Existence and stability results for a system of hybrid fractional differential problems
*Houas MohamedCorresponding authorm.houas.st@univ-dbkm.dzDepartment of Mathematics Faculty of Sciences of Matter and Computer Sciences Khemis Miliana University; Laboratory FIMA Khemis Miliana UniversityDepartment of Mathematics Faculty of Sciences of Matter and Computer Sciences Khemis Miliana UniversityKhemis Miliana, Algeria0000-0001-6256-0511View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 02 Dec 2024
- Published Online:
- 09 Mar 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2232
- Pages:
- 2097–2110
Abstract
In this paper, we are concerned with a system of hybrid fractional differential equations. We begin by establishing the existence and uniqueness results using the Banach contraction principle. Then, we derive sufficient conditions for proving an existence results by using of Leray-Schauder fixed point theorem. Also, we discuss the Ulam-Hyres stability for the mentioned hybrid system. At the end, we present an example to illustrate one of our main results.
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References
[1] B. Ahmad, S. K. Ntouyas, and A. Alsaedi, “Existence results for a system of coupled hybrid fractional differential equations,” The Scientific World Journal, vol. 2014, pp. 1-7 (2014).
[2] A. Ali, K. Shah, and R. A. Khan, “Existence of solution to a coupled system of hybrid fractional differential equations,” Bulletin of Mathematical Analysis and Applications, vol. 9, no. 1, pp. 9-18 (2017).
[3] J. Alzabut, M. Houas, and M. I. Abbas, “Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives,” Demonstratio Mathematica, vol. 56, no. 1, pp. 1-16 (2023).
[4] C. Bai and J. X. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 150, pp. 611-621 (2004).
[5] Z. Bai and H. Lu, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, pp. 495-505 (2005).
[6] D. Baleanu, S. Etemad, S. Pourrazi, and Sh. Rezapour“On the new fractional hybrid boundary value problems with three-point integral hybrid conditions,” Advances in Difference Equations, Art. no. 473, pp. 1-21 (2019).
[7] T. Bashiri, S. M. Vaezpour, and C. Park, “Existence results for fractional hybrid differential systems in Banach algebras,” Advances in Difference Equations, Art. no. 2016, pp. 1-13 (2016).
[8] S. Belarbi and Z. Dahmani, “Solvability for a nonlinear coupled system of fractional differential equations,” Matematika, vol. 30, pp. 123-133 (2014).
[9] C. Derbaz, H. Hammouche, M. Benchohra, and Y. Zhou, “Fractional hybrid differential equations with three-point boundary hybrid conditions,” Advances in Difference Equations, Art. no. 125, pp. 1-11 (2019).
[10] B. C. Dhage and S. K. Ntouyas, “Existence results for boundary value problems for fractional hybrid differential inclusions,” Topological Methods in Nonlinear Analysis, vol. 44, no. 1, pp. 229-238 (2014).
[11] B. C. Dhage and V. Lakshmikantham, “Basic results on hybrid differential equations, Nonlinear Analysis: Hybrid Systems, vol. 4, no. 3, pp. 414-424 (2010).
[12] Z. Dahmani and S. Belarbi, “New results for fractional evolution equations using Banach fixed point theorem,” International Journal of Nonlinear Analysis and Applications, vol. 5, pp. 22-30 (2014).
[13] D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, pp. 429-440 (1996).
[14] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, pp. 229-248 (2002).
[15] A. M. A. El-Sayed, “Nonlinear functional differential equations of arbitrary orders,” Nonlinear Analysis, vol. 33, pp. 181-186 (1998).
[16] S. Ferraoun and Z. Dahmani, “Existence and stability of solutions of a class of hybrid fractional differential equations involving RL-operator,” Journal of Interdisciplinary Mathematics, vol. 23, no. 4, pp. 885-903 (2020).
[17] M. A. E. Herzallah and D. Baleanu, “On fractional order hybrid differential equations,” Abstract and Applied Analysis, vol. 2014, pp. 1-8 (2014).
[18] K. Hilal and A. Kajouni, “Boundary value problems for hybrid differential equations with fractional order,” Advances in Difference Equations, Art. no. 183, pp. 1-19 (2015).
[19] M. Houas, “Solvability of a system of fractional hybrid differential equations,” Communications in Optimization Theory, vol. 2018, pp. 1-9 (2018).
[20] M. Houas and A. Saadi, “Existence and uniqueness results for a coupled system of nonlinear fractional differential equations with two fractional orders,” Journal of Interdisciplinary Mathematics, vol. 23, no. 6, pp. 1047-1064 (2020).
[21] M. Houas, “Existence results for a coupled system of fractional Caputo-Langevin equations involving two fractional orders,” Journal of Interdisciplinary Mathematics, vol. 27, no. 1, pp. 1-16 (2024).
[22] M. Houas, “Solvability and stability for fractional differential equations involving two Riemann-Liouville fractional orders,” Journal of Interdisciplinary Mathematics, vol. 26, no. 8, pp. 1699-1715 (2023).
[23] M. Houas, “Existence of solutions for nonlinear Hadamard differential equations with nonlocal conditions,” Journal of Interdisciplinary Mathematics, vol. 24, no. 3, pp. 593-612 (2021).
[24] M. Houas and Z. Dahmani, “On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions,” Lobachevskii Journal of Mathematics, vol. 37, pp. 120-127 (2016).
[25] A. A. Kilbas and S. A. Marzan, “Nonlinear differential equation with the Caputo fractional derivative in the space of continuously differentiable functions,” Differential Equations, vol. 41, pp. 84-89 (2005).
[26] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, pp. 2677-2682 (2008).
[27] M. Paknazar and M. D. L. Sen, “Fractional coupled hybrid Sturm-Liouville differential equation with multi-point boundary coupled hybrid condition,” Axioms, vol. 10, no. 2, pp. 1-26 (2021).
[28] L. Podlubny, Fractional Differential Equations. New York, NY, USA: Academic Press (1999).
[29] G. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Amsterdam, The Netherlands: Gordon and Breach (1993).
[30] G. Wang, W. Liu, and C. Ren, “Existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions,” Electronic Journal of Differential Equations, no. 54, pp. 1-10 (2012).
[31] Y. Zhao, S. Sun, Z. Han, and Q. Li, “Theory of fractional hybrid differential equations,” Computers & Mathematics with Applications, vol. 62, pp. 1312-1324 (2011).
