Lyapunov stability and solvability of nonlocal fractional differential equations with generalized katugampola derivative

Mohammed Said Souid; Zoubida Bouazza; Hatıra Günerhan; Kadda Maazouz; M'hamed Bensaid; Meraa Arab; Mohammed Said Souid; Zoubida Bouazza; Hatıra Günerhan; Kadda Maazouz; M'hamed Bensaid; Meraa Arab

doi:10.3934/math.2026724

AIMS Mathematics

2026, Volume 11, Issue 6: 17766-17793. doi: 10.3934/math.2026724

Previous Article Next Article

Research article Special Issues

Lyapunov stability and solvability of nonlocal fractional differential equations with generalized katugampola derivative

1.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
2.
Department of Allied Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University, Amman, Jordan
3.
Department of Economic Sciences, Ibn Khaldoun University of Tiaret, 14000, Tiaret, Algeria
4.
Department of Computer Sciences, Ibn Khaldoun University of Tiaret, Tiaret, Algeria
5.
Department of Mathematics, Faculty of Education, Kafkas University, Kars, Turkey
6.
Department of Mathematics, Faculty of Mathematics and Computer Science, Ibn Khaldoun University of Tiaret, 14000, Tiaret, Algeria
7.
Department of Mathematics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa, 31982, Saudi Arabia

Received: 15 April 2026 Revised: 02 June 2026 Accepted: 08 June 2026 Published: 17 June 2026
MSC : 26A33, 34A08, 34B15

This paper investigates a class of fractional differential equations (FDEs) that involve the generalized Katugampola fractional derivative (FD) subject to nonlocal boundary conditions. By transforming the considered boundary value problems (BVPs) into equivalent integral equations, we establish several results concerning the existence and uniqueness of solutions. The analysis is carried out using classical fixed point (FP) techniques, including the Banach contraction principle(BC), as well as Schaefer's FP theorems under appropriate assumptions. In addition, we examine the Lyapunov stability of nontrivial solutions and derive sufficient conditions to ensure asymptotic stability. The obtained results extend and complement the existing contributions in the literature on fractional BVPs with nonlocal conditions. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical findings.
Citation: Mohammed Said Souid, Zoubida Bouazza, Hatıra Günerhan, Kadda Maazouz, M'hamed Bensaid, Meraa Arab. Lyapunov stability and solvability of nonlocal fractional differential equations with generalized katugampola derivative[J]. AIMS Mathematics, 2026, 11(6): 17766-17793. doi: 10.3934/math.2026724

Related Papers:

Abstract

This paper investigates a class of fractional differential equations (FDEs) that involve the generalized Katugampola fractional derivative (FD) subject to nonlocal boundary conditions. By transforming the considered boundary value problems (BVPs) into equivalent integral equations, we establish several results concerning the existence and uniqueness of solutions. The analysis is carried out using classical fixed point (FP) techniques, including the Banach contraction principle(BC), as well as Schaefer's FP theorems under appropriate assumptions. In addition, we examine the Lyapunov stability of nontrivial solutions and derive sufficient conditions to ensure asymptotic stability. The obtained results extend and complement the existing contributions in the literature on fractional BVPs with nonlocal conditions. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical findings.

References

[1]	Z. Chen, Y. Hou, R. Huang, Q. Cheng, Neural network compensator-based robust iterative learning control scheme for mobile robots nonlinear systems with disturbances and uncertain parameters, Appl. Math. Comput., 469 (2024), 128549. http://doi.org/10.1016/j.amc.2024.128549 doi: 10.1016/j.amc.2024.128549
[2]	S. Boulaaras, V. T. Pham, R. Jan, Editorial: Special issue on application of fractional calculus: Mathematical modeling and control—Part Ⅰ, Fractals, 33 (2025), 2502003. http://doi.org/10.1142/S0218348X25020037 doi: 10.1142/S0218348X25020037
[3]	Z. A. Khan, M. I. Liaqat, A. Akgül, J. A. Conejero, Qualitative analysis of stochastic Caputo–Katugampola fractional differential equations, Axioms, 13 (2024), 808. http://doi.org/10.3390/axioms13110808 doi: 10.3390/axioms13110808
[4]	C. Li, Z. Gong, D. Qian, Y. Chen, On the bound of the Lyapunov exponents for the fractional differential systems, Chaos, 20 (2010), 013127. http://doi.org/10.1063/1.3314277 doi: 10.1063/1.3314277
[5]	S. Krim, S. Abbas, M. Benchohra, E. Karapinar, Terminal value problem for implicit Katugampola fractional differential equations in $b$-metric spaces, J. Funct. Spaces, 2021 (2021), 5535178. http://doi.org/10.1155/2021/5535178 doi: 10.1155/2021/5535178
[6]	A. Refice, M. Bensaid, M. S. Souid, S. Boulaaras, A. Amara, Taha Radwan, Innovative approaches to initial and terminal value problems of fractional differential equations with two different derivative orders, Fixed Point Theory Algorithms Sci. Eng., 2025 (2025), 32. http://doi.org/10.1186/s13663-025-00812-6 doi: 10.1186/s13663-025-00812-6
[7]	Z. Chen, Z. Zhang, J. Yan, M. Zhong, L. Lv, Y. Hou, Neural network–based reinforcement iterative learning fault estimation scheme for nonlinear uncertain manipulator systems with time-delay, IEEE T. Ind. Inform., 21 (2025), 7287–7298. http://doi.org/10.1109/TII.2025.3575123 doi: 10.1109/TII.2025.3575123
[8]	M. S. Souid, Z. Bouazza, M. Bensaid, K. S. Mozhi, M. Mokhtari, J. K. K. Asamoah, Analytical study of variable-order fractional differential equations with initial and terminal antiperiodic boundary conditions, J. Appl. Math., 2025 (2025), 8863599. http://doi.org/10.1155/jama/8863599 doi: 10.1155/jama/8863599
[9]	M. Bensaid, M. S. Souid, A. Benkerrouche, S. Guedim, A. Amara, Initial and terminal conditions for differential equations of fractional derivative via non-autonomous variable order, Palest. J. Math., 14 (2025), 187–205.
[10]	M. Awadalla, M. Subramanian, K. Abuasbeh, M. Manigandan, On the generalized Liouville–Caputo type fractional differential equations supplemented with Katugampola integral boundary conditions, Symmetry, 14 (2022), 2273. http://doi.org/10.3390/sym14112273 doi: 10.3390/sym14112273
[11]	S. Youcefi, S. Pinelas, O. Oqilat, M. S. Souid, M. Bensaid, Existence and compactness of the solution set for a coupled Caputo fractional system with $\phi$-Laplacian operators and nonlocal boundary conditions, Mathematics, 14 (2026), 1112. http://doi.org/10.3390/math14071112 doi: 10.3390/math14071112
[12]	S. Etemad, M. S. Souid, B. Telli, M. K. A. Kaabar, S. Rezapour, Investigation of the neutral fractional differential inclusions of Katugampola-type involving both retarded and advanced arguments via Kuratowski MNC technique, Adv. Differ. Equ., 2021 (2021), 214. http://doi.org/10.1186/s13662-021-03377-x doi: 10.1186/s13662-021-03377-x
[13]	U. N. Katugampola, New approach to generalized fractional derivative, Bull. Math. Anal. Appl., 2014, arXiv: 1106.0965. https://doi.org/10.48550/arXiv.1106.0965
[14]	K. Shah, T. Abdeljawad, N. Mlaiki, H. Alrabaiah, Spectral numerical method for fractional order partial differential equations using Katugampola derivative, Partial Differ. Equ. Appl. Math., 17 (2026), 101338. http://doi.org/10.1016/j.padiff.2026.101338 doi: 10.1016/j.padiff.2026.101338
[15]	S. P. Bhairat, M. E. Samei, Nonexistence of global solutions for a Hilfer–Katugampola fractional differential problem, Partial Differ. Equ. Appl. Math., 7 (2023), 100495. http://doi.org/10.1016/j.padiff.2023.100495 doi: 10.1016/j.padiff.2023.100495
[16]	D. Boucenna, A. B. Makhlouf, M. A. Hammami, On Katugampola fractional order derivatives and Darboux problem for differential equations, Cubo, 22 (2020), 125–136. http://doi.org/10.4067/S0719-06462020000100125 doi: 10.4067/S0719-06462020000100125
[17]	R. A. El-Nabulsi, W. Anukool, R. Valarmathi, C. Thangaraj, Nonlocal fractal diffusion-advection models with variable coefficients and nonlocal time delay: Existence of solutions, Lyapunov function, Hopf bifurcation, and stability, J. Peridyn. Nonlocal Model., 8 (2026), 6. http://doi.org/10.1007/s42102-026-00142-0 doi: 10.1007/s42102-026-00142-0
[18]	R. A. El-Nabulsi, W. Anukool, R. Valarmathi, C. Thangaraj, Extended spin-orbit modeling of unstable discrete fractional Hamiltonian systems: numerical investigation of chaotic orbits for Mercury, Mars, Triton, and Sedna-like trans-Neptunian objects, Adv. Space Res., 77 (2026), 7183–7219. http://doi.org/10.1016/j.asr.2026.01.058 doi: 10.1016/j.asr.2026.01.058
[19]	R. Allogmany, S. S. Alzahrani, Dynamic, bifurcation, and Lyapunov analysis of fractional Rössler chaos using two numerical methods, Mathematics, 13 (2025), 3642. http://doi.org/10.3390/math13223642 doi: 10.3390/math13223642
[20]	I. Ahmed, P. Kumam, F. Jarad, P. Borisut, K. Sitthithakerngkiet, A. Ibrahim, Stability analysis for boundary value problems with generalized nonlocal condition via Hilfer–Katugampola fractional derivative, Adv. Differ. Equ., 2020 (2020), 225. http://doi.org/10.1186/s13662-020-02681-2 doi: 10.1186/s13662-020-02681-2
[21]	S. Harikrishnan, R. Ibrahim, K. Kanagarajan, Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator, Univ. J. Math. Appl., 1 (2018), 106–112. http://doi.org/10.32323/ujma.419363 doi: 10.32323/ujma.419363
[22]	R. W. Ibrahim, S. Harikrishnan, K. Kanagarajan, Existence and stability of Langevin equations with two Hilfer-Katugampola fractional derivatives, Stud. Univ. Babeş-Bolyai Math., 63 (2018), 291–302. http://doi.org/10.24193/subbmath.2018.3.01
[23]	S. T. M. Thabet, I. Kedim, An investigation of a new Lyapunov-type inequality for Katugampola–Hilfer fractional BVP with nonlocal and integral boundary conditions, J. Inequal. Appl., 2023 (2023), 162. http://doi.org/10.1186/s13660-023-03070-5 doi: 10.1186/s13660-023-03070-5
[24]	M. Sarmah, M. Gabeleh, A. Das et al., Best proximity point theorems on the optimum solution for coupled system of Katugampola fractional integral equations via measure of noncompactness, J. Appl. Math. Comput., 71 (2025), 7249–7269. http://doi.org/10.1007/s12190-025-02515-y doi: 10.1007/s12190-025-02515-y
[25]	M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1–12.
[26]	J. Wang, L. Lv, Y. Zhou, Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces, J. Appl. Math. Comput., 38 (2012), 209–224. https://doi.org/10.1007/s12190-011-0474-3 doi: 10.1007/s12190-011-0474-3
[27]	D. Vivek, K. Kanagarajan, S. Harikrishnan, Theory and analysis of impulsive type pantograph equations with Katugampola fractional derivative, J. Vib. Test. Syst. Dyn., 2 (2018), 9–20. http://doi.org/10.5890/JVTSD.2018.03.002 doi: 10.5890/JVTSD.2018.03.002
[28]	M. S. Souid, M. Inc, A. Benkerrouche, Nonlinear fractional differential equations of variable order, CRC Press, 2026. http://doi.org/10.1201/9781003687900
[29]	J. E. Ante, S. O. Essang, A. Otobi, S. E. Fadugba, C. S. Akpan, S. I. Okeke, et al., On the new robust Lyapunov uniform stability approach for nonlinear impulsive Caputo fractional differential equations with nonlocal conditions, Eur. J. Math. Appl., 5 (2025), 18. http://doi.org/10.28919/ejma.2025.5.18 doi: 10.28919/ejma.2025.5.18
[30]	R. K. Pandey, K. S. Nisar, Modeling and analysis of fascioliasis disease with Katugampola fractional derivative: A memory-incorporated epidemiological approach, Sci. Rep., 15 (2025), 37849. http://doi.org/10.1038/s41598-025-21788-8 doi: 10.1038/s41598-025-21788-8
[31]	N. Mlaiki, Z. Bekri, V. S. Erturk, M. E. Samei, S. Haque, On analysis of single solution for a class of BVP with generalized Caputo-Katugampola fractional derivative, Eur. J. Pure Appl. Math., 18 (2025), 6138. http://doi.org/10.29020/nybg.ejpam.v18i2.6138 doi: 10.29020/nybg.ejpam.v18i2.6138
[32]	A. Berhail, N. Tabouche, J. Alzabut, M. E. Samei, Using the Hilfer–Katugampola fractional derivative in initial-value Mathieu fractional differential equations with application to a particle in the plane, Adv. Cont. Discr. Mod., 2022 (2022), 44. http://doi.org/10.1186/s13662-022-03716-6 doi: 10.1186/s13662-022-03716-6
[33]	R. A. El-Nabulsi, C. Thangaraj, R. Valarmathi, W. Anukool, Chaotic dynamics and fractal analysis of nonstandard Hamiltonian systems, Chaos Soliton. Fract., 200 (2025), 116974. http://doi.org/10.1016/j.chaos.2025.116974 doi: 10.1016/j.chaos.2025.116974
[34]	J. Phukan, H. Dutta, Dynamic analysis of a fractional order SIR model with specific functional response and Holling type Ⅱ treatment rate, Chaos Soliton. Fract. 175 (2023), 114005. http://doi.org/10.1016/j.chaos.2023.114005
[35]	H. Li, Y. Shen, Y. Han, J. Dong, J. Li, Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle, Chaos Soliton. Fract., 168 (2023), 113167. http://doi.org/10.1016/j.chaos.2023.113167 doi: 10.1016/j.chaos.2023.113167
[36]	A. E. Matouk, Chaos and hidden chaos in a 4D dynamical system using the fractal-fractional operators, AIMS Mathematics, 10 (2025), 6233–6257. http://doi.org/10.3934/math.2025284 doi: 10.3934/math.2025284

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)