Positive solutions for a Caputo-type fractional differential equation with a Riemann-Stieltjes integral boundary condition

Haixia Li; Xin Liu; Haixia Li; Xin Liu

doi:10.3934/era.2025179

Electronic Research Archive

2025, Volume 33, Issue 6: 4027-4044. doi: 10.3934/era.2025179

Previous Article Next Article

Research Article

Positive solutions for a Caputo-type fractional differential equation with a Riemann-Stieltjes integral boundary condition

Haixia Li ^,,
Xin Liu

School of Basic Teaching Department, Shandong Water Conservancy Vocational College, Rizhao 276826, China

Received: 28 February 2025 Revised: 26 May 2025 Accepted: 18 June 2025 Published: 25 June 2025

In this paper, we study the solvability of positive solutions for a Caputo-type fractional-order Riemann-Stieltjes integral boundary value problem. Under some monotonicity conditions for the nonlinearity, we use the upper-lower solution method to obtain two existence theorems.
- Caputo-type fractional-order differential equations,
- integral boundary value problems,
- positive solutions,
- upper-lower solution method
Citation: Haixia Li, Xin Liu. Positive solutions for a Caputo-type fractional differential equation with a Riemann-Stieltjes integral boundary condition[J]. Electronic Research Archive, 2025, 33(6): 4027-4044. doi: 10.3934/era.2025179

Related Papers:

Abstract

In this paper, we study the solvability of positive solutions for a Caputo-type fractional-order Riemann-Stieltjes integral boundary value problem. Under some monotonicity conditions for the nonlinearity, we use the upper-lower solution method to obtain two existence theorems.

References

[1]	A. Ahmadkhanlu, On the existence and multiplicity of positive solutions for a $p$-Laplacian fractional boundary value problem with an integral boundary condition, Filomat, 37 (2023), 235–250. https://doi.org/10.2298/FIL2301235A doi: 10.2298/FIL2301235A
[2]	A. Ali, M. Sarwar, M. B. Zada, K. Shah, Degree theory and existence of positive solutions to coupled system involving proportional delay with fractional integral boundary conditions, Math. Methods Appl. Sci., 47 (2014), 10582–10594. https://doi.org/10.1002/mma.6311 doi: 10.1002/mma.6311
[3]	Y. Feng, Z. Bai, Solvability of some nonlocal fractional boundary value problems at resonance in $\mathbb{R}^n$, Fractal Fractional, 6 (2022), 25. https://doi.org/10.3390/fractalfract6010025 doi: 10.3390/fractalfract6010025
[4]	S. Song, H. Li, Y. Zou, Monotone iterative method for fractional differential equations with integral boundary conditions, J. Funct. Spaces, 7 (2020), 7319098. https://doi.org/10.1155/2020/7319098 doi: 10.1155/2020/7319098
[5]	B. R. Sontakke, A. S. Shaikh, Properties of Caputo operator and its applications to linear fractional differential equations, Int. J. Eng. Res. Appl., 5 (2015), 22–27.
[6]	A. Ahmadkhanlu, S. Jamshidzadeh, Existence and uniqueness of positive solutions for a Hadamard fractional integral boundary value problem, Comput. Methods Differ. Equations, 12 (2024), 741–748. https://doi.org/10.22034/cmde.2023.51601.2150 doi: 10.22034/cmde.2023.51601.2150
[7]	K. K. Ali, K. R. Raslan, A. A. E. Ibrahim, M. S. Mohamed, On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type, AIMS Math., 8 (2023), 18206–18222. https://doi.org/10.3934/math.2023925 doi: 10.3934/math.2023925
[8]	A. E. Allaoui, L. Mbarki, Y. Allaoui, J. V. da C. Sousa, Solvability of Langevin fractional differential equation of higher-order with integral boundary conditions, J. Appl. Anal. Comput., 15 (2025), 316–332. https://doi.org/10.11948/20240092 doi: 10.11948/20240092
[9]	A. Frioui, A. Guezane-Lakoud, A. Bragdi, On a sequence of Caputo fractional differential equations with an integral condition, Nonlinear Stud., 30 (2023), 63–71.
[10]	B. Gogoi, U. K. Saha, B. Hazarika, R. P. Agarwal, Existence of positive solutions of a fractional dynamic equation involving integral boundary conditions on time scales, Iran. J. Sci., 48 (2024), 1463–1472. https://doi.org/10.1007/s40995-024-01691-z doi: 10.1007/s40995-024-01691-z
[11]	A. Hamrouni, S. Beloul, Existence of solutions for fractional integro-differential equations with integral boundary conditions, Mathematica, 65 (2023), 249–262. https://doi.org/10.24193/mathcluj.2023.2.11 doi: 10.24193/mathcluj.2023.2.11
[12]	M. Helal, M. Kerfouf, F. Semari, Boundary value problems for fractional differential equations via Riemann-Liouville derivative and nonlinear integral conditions, Nonlinear Stud., 31 (2024), 977–986.
[13]	K. Iatime, L. Guedda, S. Djebali, System of fractional boundary value problems at resonance, Fractional Calculus Appl. Anal., 26 (2023), 1359–1383. https://doi.org/10.1007/s13540-023-00157-0 doi: 10.1007/s13540-023-00157-0
[14]	I. Kaddoura, Y. Awad, Stability results for nonlinear implicit $\vartheta$-Caputo fractional differential equations with fractional integral boundary conditions, Int. J. Differ. Equations, 22 (2023), 5561399. https://doi.org/10.1155/2023/5561399 doi: 10.1155/2023/5561399
[15]	H. N. A. Khan, A. Zada, I. Khan, Analysis of a coupled system of $\Psi$-Caputo fractional derivatives with multipoint-multistrip integral type boundary conditions, Qual. Theory Dyn. Syst., 23 (2024), 129. https://doi.org/10.1007/s12346-024-00987-0 doi: 10.1007/s12346-024-00987-0
[16]	S. Muthaiah, M. Murugesan, S. Ramasamy, N. G. Thangaraj, On fractional integro-differential equation involving Caputo-Hadamard derivative with Hadamard fractional integral boundary conditions, Southeast Asian Bull. Math., 47 (2023), 367–380.
[17]	T. S. Cerdik, Solvability of a Hadamard fractional boundary value problem with multi-term integral and Hadamard fractional derivative boundary conditions, Math. Methods Appl. Sci., 47 (2024), 12946–12960. https://doi.org/10.1002/mma.10475 doi: 10.1002/mma.10475
[18]	H. Si, W. Jiang, G. Li, Solvability of Hilfer fractional differential equations with integral boundary conditions at resonance in $\mathbb{R}^M$, J. Appl. Anal. Comput., 15 (2025), 39–55. https://doi.org/10.11948/20230410 doi: 10.11948/20230410
[19]	S. N. Srivastava, S. Pati, S. Padhi, A. Domoshnitsky, Lyapunov inequality for a Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions, Math. Methods Appl. Sci., 46 (2023), 13110–13123. https://doi.org/10.1002/mma.9238 doi: 10.1002/mma.9238
[20]	R. K. Vats, K. Dhawan, V. Vijayakumar, Analyzing single and multi-valued nonlinear Caputo two-term fractional differential equation with integral boundary conditions, Qual. Theory Dyn. Syst., 23 (2024), 174. https://doi.org/10.1007/s12346-024-01026-8 doi: 10.1007/s12346-024-01026-8
[21]	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. Methods Appl. Sci., 47 (2024), 3582–3595. https://doi.org/10.1002/mma.8880 doi: 10.1002/mma.8880
[22]	N. Wang, Z. Zhou, Existence of solutions for fractional boundary value problems with $\Psi$-Caputo derivative and Stieltjes integral boundary conditions, J. Jilin Univ. Sci., 61 (2023), 469–476. https://doi.org/10.13413/j.cnki.jdxblxb.2022299 doi: 10.13413/j.cnki.jdxblxb.2022299
[23]	X. Luo, Y. Xu, Existence of positive solutions for boundary value problems of fractional differential equations with parameters, J. Guangxi Normal Univ., 42 (2014), 177–185. https://doi.org/10.16088/j.issn.1001-6600.2023112205 doi: 10.16088/j.issn.1001-6600.2023112205
[24]	Y. Yang, Positive solution for a fractional switched system involving Riemann-Stieltjes integral, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 86 (2024), 57–68.
[25]	J. Zhang, S. Haq, A. Zada, I. L. Popa, Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation, AIMS Math., 8 (2023), 28413–28434. https://doi.org/10.3934/math.20231454 doi: 10.3934/math.20231454
[26]	N. Abdellouahab, K. Bouhali, L. Alkhalifa, K. Zennir, Existence and stability analysis of a problem of the Caputo fractional derivative with mixed conditions, AIMS Math., 10 (2025), 6805–6826. https://doi.org/10.3934/math.2025312 doi: 10.3934/math.2025312
[27]	B. Dehda, F. Yazid, F. S. Djeradi, K. Zennir, K. Bouhali, T. Radwan, Numerical approach based on the haar wavelet collocation method for solving a coupled system with the Caputo-Fabrizio fractional derivative, Symmetry, 16 (2024), 713. https://doi.org/10.3390/sym16060713 doi: 10.3390/sym16060713
[28]	M. Kouidri, B. Tellab, A. Amara, K. Zennir, S. Zibar, A single and multi-valued problems involving mixed $\left(k_1, \eta\right)$-Hilfer and ($k_2, \phi$)-Hilfer fractional derivatives for the fractional navier problem, Math. Methods Appl. Sci., 2025 (2025). https://doi.org/10.1002/mma.10993
[29]	A. Kilbas, H Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006. https://doi.org/10.3182/20060719-3-PT-4902.00008
[30]	I. Podlubny, Fractional Differential Equations, Acad. Press, 1999.

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)