Solvability and stability of nonlinear variable-order fractional boundary value problems via new infinite-partition

Insaf Naima Telli; Benoumran Telli; Mohammed Said Souid; Sandra Pinelas; Insaf Naima Telli; Benoumran Telli; Mohammed Said Souid; Sandra Pinelas

doi:10.3934/math.2026918

AIMS Mathematics

2026, Volume 11, Issue 7: 22748-22769. doi: 10.3934/math.2026918

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Solvability and stability of nonlinear variable-order fractional boundary value problems via new infinite-partition

1.
Department of Mathematics, Faculty of Mathematics and computer science, University of Tiaret, Artificial Intelligence and Systems Research Laboratory, Tiaret 14000, Algeria; Email: naimainsaf.telli@univ-tiaret.dz
2.
Department of Mathematics, Faculty of Mathematics and computer science, Ibn Khaldoun University of Tiaret, Tiaret 14000, Algeria
3.
Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 02105, Tamil Nadu, India
4.
Department of Allied Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University, Amman, Jordan
5.
Department of Economic Sciences, Ibn Khaldoun University of Tiaret, Tiaret 14000, Algeria
6.
Departamento de Ciencias Exatas e Engenharia, Academia Militar, Av. Conde Castro Guimaraes, 2720-113 Amadora, Portugal
7.
Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193 Aveiro, Portugal

Received: 14 April 2026 Revised: 08 July 2026 Accepted: 15 July 2026 Published: 28 July 2026
MSC : 26A33, 34A08, 34K37

This paper studies a class of nonlinear variable-order fractional boundary value problems involving the Riemann–Liouville fractional derivative. An infinite-partition approximation is introduced by replacing the variable-order function with a sequence of piecewise constant functions, transforming the original problem into a family of constant-order fractional boundary value problems. For each approximating problem, an equivalent integral equation and the corresponding Green function are derived. Existence, uniqueness, and Ulam–Hyers stability of solutions are established using Schauder's fixed-point theorem and the Banach contraction principle. It is further proved that the approximating solutions converge uniformly to a solution of the original variable-order problem, providing a rigorous connection between constant-order and variable-order fractional models. A numerical example illustrates the theoretical results.
Citation: Insaf Naima Telli, Benoumran Telli, Mohammed Said Souid, Sandra Pinelas. Solvability and stability of nonlinear variable-order fractional boundary value problems via new infinite-partition[J]. AIMS Mathematics, 2026, 11(7): 22748-22769. doi: 10.3934/math.2026918

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Abstract

This paper studies a class of nonlinear variable-order fractional boundary value problems involving the Riemann–Liouville fractional derivative. An infinite-partition approximation is introduced by replacing the variable-order function with a sequence of piecewise constant functions, transforming the original problem into a family of constant-order fractional boundary value problems. For each approximating problem, an equivalent integral equation and the corresponding Green function are derived. Existence, uniqueness, and Ulam–Hyers stability of solutions are established using Schauder's fixed-point theorem and the Banach contraction principle. It is further proved that the approximating solutions converge uniformly to a solution of the original variable-order problem, providing a rigorous connection between constant-order and variable-order fractional models. A numerical example illustrates the theoretical results.

References

[1]	B. Telli, M. S. Souid, I. Stamova, Boundary-value problem for nonlinear fractional differential equations of variable order with finite delay via Kuratowski measure of noncompactness, Axioms, 12 (2023). http://dx.doi.org/10.3390/axioms12010080 doi: 10.3390/axioms12010080
[2]	K. Shah, G. Ali, K. J. Ansari, T. Abdeljawad, M. Meganathan, B. Abdalla, On qualitative analysis of boundary value problem of variable order fractional delay differential equations, Boundary Value Probl., 2023 (2023). http://dx.doi.org/10.1186/s13661-023-01728-z doi: 10.1186/s13661-023-01728-z
[3]	M. S. Souid, M. Inc, A. Benkerrouche, Nonlinear fractional differential equations of variable order, 1st Ed., CRC Press, 2026. http://dx.doi.org/10.1201/9781003687900
[4]	M. S. Souid, Z. Bouazza, H. Günerhan, K. Maazouz, M. Bensaid, M. Arab, Lyapunov stability and solvability of nonlocal fractional differential equations with generalized Katugampola derivative, AIMS Math., 11 (2026), 17766–17793. http://dx.doi.org/10.3934/math.2026724 doi: 10.3934/math.2026724
[5]	M. S. Souid, S. Guedim, S. Boulaaras, A. Benkerrouche, A. Amara, T. Radwan, et al., New approaches to solving initial value problems of mixed differential equations, Filomat, 40 (2026), 649–659.
[6]	A. Benkerrouche, M. S. Souid, E. Karapınar, A. Hakem, On the boundary value problems of Hadamard fractional differential equations of variable order, Math. Method. Appl. Sci., 46 (2023), 3187–3203. http://dx.doi.org/10.1002/mma.8306 doi: 10.1002/mma.8306
[7]	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://dx.doi.org/10.1155/jama/8863599 doi: 10.1155/jama/8863599
[8]	A. Refice, M. Bensaid, M. S. Souid, S. Boulaaras, A. Amara, T. Radwan, Innovative approaches to initial and terminal value problems of fractional differential equations with two different derivative orders, Fixed Point Theor. Al., 2025 (2025). http://dx.doi.org/10.1186/s13663-025-00812-6 doi: 10.1186/s13663-025-00812-6
[9]	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://dx.doi.org/10.3390/math14071112 doi: 10.3390/math14071112
[10]	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, Palestine J. Math., 14 (2025), 187–205.
[11]	S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Transf. Spec. F., 1 (1993), 277–300.
[12]	S. G. Samko, Fractional integration and differentiation of variable order, Anal. Math., 21 (1995), 213–236. https://doi.org/10.1007/BF01911126 doi: 10.1007/BF01911126
[13]	D. Valério, J. S. Costa, Variable-order fractional derivatives and their numerical approximations, Signal Proc., 91 (2011), 470–483. https://doi.org/10.1016/j.sigpro.2010.04.006 doi: 10.1016/j.sigpro.2010.04.006
[14]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam, 2006.
[15]	S. Zhang, Existence of solutions for two point boundary value problems with singular differential equations of variable order, Electron. J. Differ. Eq., 2013 (2013), 1–16.
[16]	S. Zhang, L. Hu, The existence and uniqueness result of solutions to initial value problems of nonlinear diffusion equations involving the conformable variable, Azerbaijan J. Math., 9 (2019), 22–45.
[17]	S. Zhang, S. Sun, L. Hu, Approximate solutions to initial value problem for differential equation of variable order, J. Fract. Calc. Appl., 9 (2018), 93–112.
[18]	C. F. Lorenzo, T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), 57–98. https://doi.org/10.1016/S0315-0860(02)92347-5 doi: 10.1016/S0315-0860(02)92347-5

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