International Journal of Differential Equations

Volume 2025, Issue 1 1202608

Research Article

Open Access

Existence and Uniqueness Results for the Coupled Pantograph System With Caputo Fractional Operator and Hadamard Integral

Gunaseelan Mani

orcid.org/0000-0002-4191-3338

Department of Mathematics , Saveetha School of Engineering , Saveetha Institute of Medical and Technical Sciences , Saveetha University , Chennai , 602105 , Tamil Nadu, India , saveetha.com

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Vasu Lakshmanan,

Vasu Lakshmanan

Department of Mathematics , Easwari Engineering College , 18 Bharathi Salai Ramapuram, Chennai , 600089 , Tamil Nadu, India , srmeaswari.ac.in

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Abdul Razak Kachu Mohideen,

Abdul Razak Kachu Mohideen

Department of Mathematics , K. Ramakrishnan College of Engineering (Autonomous) , Trichy , 621112 , Tamil Nadu, India

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Homan Emadifar,

Corresponding Author

Homan Emadifar

[email protected]

orcid.org/0000-0002-8034-1475

Department of Mathematics , Saveetha School of Engineering , Saveetha Institute of Medical and Technical Sciences , Saveetha University , Chennai , 602105 , Tamil Nadu, India , saveetha.com

Department of Mathematics , Hamedan Branch , Islamic Azad University , Hamedan , Iran , azad.ac.ir

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Gunaseelan Mani,

Gunaseelan Mani

orcid.org/0000-0002-4191-3338

Department of Mathematics , Saveetha School of Engineering , Saveetha Institute of Medical and Technical Sciences , Saveetha University , Chennai , 602105 , Tamil Nadu, India , saveetha.com

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Vasu Lakshmanan,

Vasu Lakshmanan

Department of Mathematics , Easwari Engineering College , 18 Bharathi Salai Ramapuram, Chennai , 600089 , Tamil Nadu, India , srmeaswari.ac.in

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Abdul Razak Kachu Mohideen,

Abdul Razak Kachu Mohideen

Department of Mathematics , K. Ramakrishnan College of Engineering (Autonomous) , Trichy , 621112 , Tamil Nadu, India

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Homan Emadifar,

Corresponding Author

Homan Emadifar

[email protected]

orcid.org/0000-0002-8034-1475

Department of Mathematics , Saveetha School of Engineering , Saveetha Institute of Medical and Technical Sciences , Saveetha University , Chennai , 602105 , Tamil Nadu, India , saveetha.com

Department of Mathematics , Hamedan Branch , Islamic Azad University , Hamedan , Iran , azad.ac.ir

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First published: 12 May 2025

https://doi.org/10.1155/ijde/1202608

Academic Editor: Patricia J. Y. Wong

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Abstract

The main objective of this research involves studying a new novel coupled pantograph system with fractional operators together with nonlocal antiperiodic integral boundary conditions. The system consists of nonlinear pantograph fractional equations which integrate with Caputo fractional operators and Hadamard integrals. A fixed-point approach has served to study both existence and uniqueness aspects of solutions for the proposed model. The paper investigates Hyers–Ulam stability properties of the developed solution. An appropriate example was provided to confirm the theoretical findings.

1. Introduction

Recent studies have shown that some real-world processes cannot be effectively described by classical differential equations formulated using standard derivatives and integrals, integer order ordinary DEs or partial . Fractional calculus is a powerful tool for simulating these complex processes, since it extends the idea of integer order to arbitrary order. Due to its numerous applications across a variety of scientific domains, fractional calculus has attracted a lot of attention during the past few decades. It has been applied to a variety of scientific disciplines, including polymer rheology, biology, capacitor theory, the blood-Bow phenomenon, nonlinear seismic oscillation, image processing, aerodynamics, geophysics, and electrical circuits. We cite some publications [1–4] for other applications of fractional DEs .

Among the most famous equations, there is the pantograph equation, for which we can state that the pantograph phenomenon is an essential component of electric trains which collects electrical current from overload lines. This equation has been modeled by Ockendon and Tayler [5]. In recent years, many researchers have proposed several fractional variants of the above pantograph equation. For more details, see for instance the research works [6–17] and references therein the evolution of physical events in an environment that is fluctuating is described by the pantograph equation. On the other side, in recent years, we have seen comprehensive discussions on FDEs with different boundary conditions [18–27]. In between, many researchers have conducted an analysis on dynamics of the LE. Yu et al. [28] examine the existence results obtained using the fixed point criteria for in the Langevin settings with various types of boundary conditions. By employing upper and lower solution technique analyses, Baghani et al. [29] connected some results for coupled Langevin systems with various forms of boundary conditions. Discussion on with nonlocal fractional boundary conditions is provided by Fazli and Nieto [30] in 2018.

A close approximation of the exact solutions for

is appeared in the notion of the Hyers–Ulam

stability. Rizwan and Zada investigated the stability and existence results in relation to the LEs in their article [31]. Later in another study, the stability and existence of unique solution were addressed for the mentioned equation by Wang and Li [32]. After that, Wang and Wang [33] turned to similar theorems on the existence notion, but with strip conditions that include the p-Laplacian operator. Matar et al. [34] presented their stability theorems in the version of the fractional system of the coupled

. Khan et al. [35] analyzed positive solutions together with the Hyers–Ulam stability for a general nonlinear Atangana–Baleanu–Caputo FDE with singularity and nonlinear p-Laplacian operator in Banach’s space. Khan et al. [36] studied approximate numerical solutions for the newly developed difference Caputo order Cancer model through their application of the iterative numerical method together with neural networks. The spread of worms in wireless sensor systems has been examined using nonlinear discrete fractional-order mathematical models according to Khan et al. [37]. Several authors analyzed fractional-order differential equations by applying various boundary conditions. The literature contains various studies which investigate classical as well as initial value problems together with periodic/antiperiodic and nonlocal and multipoint and integral boundary conditions and Integral Fractional Boundary Condition through works like those of Ahmed et al. [38] and their colleagues’ monographs and Benchohra et al. [39] and the monograph of Benhamida et al. [40] and Chergui et al. [41] and Chen et al. [42] and Goodrich et al. [43]. Hammad et al. [44] analyzed Riemann-Liouville type two-term FDEs that incorporate fractional bi-order schemes. Motivated by the above authors, we investigate the existence of solutions and stability results for the following BVP incorporating nonlinear coupled pantograph with Caputo fractional derivatives and Hadamard integrals are investigated:

()

where

refer to the Caputo derivatives of fractional orders ξ, ξ^∗, ω, ω^∗, respectively. Further

and

. Also,

. This version of BVP combines two fractional operators in the main pantograph system and its boundary conditions. Also, the investigation of the qualitative properties of solutions for such a coupled pantograph system is the main goal of this paper. Despite the current interest in the topic of

with boundary conditions, no articles specifically addressing this type of combined coupled pantograph system have been published.

The rest of the paper is organized as follows. Basic definitions and lemmas are found in Section 2. Section 3 contains the existence and uniqueness of the solution to the considered problem. -stability of the solution is discussed in the next section. We propose an application to provide a more transparent result in Section 5.

2. Preliminaries

First of all, the aim of this section is to present some important basics that help the readers in understanding our study.

Definition 1 (see [2].)Let be a continuous and integrable function , then the Hadamard integral of fractional order σ > 0 is defined as

()

Definition 2 (see [2].)Let be a , σ ≥ 0 and . The Hadamard derivative of fractional order σ is described as

()

Definition 3 (see [2].)Let be a -times continuously differentiable function. The Caputo derivative of order σ > 0 is formulated as

()

where

and

Lemma 1 (see [2].)Suppose that ϰ > 0, κ = [Re(ϰ)] + 1 and Re(κ) > 0, then

()

The next theorem is a fixed point criterion for proving our main existence theorem.

Theorem 1 (see [2].)Assume that Δ ≠ ∅ is a convex, closed and bounded set in a Banach space Ξ. Assume also that Ω₁, Ω₂ : Δ⟶Ξ are operators such that

1.
for ϱ, ϑ ∈ Δ, Ω₁ϱ + Ω₂ϑ ∈ Δ;
2.
Ω₂ is contraction;
3.
Ω₂ is compact and continuous.

Then, ∃ϱ^∗ ∈ Δ such that ϱ^∗ = Ω₁ϱ^∗ + Ω₂ϱ^∗.

3. Existence and Uniqueness Results

We begin this part with the proof of an auxiliary lemma which gives us the main integral form of the solutions of the coupled pantograph system of equation (1).

Lemma 2. Assume that and the BVP including the coupled pantograph fractional is given by equation (1). Then the form of the solution to problem equation (1) is

()

and

()

where

and

Proof 1. Applying to both sides of the first equation of (1), one can get

()

with the real constants

. Again, using

to both sides of equation (8), one can get

()

where

and

are arbitrary constants. From the conditions ϱ(1) = 0 and

, we have

. Hence, equation (9) reduces to

()

Applying boundary conditions , we obtain that

()

where

. In view of equation (11) and inserting it in equation (10), we obtain equation (6). Similarly, using the same method as above and using conditions

and

, we get equation (7).

Assume that is the space of all continuous functions from to . Define the space Θ = {ϱ(ν) : ϱ(ν) ∈ ℧} equipped with . Also, consider Υ = {ϑ(ν) : ϑ(ν) ∈ ℧} equipped with . Obviously, both (Θ, ‖·‖) and (Υ, ‖·‖) are Banach spaces. Clearly, the product (Θ × Υ, ‖·‖) is a Banach space endowed with ‖(ϱ, ϑ)‖ = ‖ϱ‖ + ‖ϑ‖.

Using Lemma 2, we apply the operator Ω : Θ × Υ⟶Θ × Υ as follows:

()

where

()

and

()

Thus, the existence of the solution for the nonlinear coupled pantograph system (1) is limited to finding the fixed point of equation (12).

Next, we state the necessary hypotheses for continuing this study.

(S1) is continuous.
(S2) For all , and , there exist constants such that
()
(S3) For , there exist such that
()
(S4) For all , there exist such that
()

We now turn to the uniqueness property for solutions of the nonlinear coupled pantograph system (1).

Theorem 2. Suppose that (S1)-(S4) to be held and if h < 1. Then the nonlinear coupled pantograph system (1) has a unique solution, where

()

Proof 2. Let be closed bounded and convex subsets of Θ × Υ, where

()

where

and

. To finish the proof, it is enough to investigate the boundedness of

, that is,

and Ω is a contraction mapping. For any

, estimate

()

and

()

Now, we have

()

In the same manner, we obtain

()

From equations (22) and (23), we obtain

()

Hence, . Finally, for completing the next part of the proof, for each , we have

()

where

()

Similarly,

()

where

()

Therefore,

()

it follows that

()

Since h < 1, then Ω is a contraction. Consequently, an operator Ω has a unique fixed point when utilizing the Banach contraction theorem. Therefore, there is a unique solution to the nonlinear coupled pantograph system (1).

Now, everything is ready to turn to the main existence theorem.

Theorem 3. Assume that the hypotheses (S1)-(S3) hold, then there exists at least one solution to the nonlinear coupled pantograph system (1) on if h₅ < 1, where h₅ = max{h₃, h₄}.

Proof 3. Let be closed bounded and convex subsets of Θ × ϒ, where

()

where

and

. For any

, define the operators Ω, Ω^∗ : Θ × ϒ⟶Θ × ϒ as follows:

()

and

()

where

()

We split the proof into the following steps:

Step 1: We shall show that , for all . For this,
()
In the same manner, we obtain
()
From equations (35) and (36), we obtain
()
Step 2: Ω^∗ is contraction: For this regard, let . We have
()
where
()
which implies that
()
Similarly, we can prove that
()
where
()
Therefore,
()
it follows that
()
Since h₅ < 1, then Ω^∗ is a contraction.
Step 3: Ω is continuous and compact: Since , then it is continuous on . Hence, Ω is continuous. To show that Ω is compact, it is sufficient to claim that Ω is relatively compact and uniformly bounded on . Now, for , we have
()

Similarly, we can write

()

It follows from equations (45) and (46) that Ω is uniformly bounded on . Consider

()

which yields

()

From Lagrange mean value theorem, we can write

()

where u₁ and u₂ are independent of ν. Applying equations (49) in (48) and letting ν₂⟶ν₁, we conclude that

()

By the same technique, we have

()

Therefore, Ω is equicontinuous. Thank to Arzela–Ascoli theorem, Ω is compact on . This completes condition (3) in Theorem 1. Thus, there exists at least one solution to the nonlinear coupled pantograph system (1).

4. Stability Analysis

The -stability of the nonlinear coupled pantograph system (1) will be discussed in this step. First, the -stability of the considered nonlinear coupled pantograph system is defined as follows.

Definition 4. The solution of the nonlinear coupled pantograph system (1) is called -stable if such that for every and (ϱ, ϑ) ∈ ℧×℧ as a solution to the system

()

for all

, there exists a unique solution

to the nonlinear coupled pantograph system (1) such that

()

where

Remark 1. The pair (ϱ, ϑ) ∈ ℧×℧ is called a solution to the nonlinear coupled pantograph system (1) if and only if there exist such that for all ,

i.
and ;
ii.
,
iii.
.

Lemma 3. Based on Remark 1, for each , the solution of the system

()

where

and

, under boundary conditions

()

is given by

()

and

()

Proof 4. From Lemma 2, the desired relations are proved.

Theorem 4. If the assumptions (S1)-(S3) hold, then the solution to the nonlinear coupled pantograph system (1) is -stable if

()

where the latter constants will be introduced later.

Proof 5. Thank to Lemma 3, if (ϱ, ϑ) is a solution to equation (52), and is a solution to equation (1), then, we get

()

which implies that

()

Therefore,

()

where

()

and

()

Hence,

()

Setting ∇₁ = (2b₁ζ_Λ/1 − (2b₁ζ_Λ + b₂)) and ∇₂ = (b₃/1 − (2b₁ζ_Λ + b₂))), we can write the above inequality as

()

Similarly, we have

()

where

and

The inequalities equations (65) and (66) can be written as

()

Using the inverse of the matrix, we get

()

where

. Based on equation (68), we have

()

Let us consider and

()

Putting

()

then we have

. This proves that the solution of the nonlinear coupled pantograph system (1) is

-stable.

5. An Application

This part is devoted to presenting an illustrative example in the form of the given nonlinear coupled pantograph system to support the results.

Example 1. Consider the nonlinear coupled pantograph system

()

Problem equation (72) is a special case of equation (1) with (1/7) = β ≠ γ = (1/6), (1/6) = β^∗ ≠ γ^∗ = (1/5), ξ = (5/3), ω = (1/7), ξ^∗ = (5/4), ω^∗ = (1/4), ξ + ω ∈ (1, 3], ξ^∗ + ω^∗ ∈ (1, 3], γ = (1/6) > 0, γ^∗ = (1/5) > 0. Moreover,

()

Clearly,

()

with

and

. Hence, the assumptions (S1)-(S4) are satisfied. By doing some calculations, we find that

()

Hence, h ≈ 0.5023 < 1. Therefore all requirements of Theorem 2 are fulfilled. Then the nonlinear coupled pantograph system (72) has a unique solution on

. Moreover,

()

such that

. Thus, Theorem 4 is satisfied. Then the unique solution of equation (72) is

-stable.

6. Conclusion

In some applicable areas of sciences like physics, chemistry, and electrical engineering, fractional pantograph have been an important subject. Many researchers substituted FDEs instead of the standard classical DEs in order to reduce noise and staircase effects. Therefore, in this paper, by defining a coupled system of such pantograph in its fractional form, we discussed on some qualitative properties. With the help of the Krasnoselskii’s fixed point theorem and the Banach principle, the existence and uniqueness results of the pantograph fractional DEs have been investigated under the Caputo derivatives. Despite the current interest in the topic of FDEs with boundary conditions, no articles specifically addressing this type of combined coupled pantograph system have been published. Stability property was also checked completely in relation to the solutions of the mentioned system. An application that describes and illustrates the validity of the results was provided as a numerical example.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

No financial support was received.

Open Research

Data Availability Statement

No data were used for the research described in this article.

References

1 Podlubny I., Fractional Differential Equations, 1999, Academic Press.
Web of Science® Google Scholar
2 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, Elsevier Science Limited.
10.1016/S0304-0208(06)80001-0
Google Scholar
3 Sabatier J., Agrawal O. P., and Machado J. A. T., Advances in Fractional Calculus, 2007, 4, Springer.
10.1007/978-1-4020-6042-7
Google Scholar
4 Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 2010, World Scientific.
10.1142/p614
Google Scholar
5 Ockendon J. R. and Tayler A. B., The Dynamics of a Current Collection System for an Electric Locomotive, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. (1971) 322, no. 1551, 447–468.
10.1098/rspa.1971.0078
Web of Science® Google Scholar
6 Afshri H., Marasi H. R., and Alzabut J., Applications of New Contraction Mappings on Existence and Uniqueness Results for Implicit ϕ-Hilfer Fractional Pantograph Differential Equations, Journal of Inequalities and Applications. (2021) 185, 1–14.
Google Scholar
7 Ahmad I., Nieto J. J., Rahman G. U., and Shah K., Existence and Stability for Fractional Order Pantograph Equations With Nonlocal Conditions, The Electronic Journal of Differential Equations. (2020) 2020, 132–216, https://doi.org/10.58997/ejde.2020.132.
10.58997/ejde.2020.132
Google Scholar
8 George R., Houas M., Ghaderi M., Rezapour S., and Elagan S. K., On a Coupled System of Pantograph Problem With Three Sequential Fractional Derivatives by Using Positive Contraction-Type Inequalities, Results in Physics. (2022) 39, 105687–687, https://doi.org/10.1016/j.rinp.2022.105687.
10.1016/j.rinp.2022.105687
Google Scholar
9 Guida K., Ibnelazyz L., Hilal K., and Melliani S., Existence and Uniqueness Results for Sequential ψ-Hilfer Fractional Pantograph Differential Equations With Mixed Nonlocal Boundary Conditions, AIMS Mathematics. (2021) 6, no. 8, 8239–8255, https://doi.org/10.3934/math.2021477.
10.3934/math.2021477
Google Scholar
10 Houas M., Sequential Fractional Pantograph Differential Equations With Nonlocal Boundary Conditions: Uniqueness and Ulam-Hyers-Rassias Stability, Results in Nonlinear Analysis. (2022) 1, 29–41.
Google Scholar
11 Houas M., Existence and Stability of Fractional Pantograph Differential Equations With Caputo Hadamard Type Derivative, Turkish Journal of Inequalities. (2020) 4, no. 2, 29–38.
Google Scholar
12 Houas M., Devi A., and Kumar A., Existence and Stability Results for Fractional-Order Pantograph Differential Equations Involving Riemann-Liouville and Caputo Fractional Operators, International Journal of Dynamics and Control. (2023) 11, no. 3, 1386–1395, https://doi.org/10.1007/s40435-022-01005-4.
10.1007/s40435-022-01005-4
Web of Science® Google Scholar
13 Houas M., Kaushik K., Kumar A., Khan A., and Abdeljawad T., Existence and Stability Results of Pantograph Equation With Three Sequential Fractional Derivatives, AIMS Mathematics. (2022) 8, no. 3, 5216–5232, https://doi.org/10.3934/math.2023262.
10.3934/math.2023262
Google Scholar
14 Khan M. B. A., Abdeljawad T., Shah K., Ali G., Khan H., and Khan A., Study of a Nonlinear Multi-Terms Boundary Value Problem of Fractional Pantograph Differential Equations, Advances in Difference Equations. (2021) 143, 1–15.
Google Scholar
15 Vivek D., Kanagarajan K., and Sivasundaram S., Dynamics and Stability of Pantograph Equations via Hilfer Fractional Derivative, Nonlinear Studies. (2016) 23, 685–698.
Google Scholar
16 Vivek D., Elsayed E. M., and Kanagarajan K., Existence and Ulam Stability Results for a Class of Boundary Value Problem of Neutral Pantograph Equations With Complex Order, SeMA Journal. (2020) 77, no. 3, 243–256, https://doi.org/10.1007/s40324-020-00214-1.
10.1007/s40324-020-00214-1
Google Scholar
17 Zhang Y. and Li L., Stability of Numerical Method for Semi-Linear Stochastic Pantograph Differential Equations, Journal of Inequalities and Applications. (2016) 2016, 30–11, https://doi.org/10.1186/s13660-016-0971-x, 2-s2.0-84956537534.
10.1186/s13660-016-0971-x
Google Scholar
18 Gambo Y. Y., Jarad F., Baleanu D., and Abdeljawad T., On Caputo Modification of the Hadamard Fractional Derivatives, Advances in Differential Equations. (2014) 2014, 10–12, https://doi.org/10.1186/1687-1847-2014-10, 2-s2.0-84899642427.
10.1186/1687-1847-2014-10
Web of Science® Google Scholar
19 Hammad H. A., De la Sen M., and Aydi H., Analytical Solution for Differential and Nonlinear Integral Equations via Fϖ_e-Suzuki Contractions in Modified ϖ_e-metric-like Spaces, Journal of Function Spaces. (2021) 2021.
10.1155/2021/6128586
Google Scholar
20 Hammad H. A., Aydi H., Işık H., and De la Sen M., Existence and Stability Results for a Coupled System of Impulsive Fractional Differential Equations with Hadamard Fractional Derivatives, AIMS Mathematics. (2023) 8, no. 3, 6913–6941, https://doi.org/10.3934/math.2023350.
10.3934/math.2023350
Web of Science® Google Scholar
21 Khan H., Alzabut J., Baleanu D., Alobaidi G., and Rehman M. U., Existence of Solutions and a Numerical Scheme for a Generalized Hybrid Class of N-Coupled Modified ABC-Fractional Differential Equations With an Application, AIMS Mathematics. (2023) 8, no. 3, 6609–6625, https://doi.org/10.3934/math.2023334.
10.3934/math.2023334
Web of Science® Google Scholar
22 Adjimi N., Boutiara A., Abdo M. S., and Benbachir M., Existence Results for Nonlinear Neutral Generalized Caputo Fractional Differential Equations, Journal of Pseudo-Differential Operators and Applications. (2021) 12, no. 2, 25–17, https://doi.org/10.1007/s11868-021-00400-3.
10.1007/s11868-021-00400-3
Web of Science® Google Scholar
23 Boutiara A., Etemad S., Hussain A., and Rezapour S., The Generalized U–H and U–H Stability and Existence Analysis of a Coupled Hybrid System of Integro-Differential IVPs Involving ϕ-Caputo Fractional Operators, Advances in Differential Equations. (2021) 2021, no. 1, https://doi.org/10.1186/s13662-021-03253-8.
10.1186/s13662-021-03253-8
Google Scholar
24 Boutiara A., Benbachir M., and Guerbati K., Boundary Value Problem for Nonlinear Caputo–Hadamard Fractional Differential Equation With Hadamard Fractional Integral and Antiperiodic Conditions, Facta University Series, Mathematical Information. (2021) 36, no. 1, 735–748.
Google Scholar
25 Hammad H. A. and Zayed M., Solving Systems of Coupled Nonlinear Atangana–Baleanu-Type Fractional Differential Equations, Boundary Value Problems. (2022) 2022, no. 1, https://doi.org/10.1186/s13661-022-01684-0.
10.1186/s13661-022-01684-0
PubMed Web of Science® Google Scholar
26 Humaira, Hammad H. A., Sarwar M., and De la Sen M., Existence Theorem for a Unique Solution to a Coupled System of Impulsive Fractional Differential Equations in Complex-Valued Fuzzy Metric Spaces, Advances in Differential Equations. (2021) 2021, no. 1, https://doi.org/10.1186/s13662-021-03401-0.
10.1186/s13662-021-03401-0
Google Scholar
27 Hammad H. A., Rashwan R. A., Nafea A., Samei M. E., and Noeiaghdam S., Stability Analysis for a Tripled System of Fractional Pantograph Differential Equations With Nonlocal Conditions, Journal of Vibration and Control. (2023) 30, no. 3-4, 632–647, https://doi.org/10.1177/10775463221149232.
10.1177/10775463221149232
Web of Science® Google Scholar
28 Yu T., Deng K. E., and Luo M., Existence and Uniqueness of Solutions of Initial Value Problems for Nonlinear Langevin Equation Involving Two Fractional Orders, Communications in Nonlinear Science and Numerical Simulation. (2014) 19, no. 6, 1661–1668, https://doi.org/10.1016/j.cnsns.2013.09.035, 2-s2.0-84890797227.
10.1016/j.cnsns.2013.09.035
Web of Science® Google Scholar
29 Baghani H., Alzabut J., and Nieto J. J., A Coupled System of Langevin Differential Equations of Fractional Order and Associated to Antiperiodic Boundary Conditions, Mathematical Methods in the Applied Sciences. (2020) 47, no. 13, 10900–10910, https://doi.org/10.1002/mma.6639.
10.1002/mma.6639
Web of Science® Google Scholar
30 Fazli H. and Nieto J. J., Fractional Langevin Equation With Anti-Periodic Boundary Conditions, Chaos, Solitons & Fractals. (2018) 114, 332–337, https://doi.org/10.1016/j.chaos.2018.07.009, 2-s2.0-85050616271.
10.1016/j.chaos.2018.07.009
Web of Science® Google Scholar
31 Rizwan R. and Zada A., Existence Theory and Ulam’s Stabilities of Fractional Langevin Equation, Qualitative Theory of Dynamical Systems. (2021) 20, no. 2, https://doi.org/10.1007/s12346-021-00495-5.
10.1007/s12346-021-00495-5
PubMed Web of Science® Google Scholar
32 Wang J. and Li X., Ulam–Hyers Stability of Fractional Langevin Equations, Applied Mathematics and Computation. (2015) 258, 72–83, https://doi.org/10.1016/j.amc.2015.01.111, 2-s2.0-84923280073.
10.1016/j.amc.2015.01.111
Web of Science® Google Scholar
33 Wang G. and Wang T., On a Nonlinear Hadamard Type Fractional Differential Equation With P-Laplacian Operator and Strip Condition, The Journal of Nonlinear Science and Applications. (2016) 9, no. 7, 5073–5081, https://doi.org/10.22436/jnsa.009.07.10.
10.22436/jnsa.009.07.10
Google Scholar
34 Matar M. M., Alzabut J., and Jonnalagadda J. M., A Coupled System of Nonlinear Caputo Hadamard Langevin Equations Associated With Nonperiodic Boundary Conditions, Mathematical Methods in the Applied Sciences. (2020) 44, no. 3, 2650–2670, https://doi.org/10.1002/mma.6711.
10.1002/mma.6711
Web of Science® Google Scholar
35 Khan A., Khan H., Gómez-Aguilar J. F., and Abdeljawad T., Existence and Hyers-Ulam Stability for a Nonlinear Singular Fractional Differential Equations With Mittag-Leffler Kernel, Chaos, Solitons & Fractals. (2019) 127, 422–427, https://doi.org/10.1016/j.chaos.2019.07.026, 2-s2.0-85069616688.
10.1016/j.chaos.2019.07.026
Web of Science® Google Scholar
36 Khan A., Abdeljawad T., Abdel-Aty M., and Almutairi D. K., Digital Analysis of Discrete Fractional Order Cancer Model by Artificial Intelligence, Alexandria Engineering Journal. (2025) 118, 115–124, https://doi.org/10.1016/j.aej.2025.01.036.
10.1016/j.aej.2025.01.036
Web of Science® Google Scholar
37 Khan A., Abdeljawad T., and Alkhawar H. M., Digital Analysis of Discrete Fractional Order Worms Transmission in Wireless Sensor Systems: Performance Validation by Artificial Intelligence, Modeling Earth Systems and Environment. (2025) 11, no. 1, https://doi.org/10.1007/s40808-024-02237-3.
10.1007/s40808-024-02237-3
PubMed Web of Science® Google Scholar
38 Agarwal R. P., Ahmad B., and Alsaedi A., Fractional-Order Differential Equations With Anti-Periodic Boundary Conditions: A Survey, Bound, Value Problems. (2017) 2017, 1–27.
Google Scholar
39 Benchohra M., Hamani S., and Ntouyas S. K., Boundary Value Problems for Differential Equations With Fractional Order, Surveys in Mathematics and its Applications. (2008) 3, 1–12.
Google Scholar
40 Benhamida W., Graef J. R., and Hamani S., Boundary Value Problems for Fractional Differential Equations With Integral and Anti-Periodic Conditions in a Banach Space, Progress in Fractional Differentiation and Applications. (2018) 4, no. 2, 65–70, https://doi.org/10.18576/pfda/040201, 2-s2.0-85051075173.
10.18576/pfda/040201
Google Scholar
41 Chergui D., Eddine Oussaeif T., and Ahcene M., Existence and Uniqueness of Solutions for Nonlinear Fractional Differential Equations Depending on Lower-Order Derivative With Non-Separated Type Integral Boundary Conditions, AIMS Mathematics. (2019) 4, no. 1, 112–133, https://doi.org/10.3934/math.2019.1.112, 2-s2.0-85073334227.
10.3934/Math.2019.1.112
Web of Science® Google Scholar
42 Chen T. and Liu W., An Anti-Periodic Boundary Value Problem for the Fractional Differential Equations With a P-Laplacian Operator, Applied Mathematics Letters. (2012) 25, no. 11, 1671–1675, https://doi.org/10.1016/j.aml.2012.01.035, 2-s2.0-84865642374.
10.1016/j.aml.2012.01.035
Web of Science® Google Scholar
43 Goodrich C., Existence and Uniqueness of Solutions to a Fractional Difference Equation With Nonlocal Conditions, Computers & Mathematics With Applications. (2011) 61, no. 2, 191–202, https://doi.org/10.1016/j.camwa.2010.10.041, 2-s2.0-78651230726.
10.1016/j.camwa.2010.10.041
Web of Science® Google Scholar
44 Hammad H. A., Aydi H., and De la Sen M., The Existence and Stability Results of Multi-Order Boundary Value Problems Involving Riemann-Liouville Fractional Operators, AIMS Mathematics. (2023) 8, no. 5, 11325–11349, https://doi.org/10.3934/math.2023574.
10.3934/math.2023574
Google Scholar

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Existence and Uniqueness Results for the Coupled Pantograph System With Caputo Fractional Operator and Hadamard Integral

Abstract

1. Introduction

2. Preliminaries

3. Existence and Uniqueness Results

4. Stability Analysis

5. An Application

6. Conclusion

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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Existence and Uniqueness Results for the Coupled Pantograph System With Caputo Fractional Operator and Hadamard Integral

Abstract

1. Introduction

2. Preliminaries

3. Existence and Uniqueness Results

4. Stability Analysis

5. An Application

6. Conclusion

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

Related

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