Fixed Point Approach for Fractional Order Differential Equation Systems

Lale Cona; Alperen Hasan Kocağ

doi:10.54187/jnrs.1671939

Fixed Point Approach for Fractional Order Differential Equation Systems

Abstract

This paper examines the existence and uniqueness of solutions to a nonlinear system of fractional differential equations involving the Atangana–Baleanu fractional derivative. The system under consideration is analyzed through a fixed-point approach by means of the Perov sense. The Atangana–Baleanu fractional derivative, characterized by a non-local and non-singular kernel, provides a more suitable framework for modeling various physical phenomena. The main results are illustrated through an example, which demonstrates the applicability and reliability of the proposed approach.

Keywords

Atangana-Baleanu Caputo fraction order derivative, initial value problem, fraction order differential equation systems, fixed point theorem

References

K. B. Oldham, J. Spanier, The fractional calculus – Theory and applications of differentiation and integration to arbitrary order, Vol. 111 of Mathematics in Science and Engineering, 1st Edition, Elseiver Science, New York, 1974.
J. Padovan, Computational algorithms for FE formulations involving fractional operators, Computational Mechanics 2 (4) (1987) 271–287.
K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, New York, 1993.
I. Podlubny, Fractional differential equations – An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198 of Mathematics in Science and Engineering, 1st Edition, Academic Press, San Diego, 1999.
H. Bilgil, S. Yüksel, Comparision of conformable and Caputo fractional grey models, Journal of Computational and Applied Mathematics 463 (2025) 116500.
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications 1 (2) (2015) 73–85.
A. Atangana, B. Dumitru, New fractional derivatiands with nonlocal and non-singular kernel: Theroy and aplicaion to heat transfer model, Thermal Sciences 20 (2) (2016) 763–769.
A. Chidouh, A. Guezane-Lakoud, R. Bebbouchi, Positive solutions of the fractional relaxation equation using lower and upper solutions, Vietnam Journal of Mathematics 44 (4) (2016) 739–748.
M. ur Rahman, M. Arfan, D. Baleanu, Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions, Bulletin of Biomathematics 1 (1) (2023) 1–23.
N. A. Shah, D. Vieru, C. Fetecau, Effects of the fractional order and magnetic field on the blood flow in cylindrical domains, Journal of Magnetism and Magnetic Materials 409 (2016) 10–19.

Z. Ali, S. N. Nia, F. Rabiei, K. Shah, M. K. Tan, A semi-analytical approach for the solution of time-fractional Navier-Stokes equation, Advances in Mathematical Physics 2021 (2021) 5547804.
V. Tripathi, S. Das, Time-optimal feedback control of nonlocal Hilfer fractional state-dependent delay inclusion with Clarke’s subdifferential, Mathematical Methods in the Applied Sciences 47 (9) (2024) 7637–7657.
M. Hajipour, A. Jajarmi, D. Baleanu, H. Sun, On an accurate discretization of a variable-order fractional reaction-diffusion equation, Communications in Nonlinear Science and Numerical Simulation 69 (2019) 119–133.
U. B. Odionyenma, N. Ikenna, B. Bolaji, Analysis of a model to controlthe co-dynamics of Chlamydia and Gonorrhea using Caputo fractional derivative, Mathematical Modelling and Numerical Simulation with Applications 3 (2) (2023) 111–140.
A. Jajarmi, M. Hajipour, D. Baleanu, New aspects of the adaptive synchronization and hyperchaos suppression of a financial model, Chaos, Solitons & Fractals 99 (2017) 285–296.
H. Joshi, B. K. Jha, M. Yavuz, Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data, Mathematical Biosciences and Engineering 20 (1) (2023) 213-240.
M. Hajipour, A. Jajarmi, D. Baleanu, An efficient nonstandard finite difference scheme for a class of fractional chaotic systems, Journal of Computational and Nonlinear Dynamics 13 (2) (2018) 021013.
M. Wang, S. Wang, X. Ju, Y. Wang, Image denoising method relying on iterative adaptive weight-mean filtering, Symmetry 15 (2023) 1181.
M. Hashemi, E. Ashpazzadeh, M. Moharrami, M. Lakestani, Fractional order Alpert multiwavelets for discretizing delay fractional differential equation of pantograph type, Applied Numerical Mathematics 170 (2021) 1–13.
D. Baleanua, M. Hajipourc, A. Jajarmid, An accurate finite difference formula for the numerical solution of delay-dependent fractional optimal control problems, An International Journal of Optimization and Control: Theories & Applications 14 (3) (2024) 183–192.
N. H. Can, H. Jafari, M. N. Ncube, Fractional calculus in data fitting, Alexandria Engineering Journal 59 (2020) 3269–3274.
N. A. Shah, D. Vieru, C. Fetecau, Effects of the fractional order and magnetic field on the blood flow in cylindrical domains, Journal of Magnetism and Magnetic Materials 409 (2016) 10–19.
Z. Ali, S. N. Nia, F. Rabiei, K. Shah, M. K. Tan, A semi-analytical approach for the solution of time-fractional Navier-Stokes equation, Advances in Mathematical Physics 2021 (2021) 5547804.
Y. Jiang, B. Zhang, X. Shu, Z. Wei, Fractional-order autonomous circuits with order larger than one, Journal of Advanced Research 25 (2020) 217–225.
O. W. Abdulwahhab, N. H. Abbas, A new method to tune a fractional-order PID controller for a twin rotor aerodynamic system, Arabian Journal for Science and Engineering 42 (2017) 5179–5189.
H. Gu, Forecasting carbon dioxide emission regional difference in China by damping fractional grey model, Fractal and Fractional 8 (2024) 597.
H. Bilgil, Ü. Erdinç, China total energy consumption forecast with optimized continuous conformable fractional grey model, Alphanumeric Journal 12 (2024) 157–168.
Y. Zhou, Existence and uniqueness of solitions for a system of fractional differential equations, Fractional Calculus and Applied Analysis 12 (2) (2009) 195–204.
B. Ahmad, J. J. Nieto, A. Alsaedi, Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions, Acta Mathematica Scientia 31 (6) (2011) 2122–2130.
B. Ahmad, S. K. Ntouyas, Fractional differential inclusions with fractional separated boundary conditions, Fractional Calculus and Applied Analysis 15 (2012) 362–382.
X. Liu, Z. Liu, Separated boundary value problem for fractional differential equations depending on lower-order derivative, Advances in Difference Equations (2013) 78.
N. I. Mahmudov, A. Al-Khateeb, Stability, existence and uniqueness of boundary value problems for a coupled system of fractional differential equations, Mathematics 7 (2019) 1–14.
D. Chergui, T. E. Oussaeif, M. Ahcene, Existence and uniqueness of solutions for nonlinear fractional differential equations depending on lower-order derivative with non-separated type integral boundary conditions, AIMS Mathematics 4 (1) (2019) 112–133.
C. Guendouz, J. E. Lazreg, J. J. Nieto, A. Ouahab, Existence and compactness results for a system of fractional differential equations, Journal of Function Spaces (2020) 5735140.
A. Y. Madani, M. N. A. Rabih, F. A. Alqarni, Z. Ali, K. A. Aldwoah, M. Hleili, Existence, uniqueness, and stability of a nonlinear tripled Fractional Order Differential System, Fractal and Fractional 8 (416) (2024) 1–19.
A. Ashyralyev, Y. A. Sharifov, Existence and uniqueness of solutions for the system of nonlinear fractional differential equations with nonlocal and integral boundary conditions, Abstract and Applied Analysis (2012) 594802.
M. I. Syam, M. Al-Refai, Fractional differential equations with Atangana–Baleanu fractional derivatiand: Analysis and applications, Chaos, Solitons & Fractals:X 2 (2019) 100013.
V. Berinde, Iterative approximation of fixed points, Vol. 1912 of Lecture Notes in Mathematics, 2nd Edition, Springer Berlin, Heidelberg, 2007.
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae 3 (1) (1922) 133–181.
R. S. Varga, Matrix iteratiand analysis, Vol. 27 of Springer Series in Computational Mathematics, 2nd Edition, Springer Berlin, Heidelberg, 2000.
I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001.
A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Približ. Metod. Rešen. Differencial. Uravnen. 2 (1964) 115–134.
J. R. Graef, J. Henderson, A. Ouahab, Topological methods for differential equations and inclusions, Monographs and Research Notes in Mathematics, 1st Edition, Chapman & Hall/CRC Press, 2018.
G. M. Mittag-Leffler, Sur la nouandlle fonction E_\alpha\left(x\right), Comptes Rendus de l’Academie des Sciences Paris 137 (2) (1903) 554–558.
A. Wiman, Über die nullstellun der funktionen E_\alpha\left(x\right), Acta Mathematica 29 (1905) 217–234.
W. R. Schineider, Completely monotone generalized Mittag-Leffler functions, Expositiones Mathematicae 14 (1996) 3–16.
K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, in: J. -M. Morel, F. Takens, B. Teissier (Eds.), Vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 2010.
M. Hassouna, E. H. El Kinani, A. Ouhadan, Global existence and uniqueness of solution of Atangana–Baleanu Caputo fractional differential equation with nonlinear term and approximate solutions, International Journal of Differential Equations 2021 (2021) 1–11.

Details

Primary Language

English

Subjects

Approximation Theory and Asymptotic Methods

Journal Section

Research Article

Authors

Lale Cona ^*
0000-0002-2744-1960
Türkiye

Alperen Hasan Kocağ
0000-0003-2441-1405
Türkiye

Publication Date

August 31, 2025

Submission Date

April 8, 2025

Acceptance Date

June 19, 2025

Published in Issue

Year 2025 Volume: 14 Number: 2

DOI

https://doi.org/10.54187/jnrs.1671939

IZ

https://izlik.org/JA92TA36PU

APA

Cona, L., & Kocağ, A. H. (2025). Fixed Point Approach for Fractional Order Differential Equation Systems. Journal of New Results in Science, 14(2), 124-137. https://doi.org/10.54187/jnrs.1671939

AMA

1.Cona L, Kocağ AH. Fixed Point Approach for Fractional Order Differential Equation Systems. JNRS. 2025;14(2):124-137. doi:10.54187/jnrs.1671939

Chicago

Cona, Lale, and Alperen Hasan Kocağ. 2025. “Fixed Point Approach for Fractional Order Differential Equation Systems”. Journal of New Results in Science 14 (2): 124-37. https://doi.org/10.54187/jnrs.1671939.

EndNote

Cona L, Kocağ AH (August 1, 2025) Fixed Point Approach for Fractional Order Differential Equation Systems. Journal of New Results in Science 14 2 124–137.

IEEE

[1]L. Cona and A. H. Kocağ, “Fixed Point Approach for Fractional Order Differential Equation Systems”, JNRS, vol. 14, no. 2, pp. 124–137, Aug. 2025, doi: 10.54187/jnrs.1671939.

ISNAD

Cona, Lale - Kocağ, Alperen Hasan. “Fixed Point Approach for Fractional Order Differential Equation Systems”. Journal of New Results in Science 14/2 (August 1, 2025): 124-137. https://doi.org/10.54187/jnrs.1671939.

JAMA

1.Cona L, Kocağ AH. Fixed Point Approach for Fractional Order Differential Equation Systems. JNRS. 2025;14:124–137.

MLA

Cona, Lale, and Alperen Hasan Kocağ. “Fixed Point Approach for Fractional Order Differential Equation Systems”. Journal of New Results in Science, vol. 14, no. 2, Aug. 2025, pp. 124-37, doi:10.54187/jnrs.1671939.

Vancouver

1.Lale Cona, Alperen Hasan Kocağ. Fixed Point Approach for Fractional Order Differential Equation Systems. JNRS. 2025 Aug. 1;14(2):124-37. doi:10.54187/jnrs.1671939