Existence, uniqueness, and stability analysis of a fractional differential equation

Document Type : Research Paper

Authors

School of Mathematics and Computer Sciences, Iran University of Science and Technology, Narmak, Tehran, Iran

Abstract

This research is dedicated to establishing the existence and uniqueness of solutions for a Caputo-Fabrizino fractional differential system. Additionally, it explores the Hyers-Ulam-Rassias and Hyers-Ulam-Mittag-Leffler stability of these solutions. This study utilizes the alternative fixed point theorem as a fundamental tool in its analysis. In recent papers, authors used the Schauder fixed point theorem and the Laplace transform to prove the stability of Caputo-Fabrizio equations, but we use the alternative fixed point theorem to prove the stability of these equations.

Keywords

[1] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Sci. 20 (2016), no. 2, 763.
[2] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fractional Differ. Appl. 1 (2015), no. 2, 73–85.
[3] J.B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), no. 6, 305–309.
[4] L. Herrera Diez, R. Kruk, K. Leistner, and J. Sort, Magnetoelectric materials, phenomena, and devices, APL Mater. 9 (2021), no. 5.
[5] R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, 2011.
[6] C. Jiang, F. Zhang, and T. Li, Synchronization and antisynchronization of n-coupled fractional-order complex chaotic systems with ring connection, Math. Meth. Appl. Sci. 41 (2018), no. 7, 2625–2638.
[7] D. Jiles, Introduction to Magnetism and Magnetic Materials, CRC Press, 2015.
[8] D.C. Jiles, Modelling the effects of eddy current losses on frequency dependent hysteresis in electrically conducting media, IEEE Trans. Magnet. 30 (2002), no. 6, 4326–4328.
[9] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
[10] J. Losada and J.J. Nieto, Fractional integral associated to fractional derivatives with nonsingular kernels, Prog. Fractional Differ. Appl. 7 (2021), no. 3, 137–143.
[11] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[12] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
[13] H. Qin, Z. Gu, Y. Fu, and T. Li, Existence of mild solutions and controllability of fractional impulsive integrodifferential systems with nonlocal conditions, J. Funct. Spaces 2017 (2017).
[14] T. Sardar, S. Rana, and J. Chattopadhyay, A mathematical model of dengue transmission with memory, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), no. 1-3, 511–525.
[15] N.A. Sheikh, F. Ali, M. Sagib, I. Khan, S.A.A. Jan, A.S. Alshomrani, and M.S. Alghamdi, Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results Phys. 7 (2017), 789–800.
[16] S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Chapman and Hall/CRC, 2024.
[17] P. Wang, C. Li, J. Zhang, and T. Li, Quasilinearization Method for First-Order Impulsive Integro-Differential Equations, Texas State University, Department of Mathematics, 2019.
[18] S. Wen, R. Hu, Y. Yang, T. Huang, Z. Zeng, and Y. Song, Memristor-based echo state network with online least mean square, IEEE Trans. Syst. Man Cybernet.: Syst. 49 (2018), no. 9, 1787–1796.

Articles in Press, Corrected Proof
Available Online from 23 November 2025
  • Receive Date: 15 April 2025
  • Revise Date: 23 July 2025
  • Accept Date: 22 November 2025