υ₂) + λ_ƛΛ₂(ℓ₁ + ℓ₂)] < 1. Then, the system in Equation (6) has a unique solution.

Proof. Define a closed ball with γ ≥ (λ_ℏΛ₁N_ψ + λ_ƛΛ₂N_φ)/(1 − (λ_ℏΛ₁(υ₁ + υ₂) + λ_ƛΛ₂(ℓ₁ + ℓ₂))), where .

Observe that |ψ(τ, χ, ϑ)| = |ψ(τ, χ, ϑ) − ψ(τ, 0, 0) + ψ(τ, 0, 0)| ≤ υ₁‖χ‖ + υ₂‖ϑ‖ + N_ψ ≤ (υ₁ + υ₂)γ + N_ψ.

First, we show that . For any , we have

(25)

similar to what was done above, we get

(26)

From Equation (25) and Equation (26), we deduce that ‖ϒ(χ, ϑ)‖ ≤ γ.

Next, for (χ₁, ϑ₁), (χ₂, ϑ₂) ∈ C × C, ∀τ ∈ [0, 1], we have

(27)

Similarly, we can find

(28)

Combining Equation (27) and Equation (28) yields

(29)

Equation (29) becomes ‖ϒ(χ₁, ϑ₁) − ϒ(χ₂, ϑ₂)‖ ≤ (‖χ₁ − χ₂‖ + ‖ϑ₁ − ϑ₂‖). That is, ϒ is a contraction; consequently, Banach fixed point theorem applies; thus, the uniqueness of solutions for Equation (6) holds on [0, 1].

Theorem 8. If (C1), (C3), and (C4) are satisfied, and if (λ_ℏΛ₁ω₁ + λ_ƛΛ₂θ₁) < 1 and (λ_ℏΛ₁ω₂ + λ_ƛΛ₂θ₂) < 1, then Equation (6) has at least one solution.

Proof. In the first step, we verify that the operator ϒ : C × C⟶C × C is completely continuous; obviously, the operator is continuous as a result that ℏ, ƛ, ψ, and φ are all assumed to be continuous.

With the aid of (C4), ∀(χ, ϑ) ∈ S, we have

(30)

Similarly,

(31)

Combining the inequalities (30) and (31) yields ‖ϒ(χ, ϑ)‖ ≤ λ_ℏΛ₁κ₁ + λ_ƛΛ₂κ₂, implying that ϒ is uniformly bounded.

Next, to verify the equicontinuity for the operator ϒ, we let τ₁, τ₂ ∈ [0, 1], (τ₁ < τ₂) then

(32)

(33)

(34)

The R.H.S for both inequalities (33) and (34) tend to zero as τ₁⟶τ₂, and they are both independent on (χ, ϑ). So, operator ϒ(χ, ϑ) is equicontinuous and yields; ϒ(χ, ϑ) is completely continuous.

Finally, we establish the bounded set given by ; then, ∀τ ∈ [0, 1]; the equation (χ, ϑ) = βϒ(χ, ϑ) gives

(35)

Using the hypothesis (C3), we get

(36)

Consequently, we have

(37)

Inequality (37) can be written as follows:

(38)

where Λ₀ = min{1 − (λ_ℏΛ₁ω₁ + λ_ƛΛ₂θ₁), 1 − (λ_ℏΛ₁ω₂ + λ_ƛΛ₂θ₂)}.

Inequality (38) shows that Ω is bounded. Hence, Leray-Schauder alternative applies, implying the existence of the solution for Equation (6).

4. Stability

In this part, we address the issue of stability of solutions to the system of equations defined by Equation (6) via U-H definition.

Definition 9. The system of the coupled sequential fractional differential BVPs Equation (6) is stable in U–H sense if a real number c = max(c₁, c₂) > 0 exists so that, for any ε = max(ε₁, ε₂) > 0 and for any satisfying

(39)

there exists a unique solution (χ, ϑ) ∈ C × C of (6) with

(40)

It is clear that satisfies the inequalities (39) if there exists a function (h₁, h₂) ∈ C × C (which depends on ), such that

(i)
|h₁(τ)| < ε₁ and |h₂(τ)| < ε₂, τ ∈ [0, 1]
(ii)
For τ ∈ [0, 1]

(41)

Theorem 10. Suppose that (C2) is fulfilled. Moreover

(42)

Then, the system of coupled sequential fractional differential BVPs (6) is U–H stable.

Proof. Assume that for ε₁, ε₂ > 0 a couple satisfies the inequalities (39). Introduce the following operator

(43)

Then

(44)

(45)

From Equation (44) and Equation (45), we obtain

(46)

(47)

From Equation (27) and Equation (28), we obtain

(48)

(49)

Let (χ, ϑ) ∈ C × C be a solution of Equation (6). Thanks to Lemma 6, it is equivalent to the following integral equations:

(50)

By the same arguments in Theorem 7, we get

(51)

(52)

It follows that

(53)

Representing these inequalities as matrices, we get

(54)

Solving the above inequality, we get

(55)

where Δ = (1 − λ_ℏΛ₁(υ₁ + υ₂))(1 − λ_ƛΛ₂(ℓ₁ + ℓ₂)) − λλ_ℏΛ₁(υ₁ + υ₂)λ_ƛΛ₂(ℓ₁ + ℓ₂) ≠ 0.

Thus

(56)

For ε = max(ε₁, ε₂) and

(57)

we get

(58)

Therefore, with the aid of Definition 9, the solution of the problem Equation (6) is U–H stable.

5. Example

In this part, we present an applied example to support the theoretical results we reached in the previous part, consider the following system:

(59)

Here,

(60)

Observe that

(61)

Thus, the boundary value problem Equation (59) satisfies all the conditions of Theorem 7; consequently, the uniqueness of solution of Equation (59) is satisfied on [0, 1].

In order to explain Theorem 7, it is clear that (C1) is satisfied as follows:

(62)

Also, one can easily show that (C3) holds, taking into account that τ ∈ [0, 1], then

(63)

Also, (C4) satisfied with

(64)

Finally, easy calculations with the data above give (λ_ℏΛ₁ω₁ + λ_ƛΛ₂θ₁) = 0.203275 < 1 and (λ_ℏΛ₁ω₂ + λ_ƛΛ₂θ₂) = 0.203 < 1; all conditions of Theorem 8 hold; that is, the problem (59) has at least one solution in [0, 1].

6. Conclusion

We have studied a coupled hybrid FDEs consisting of mixed fractional derivatives such as Caputo and Riemann-Liouville fractional derivatives and nonlocal boundary conditions. Existence/uniqueness results are established via a nonlinear alternative of the Leray-Schauder and Banach fixed point theorem. We also studied the Ulam-Hyers stability of these couple of hybrid FDEs. The obtained result is well illustrated by a numerical example. The result obtained in this paper is new and significantly contributes to the existing literature on the topic.

One possible direction in which to extend the results of this paper is toward different kinds of mixed fractional differential and mixed conformable fractional differential systems of higher order. Another challenge is to find out if similar results can be derived in the case of constant/variable delays in linear/nonlinear terms.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

M.A and N. M contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgments

This work was supported through the Annual Funding track by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Project No. AN000501).

Open Research

Data Availability

No data were used to support this study.

References

1 Sabatier J. A. T. M. J., Agrawal O. P., and Machado J. T., Advances in Fractional Calculus, 2007, Springer, Dordrecht, https://doi.org/10.1007/978-1-4020-6042-7, 2-s2.0-84885710343.
10.1007/978-1-4020-6042-7
Google Scholar
2 Podlubny I., Fractional Differential Equations, 1999, Academic Press, San Diego.
Web of Science® Google Scholar
3 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006, Elsevier Science B.V, Amsterdam.
Google Scholar
4 Awadalla M., Yameni Noupoue Y. Y., and Asbeh K. A., Psi-Caputo logistic population growth model, Journal of Mathematics. (2021) 2021, 9, https://doi.org/10.1155/2021/8634280, 8634280.
10.1155/2021/8634280
Web of Science® Google Scholar
5 Awadalla M., Noupoue Y. Y. Y., and Abuasbeh K., Population growth modeling via Rayleigh-Caputo fractional derivative, Journal of Statistics Applications & Probability. (2021) 10, no. 1, 11–16, https://doi.org/10.18576/jsap/100102.
10.18576/jsap/100102
Google Scholar
6 Awadalla M., Yannick Y. Y. N., and Asbeh K. A., Modeling the dependence of barometric pressure with altitude using Caputo and Caputo–Fabrizio fractional derivatives, Journal of Mathematics. (2020) 2020, 9, https://doi.org/10.1155/2020/2417681, 2417681.
10.1155/2020/2417681
Web of Science® Google Scholar
7 Ahmad B. and Nieto J. J., Sequential fractional differential equations with three-point boundary conditions, Computers & Mathematics with Applications. (2012) 64, no. 10, 3046–3052, https://doi.org/10.1016/j.camwa.2012.02.036, 2-s2.0-84868204874.
10.1016/j.camwa.2012.02.036
Web of Science® Google Scholar
8 Klimek M., Sequential fractional differential equations with Hadamard derivative, Communications in Nonlinear Science and Numerical Simulation. (2011) 16, no. 12, 4689–4697, https://doi.org/10.1016/j.cnsns.2011.01.018, 2-s2.0-79960285223.
10.1016/j.cnsns.2011.01.018
Web of Science® Google Scholar
9 Băleanu D., Mustafa O. G., and Agarwal R. P., On L^p-solutions for a class of sequential fractional differential equations, Applied Mathematics and Computation. (2011) 218, no. 5, 2074–2081, https://doi.org/10.1016/j.amc.2011.07.024, 2-s2.0-80052260699.
10.1016/j.amc.2011.07.024
Web of Science® Google Scholar
10 Aljoudi S., Ahmad B., Nieto J. J., and Alsaedi A., A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos, Solitons & Fractals. (2016) 91, 39–46, https://doi.org/10.1016/j.chaos.2016.05.005, 2-s2.0-84969203928.
10.1016/j.chaos.2016.05.005
Web of Science® Google Scholar
11 Fazli H., Nieto J. J., and Bahrami F., On the existence and uniqueness results for nonlinear sequential fractional differential equations, Applied and Computational Mathematics. (2018) 17, no. 1, 36–47.
Web of Science® Google Scholar
12 Mahmudov N. I., Awadalla M., and Abuassba K., Nonlinear sequential fractional differential equations with nonlocal boundary conditions, Advances in Difference Equations. (2017) 2017, no. 1, https://doi.org/10.1186/s13662-017-1371-3, 2-s2.0-85031430140.
10.1186/s13662-017-1371-3
Google Scholar
13 Mahmudov N. I. and Mahmoud H., Four-point impulsive multi-orders fractional boundary value problems, Journal of Computational Analysis and Applications. (2017) 22, no. 7, 1249–1260.
Google Scholar
14 Ahmad B., Ntouyas S. K., Tariboon J., Alsaedi A., and Shammakh W., Impulsive hybrid fractional quantum difference equations, Journal of Computational Analysis & Applications. (2017) 22, no. 1.
Google Scholar
15 Ahmad B., Ntouyas K., and Alsaedi A., Existence theory for nonlocal boundary value problems involving mixed fractional derivatives, Nonlinear Analysis: Modelling and Control. (2019) 24, no. 6, 937–957, https://doi.org/10.15388/na.2019.6.6.
10.15388/NA.2019.6.6
Web of Science® Google Scholar
16 Alegria-Zamudio M., Escobar-Jimenez R. F., Gomez-Aguilar J. F., Garcıa-Morales J., and Hernandez-Perez J. A., Double pipe heat exchanger temperatures estimation using fractional observers, The European Physical Journal Plus. (2019) 134, no. 10, https://doi.org/10.1140/epjp/i2019-12939-8, 2-s2.0-85073465955.
10.1140/epjp/i2019-12939-8
Web of Science® Google Scholar
17 Chávez-Vázquez S., Gómez-Aguilar J. F., Lavín-Delgado J. E., Escobar-Jiménez R. F., and Olivares-Peregrino V. H., Applications of fractional operators in robotics: a review, Journal of Intelligent & Robotic Systems. (2022) 104, no. 4, 1–40, https://doi.org/10.1007/s10846-022-01597-1.
10.1007/s10846-022-01597-1
Web of Science® Google Scholar
18 Baleanu D., Etemad S., and Rezapour S., A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Boundary Value Problems. (2020) 2020, no. 1, 1–16, https://doi.org/10.1186/s13661-020-01361-0.
10.1186/s13661-020-01361-0
Web of Science® Google Scholar
19 Dhage B. C., Existence and attractivity theorems for nonlinear hybrid fractional integrodifferential equations with anticipation and retardation, CUBO. (2020) 22, no. 3, 325–350, https://doi.org/10.4067/S0719-06462020000300325.
10.4067/S0719-06462020000300325
Google Scholar
20 Ji D. and Ge W., A nonlocal boundary value problems for hybrid ϕ-Caputo fractional integro-differential equations, AIMS Mathematics. (2020) 5, no. 6, 7175–7190, https://doi.org/10.3934/math.2020459.
10.3934/math.2020459
Google Scholar
21 Ahmad B., Ntouyas S. K., and Alsaedi A., Existence results for a system of coupled hybrid fractional differential equations, The Scientific World Journal. (2014) 2014, 6, https://doi.org/10.1155/2014/426438, 2-s2.0-84907253364, 426438.
10.1155/2014/426438
Google Scholar
22 Sutar S. T. and Kucche K. D., On nonlinear hybrid fractional differential equations with Atangana-Baleanu-Caputo derivative, Chaos, Solitons & Fractals. (2021) 143, article 110557, https://doi.org/10.1016/j.chaos.2020.110557.
10.1016/j.chaos.2020.110557
Web of Science® Google Scholar
23 Sitho S., Ntouyas S. K., and Tariboon J., Existence results for hybrid fractional integro-differential equations, Boundary Value Problems. (2015) 2015, no. 1, https://doi.org/10.1186/s13661-015-0376-7, 2-s2.0-84948445364.
10.1186/s13661-015-0376-7
Google Scholar
24 Ullah Z., Ali A., Khan R. A., and Iqbal M., Existence results to a class of hybrid fractional differential equations, Matrix Science Mathematic. (2018) 2, no. 1, 13–17, https://doi.org/10.26480/msmk.01.2018.13.17.
10.26480/msmk.01.2018.13.17
Google Scholar
25 Abbas M. I. and Ragusa M. A., On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function, Symmetry. (2021) 13, no. 2, https://doi.org/10.3390/sym13020264.
10.3390/sym13020264
Web of Science® Google Scholar
26 Khan H., Tunc C., Chen W., and Khan A., Existence theorems and Hyers–Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator, Journal of Applied Analysis & Computation. (2018) 8, no. 4, 1211–1226.
10.11948/2018.1211
Web of Science® Google Scholar
27 Baleanu D., Etemad S., Pourrazi S., and Rezapour S., On the new fractional hybrid boundary value problems with three-point integral hybrid conditions, Advances in Difference Equations. (2019) 2019, no. 1, https://doi.org/10.1186/s13662-019-2407-7.
10.1186/s13662-019-2407-7
PubMed Web of Science® Google Scholar
28 Mahmudov N. and Matar M., Existence of mild solutions for hybrid differential equations with arbitrary fractional order, TWMS Journal of Pure and Applied Mathematics. (2017) 8, 160–169.
Web of Science® Google Scholar
29 Zhao Y., Sun S., Han Z., and Li Q., Theory of fractional hybrid differential equations, Computers & Mathematcs with Applications. (2011) 62, no. 3, 1312–1324, https://doi.org/10.1016/j.camwa.2011.03.041, 2-s2.0-79961001020.
10.1016/j.camwa.2011.03.041
Web of Science® Google Scholar
30 Hilal K. and Kajouni A., Boundary value problems for hybrid differential equations with fractional order, Advances in Difference Equations. (2015) 2015, no. 1, https://doi.org/10.1186/s13662-015-0530-7, 2-s2.0-84935866638.
10.1186/s13662-015-0530-7
PubMed Google Scholar
31 Samadi A., Ntouyas S. K., and Tariboon J., Nonlocal coupled hybrid fractional system of mixed fractional derivatives via an extension of Darbo’s theorem, AIMS Mathematics. (2021) 6, no. 4, 3915–3926, https://doi.org/10.3934/math.2021232.
10.3934/math.2021232
Web of Science® Google Scholar

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On System of Mixed Fractional Hybrid Differential Equations

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Stability

5. Example

6. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

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On System of Mixed Fractional Hybrid Differential Equations

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Stability

5. Example

6. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

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