On the existence and stability of bounded solutions for a class of three-point boundary value problems of fractional type

Luís P. Castro; Edixon M. Rojas; Luís P. Castro; Edixon M. Rojas

doi:10.3934/math.2025934

AIMS Mathematics

2025, Volume 10, Issue 9: 20909-20931. doi: 10.3934/math.2025934

Previous Article Next Article

Research article Topical Sections

On the existence and stability of bounded solutions for a class of three-point boundary value problems of fractional type

Luís P. Castro ^{1
,
,},
Edixon M. Rojas ²

1.
CIDMA–Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810–193 Aveiro, Portugal
2.
Departamento de Matemáticas, Universidad Nacional de Colombia, Sede Bogotá, Colombia

Received: 22 May 2025 Revised: 15 August 2025 Accepted: 02 September 2025 Published: 11 September 2025
MSC : 26A33, 34A08, 46B70, 47H11, 47H30

This article investigates the existence and stability of solutions for a class of three-point boundary value problems involving Riemann-Liouville fractional derivatives of order less than one. We considered nonlinear fractional differential equations subject to nonlocal boundary conditions that include a singular condition at the origin and a global fractional condition at interior points. Our approach generalizes previous work, allowing for singular behavior near zero, and employs the Leray-Schauder fixed point theorem to establish the existence of bounded solutions without requiring contractive conditions on the nonlinear term. Instead, we imposed a local integrability condition known as the $ L^p $-Carathéodory condition. Furthermore, we studied the Ulam-Hyers-Rassias stability of approximate solutions by means of a Gronwall-type inequality adapted to weakly singular kernels. Two concrete examples were also included to illustrate the theory.
Citation: Luís P. Castro, Edixon M. Rojas. On the existence and stability of bounded solutions for a class of three-point boundary value problems of fractional type[J]. AIMS Mathematics, 2025, 10(9): 20909-20931. doi: 10.3934/math.2025934

Related Papers:

Abstract

This article investigates the existence and stability of solutions for a class of three-point boundary value problems involving Riemann-Liouville fractional derivatives of order less than one. We considered nonlinear fractional differential equations subject to nonlocal boundary conditions that include a singular condition at the origin and a global fractional condition at interior points. Our approach generalizes previous work, allowing for singular behavior near zero, and employs the Leray-Schauder fixed point theorem to establish the existence of bounded solutions without requiring contractive conditions on the nonlinear term. Instead, we imposed a local integrability condition known as the $ L^p $-Carathéodory condition. Furthermore, we studied the Ulam-Hyers-Rassias stability of approximate solutions by means of a Gronwall-type inequality adapted to weakly singular kernels. Two concrete examples were also included to illustrate the theory.

References

[1]	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. https://doi.org/10.2969/jmsj/00210064 doi: 10.2969/jmsj/00210064
[2]	N. Alghamdi, R. P. Agarwal, B. Ahmad, E. A. Alharbi, W. Shammakh, Existence and stability results for multi-term fractional delay differential equations equipped with nonlocal multi-point and multi-strip boundary conditions, Carpathian J. Math., 41 (2025), 651–671.
[3]	M. I. Abbas, M. A. Ragusa, Nonlinear fractional differential inclusions with non-singular Mittag-Leffler kernel, AIMS Math., 7 (2022), 20328–20340. https://doi.org/10.3934/math.20221113 doi: 10.3934/math.20221113
[4]	L. C. Becker, T. A. Burton, I. K. Purnaras, Complementary equations: a fractional differential equation and a Volterra integral equation, Electron. J. Qual. Theory Differ. Equ., 12 (2015), 1–24. https://doi.org/10.14232/ejqtde.2015.1.12 doi: 10.14232/ejqtde.2015.1.12
[5]	F. Bouchelaghem, H. Boulares, A. Ardjouni, F. Jarad, T. Abdeljawad, B. Abdalla, et al., Existence of solutions of multi-order fractional differential equations, Partial Differ. Equ. Appl. Math., 13 (2025), 101104. https://doi.org/10.1016/j.padiff.2025.101104 doi: 10.1016/j.padiff.2025.101104
[6]	L. P. Castro, A. Ramos, Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations, Banach J. Math. Anal., 3 (2009), 36–43.
[7]	L. P. Castro, A. M. Simões, Different types of Hyers-Ulam-Rassias stabilities for a class of integro-differential equations, Filomat, 31 (2017), 5379–5390. https://doi.org/10.2298/FIL1717379C doi: 10.2298/FIL1717379C
[8]	L. P. Castro, A. M. Simões, Stabilities of Ulam-Hyers type for a class of nonlinear fractional differential equations with integral boundary conditions in Banach spaces, Filomat, 39 (2025), 617–628. https://doi.org/10.2298/FIL2502617C doi: 10.2298/FIL2502617C
[9]	V. S. Ertürk, A. Ali, K. Shah, P. Kumar, T. Abdeljawad, Existence and stability results for nonlocal boundary value problems of fractional order, Bound. Value Probl., 2022 (2022), 25. https://doi.org/10.1186/s13661-022-01606-0 doi: 10.1186/s13661-022-01606-0
[10]	R. Herrmann, Fractional calculus: an introduction for physicists, 2 Eds., World Scientific, 2014.
[11]	S. Hristova, Ulam-type stability for Caputo-type fractional delay differential equations, Demonstr. Math., 58 (2025), 20250112. https://doi.org/10.1515/dema-2025-0112 doi: 10.1515/dema-2025-0112
[12]	D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27 (1941), 222–224.
[13]	D. H. He, L. G. Xu, Stability of conformable fractional delay differential systems with impulses, Appl. Math. Lett., 149 (2024), 108927. https://doi.org/10.1016/j.aml.2023.108927 doi: 10.1016/j.aml.2023.108927
[14]	A. Kochubei, Y. Luchko, Handbook of fractional calculus with applications. Volume 2: Fractional differential equations, De Gruyter, 2019.
[15]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[16]	V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems, Cambridge: Cambridge Scientific Publishers, 2009.
[17]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity, London: Imperial College Press; Singapore: World Scientific, 2010.
[18]	Y. Meng, X. R. Du, H. H. Pang, Iterative positive solutions to a coupled Riemann-Liouville fractional $q$-difference system with the Caputo fractional $q$-derivative boundary conditions, J. Funct. Spaces, 2023 (2023), 1–16. https://doi.org/10.1155/2023/5264831 doi: 10.1155/2023/5264831
[19]	B. D. Nghia, N. H. Luc, X. L. Qin, Y. Wang, On maximal solution to a degenerate parabolic equation involving a time fractional derivative, Electron. J. Appl. Math., 1 (2023), 62–80. https://doi.org/10.61383/ejam.20231129 doi: 10.61383/ejam.20231129
[20]	I. Podlubny, Fractional differential equations, Academic Press, 1999.
[21]	G. Pólya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis. Bd. I: Reihen, Integralrechnung, Funktionentheorie; Bd. II: Funktionentheorie, Nullstellen, Polynome, Determinanten, Zahlentheorie, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Berlin: Julius Springer, 1925.
[22]	R. Poovarasan, M. E. Samei, V. Govindaraj, Analysis of existence, uniqueness, and stability for nonlinear fractional boundary value problems with novel integral boundary conditions, J. Appl. Math. Comput., 71 (2025), 3771–3802. https://doi.org/10.1007/s12190-025-02378-3 doi: 10.1007/s12190-025-02378-3
[23]	T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
[24]	S. M. Ulam, A collection of mathematical problems, New York: Interscience, 1960. https://doi.org/10.1126/science.132.3428.665
[25]	J. R. L. Webb, Initial value problems for Caputo fractional equations with singular nonlinearities, Electron. J. Differ. Equ., 2019 (2019), 1–32.
[26]	J. R. L. Webb, Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl., 471 (2019), 692–711. https://doi.org/10.1016/j.jmaa.2018.11.004 doi: 10.1016/j.jmaa.2018.11.004
[27]	J. R. L. Webb, Compactness of nonlinear integral operators with discontinuous and with singular kernels, J. Math. Anal. Appl., 509 (2022), 126000. https://doi.org/10.1016/j.jmaa.2022.126000 doi: 10.1016/j.jmaa.2022.126000
[28]	L. G. Xu, B. Z. Bao, H. X. Hu, Stability of impulsive delayed switched systems with conformable fractional-order derivatives, Int. J. Syst. Sci., 56 (2025), 1271–1288. https://doi.org/10.1080/00207721.2024.2421454 doi: 10.1080/00207721.2024.2421454
[29]	L. Xu, Q. X. Dong, G. Li, Existence and Hyers-Ulam stability for three-point boundary value problems with Riemann-Liouville fractional derivatives and integrals, Adv. Differ. Equ., 2018 (2018), 458. https://doi.org/10.1186/s13662-018-1903-5 doi: 10.1186/s13662-018-1903-5
[30]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014.
[31]	Y. Zhou, Fractional evolution equations and inclusions: Analysis and control, Amsterdam: Elsevier/Academic Press, 2016.
[32]	M. Zakaria, A. Moujahid, A. Bouzelmate, Existence and uniqueness of a mild solution for a class of the fractional evolution equation with nonlocal condition involving $\varphi$-Riemann Liouville fractional derivative, Filomat, 37 (2023), 6041–6057. https://doi.org/10.2298/FIL2318041Z doi: 10.2298/FIL2318041Z

Reader Comments

Your name:*

Email:*
© 2025 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)