Analyzing the existence, uniqueness, and stability of solutions to boundary value problems involving a generalized fractional derivative

Ricardo Almeida; Natália Martins; Ricardo Almeida; Natália Martins

doi:10.3934/math.2026125

AIMS Mathematics

2026, Volume 11, Issue 2: 3142-3159. doi: 10.3934/math.2026125

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Analyzing the existence, uniqueness, and stability of solutions to boundary value problems involving a generalized fractional derivative

Ricardo Almeida ^,,
Natália Martins

Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received: 15 December 2025 Revised: 20 January 2026 Accepted: 26 January 2026 Published: 02 February 2026
MSC : 26A33, 34A08, 34D20

This paper examines tempered fractional derivatives with respect to a kernel function, extending classical operators like Caputo and tempered derivatives. We investigate fractional differential equations (FDEs) that incorporate these generalized derivatives, focusing on the existence and uniqueness of solutions for boundary value problems. Using fixed-point theorems, we establish conditions for the existence and uniqueness of solutions. Additionally, we analyze the stability of these equations under different criteria. Our approach addresses inaccuracies in previous studies and contributes to the broader theory of fractional equations with generalized derivatives.
- fractional calculus,
- fractional differential equations,
- boundary value problems,
- stability analysis
Citation: Ricardo Almeida, Natália Martins. Analyzing the existence, uniqueness, and stability of solutions to boundary value problems involving a generalized fractional derivative[J]. AIMS Mathematics, 2026, 11(2): 3142-3159. doi: 10.3934/math.2026125

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Abstract

This paper examines tempered fractional derivatives with respect to a kernel function, extending classical operators like Caputo and tempered derivatives. We investigate fractional differential equations (FDEs) that incorporate these generalized derivatives, focusing on the existence and uniqueness of solutions for boundary value problems. Using fixed-point theorems, we establish conditions for the existence and uniqueness of solutions. Additionally, we analyze the stability of these equations under different criteria. Our approach addresses inaccuracies in previous studies and contributes to the broader theory of fractional equations with generalized derivatives.

References

[1]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[2]	R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Meth. Appl. Sci., 41 (2018), 336–352. https://doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
[3]	R. Almeida, A. B. Malinowska, T. Odzijewicz, On systems of fractional differential equations with the $\psi$-Caputo derivative and their applications, Math. Meth. Appl. Sci., 44 (2021), 8026–8041. https://doi.org/10.1002/mma.5678 doi: 10.1002/mma.5678
[4]	R. Almeida, N. Martins, J. V. C. Sousa, Fractional tempered differential equations depending on arbitrary kernels, AIMS Math., 9 (2024), 9107–9127. https://doi.org/10.3934/math.2024443 doi: 10.3934/math.2024443
[5]	A. Aphithana, S. K. Ntouyas, J. Tariboon, Existence and Ulam–Hyers stability for Caputo conformable differential equations with four-point integral conditions, Adv. Differ. Equ., 2019 (2019), 139. https://doi.org/10.1186/s13662-019-2077-5 doi: 10.1186/s13662-019-2077-5
[6]	L. P. Castro, A. S. Silva, On the solution and Ulam–Hyers–Rassias stability of a Caputo fractional boundary value problem, Math. Biosci. Eng., 19 (2022), 10809–10825. https://doi.org/10.3934/mbe.2022505 doi: 10.3934/mbe.2022505
[7]	I. Colombaro, R. Garra, A. Giusti, F. Mainardi, Scott–Blair models with time-varying viscosity, Appl. Math. Lett., 86 (2018), 57–63. https://doi.org/10.1016/j.aml.2018.06.022 doi: 10.1016/j.aml.2018.06.022
[8]	J. Deng, L. Zhao, Y. Wu, Fast predictor-corrector approach for the tempered fractional ordinary differential equations, Numer. Algor., 74 (2017), 717–754. https://doi.org/10.1007/s11075-016-0169-9 doi: 10.1007/s11075-016-0169-9
[9]	Y. Y. Gambo, R. Ameen, F. Jarad, T. Abdeljawad, Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives, Adv. Differ. Equ., 2018 (2018), 134. https://doi.org/10.1186/s13662-018-1594-y doi: 10.1186/s13662-018-1594-y
[10]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8
[11]	X. Hai, Y. Yu, C. Xu, G. Ren, Stability analysis of fractional differential equations with the short-term memory property, Fract. Calc. Appl. Anal., 25 (2022), 962–994. https://doi.org/10.1007/s13540-022-00049-9 doi: 10.1007/s13540-022-00049-9
[12]	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.
[13]	I. Lamrani, H. Zitane, D. F. M. Torres, Controllability and observability of tempered fractional differential systems, Commun. Nonlinear Sci. Numer. Simul., 142 (2025), 108501. https://doi.org/10.1016/j.cnsns.2024.108501 doi: 10.1016/j.cnsns.2024.108501
[14]	C. Li, W. Deng, L Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1989–2015. https://doi.org/10.3934/dcdsb.2019026 doi: 10.3934/dcdsb.2019026
[15]	Y. Li, N. Sun, Numerical solution of fractional differential equations using the generalized block pulse operational matrix, Comput. Math. Appl., 62 (2011), 1046–1054. https://doi.org/10.1016/j.camwa.2011.03.032 doi: 10.1016/j.camwa.2011.03.032
[16]	Q. H. Ma, J. Pečarić, On some qualitative properties for solutions of a certain two-dimensional fractional differential systems, Comput. Math. Appl., 59 (2010), 1294–1299. https://doi.org/10.1016/j.camwa.2009.07.008 doi: 10.1016/j.camwa.2009.07.008
[17]	K. M. Owolabi, A. Atangana, Numerical methods for fractional differentiation, Springer Cham, 2019. https://doi.org/10.1007/978-981-15-0098-5
[18]	R. Poovarasan, J. F. Gómez–Aguilar, V. Govindaraj, Investigating the existence, uniqueness, and stability of solutions in boundary value problem of fractional differential equations, Phys. Scr., 99 (2024), 055264. https://doi.org/10.1088/1402-4896/ad3d97 doi: 10.1088/1402-4896/ad3d97
[19]	C. Promsakon, I. Ansari, M. Wetsah, A. Kumar, K. Karthikeyan, T. Sitthiwirattham, Existence and uniqueness of solutions for fractional-differential equation with boundary condition using nonlinear multi-fractional derivatives, Math. Probl. Eng., 2024 (2024), 6844686. https://doi.org/10.1155/2024/6844686 doi: 10.1155/2024/6844686
[20]	D. Qian, C. Li, R. P. Agarwal, P. J. Y. Wong, Stability analysis of fractional differential system with Riemann–Liouville derivative, Math. Comput. Model., 52 (2010), 862–874. https://doi.org/10.1016/j.mcm.2010.05.016 doi: 10.1016/j.mcm.2010.05.016
[21]	F. Sabzikar, M. M. Meerschaert, J. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14–28. https://doi.org/10.1016/j.jcp.2014.04.024 doi: 10.1016/j.jcp.2014.04.024
[22]	D. R. Smart, Fixed point theorems, Cambridge: Cambridge University Press, 1980.
[23]	J. P. Sun, L. Fang, Y. H. Zhao, Q. Ding, Existence and uniqueness of solutions for multi-order fractional differential equations with integral boundary conditions, Bound Value Probl., 2024 (2024), 5. https://doi.org/10.1186/s13661-023-01804-4 doi: 10.1186/s13661-023-01804-4
[24]	A. M. Vargas, Finite difference method for solving fractional differential equations at irregular meshes, Math. Comput. Simul., 193 (2022), 204–216. https://doi.org/10.1016/j.matcom.2021.10.010 doi: 10.1016/j.matcom.2021.10.010
[25]	O. K. Wanassi, R. Bourguiba, D. F. M. Torres, Existence and uniqueness of solution for fractional differential equations with integral boundary conditions and the Adomian decomposition method, Math. Meth. Appl. Sci., 47 (2024), 3582–-3595. https://doi.org/10.1002/mma.8880 doi: 10.1002/mma.8880
[26]	O. K. Wanassi, D. F. M. Torres, An integral boundary fractional model to the world population growth, Chaos Solit. Fract., 168 (2023), 113151. https://doi.org/10.1016/j.chaos.2023.113151 doi: 10.1016/j.chaos.2023.113151

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