Multiple positive solutions for nonlinear singular Katugampola fractional differential equations with nonlocal coupled Riemann-Stieltjes integral boundary conditions

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This study explores a system of singular nonlinear higher-order katugampola fractional differential equations (KFDEs) with nonlocal coupled Riemann-Stieltjes integral boundary conditions. First, the addressed KFDEs are reformulated as the equivalent integral equations through explicit construction of some appropriate Green’s functions. Second, through synergistic application of Schauder’s and Guo-Krasnoselskii’s fixed-point theorems, some explicit parameter interval-dependent existence criteria are established for at least one or two positive solutions to the considered KFDEs with the help of the properties of Green’s functions. Finally, some concrete examples are provided to validate the effectiveness of the main theoretical results.

Keywords

Katugampola fractional differential equations / multiple positive solutions / Green’s functions / nonlinear alternative of Leray-Schauder type / Guo-Krasnoselskii’s fixed-point theorem

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl., 62 (2011), 1251-1268. https://doi.org/10.1016/j.camwa.2011.02.039

[2] B. Jalili, P. Jalili, A. Shateri, D. D. Ganji, Rigid plate submerged in a Newtonian fluid and fractional differential equation problems via Caputo fractional derivative, Partial Differ. Equations Appl. Math., 6 (2022), 100452. https://doi.org/10.1016/j.padiff.2022.100452

[3] H. Boulares, A. Ardjouni, Y. Laskri, Positive solutions for nonlinear fractional differential equations, Positivity, 21 (2017), 1201-1212. https://doi.org/10.1007/s11117-016-0461-x

[4] R. Agarwal, D. O’Regan S. Stanĕk, Positive solutions for mixed problems of singular fractional differential equations, Math. Nachr., 285 (2012), 27-41. https://doi.org/10.1002/mana.201000043

[5] J. Jiang, D. O’Regan, J. Xu, Z. Fu, Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions, J. Ineq. Appl., 2019 (2019), 204. https://doi.org/10.1186/s13660-019-2156-x

[6] R. Luca, A. Tudorache, On a system of Hadamard fractional differential equations with nonlocal boundary conditions on an infinite interval, Fractal Fract., 7 (2023), 458. https://doi.org/10.3390/fractalfract7060458

[7] A. Tudorache, R. Luca, Positive solutions for a system of Hadamard fractional boundary value problems on an infinite interval, Axioms, 12 (2023), 793. https://doi.org/10.3390/axioms12080793

[8] A. Ardjouni, Existence and uniqueness of positive solutions for nonlinear Caputo-Hadamard fractional differential equations, Proyecciones J. Math., 40 (2021), 139-152. https://doi.org/10.22199/issn.0717-6279-2021-01-0009

[9] 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. https://doi.org/10.3390/sym14112273

[10] F. Jarad, T. Abdeljawad, D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619. http://dx.doi.org/10.22436/jnsa.010.05.27

[11] M. Medved, E. Brestovanská, Differential equations with tempered Ψ-Caputo fractional derivative, Math. Model. Anal., 26 (2021), 631-650. https://doi.org/10.3846/mma.2021.13252

[12] A. R. Sevinik, Ü. Aksoy, E. Karapinar, İ. M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Methods Appl. Sci., 47 (2024), 10928-10939. https://doi.org/10.1002/mma.6652

[13] K. Kavitha, V. Vijayakumar, R. Udhayakumar, C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control, 24 (2022), 1406-1415. https://doi.org/10.1002/asjc.2549

[14] A. Potapov, V. Beybalaev, A. Aliverdiev, On a boundary value problem for a nonlinear differential equation with a Riemann-Liouville fractional derivative of variable order and nonlocal boundary conditions, Electron. Res. Arch., 33 (2025), 5829-5844. https://doi.org/10.3934/era.2025259

[15] S. Hristova, A. Benkerrouche, M. S. Souid, A. Hakem, Boundary value problems of Hadamard fractional differential equations of variable order, Symmetry, 13 (2021), 896. https://doi.org/10.3390/sym13050896

[16] F. Berrighi, I. Medjadja, E. Karapınar, Mild solutions for conformable fractional order functional evolution equations via Meir-Keeler type fixed point theorem, Filomat, 39 (2025), 1989-2002. https://doi.org/10.2298/FIL2506989B

[17] K. Szajek, W. Sumelka, Discrete mass-spring structure identification in nonlocal continuum space-fractional model, Eur. Phys. J. Plus, 134 (2019), 448. https://doi.org/10.1140/epjp/i2019-12890-8

[18] Z. Malki, F. Berhoun, A. Ouahab, System of boundary random fractional differential equations via Hadamard derivative, Ann. Univ. Paedagog. Cracov. Stud. Math., 20 (2021), 17-41. https://doi.org/10.2478/aupcsm-2021-0002

[19] J. Jiang, L. Liu, Y. Wu, Positive solutions to singular fractional differential system with coupled boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3061-3074. https://doi.org/10.1016/j.cnsns.2013.04.009

[20] C. Yuan, D. Jiang, D. O’Regan, R. Agarwal, Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions, Electron. J. Qual. Theory Differ. Equations, 2012 (2012), 1-17. https://doi.org/10.14232/ejqtde.2012.1.13

[21] B. Ahmad, S. K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 348-360. https://doi.org/10.2478/s13540-014-0173-5

[22] B. Ahmad, S. K. Ntouyas, A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Soliton. Fract., 83 (2016), 234-241. https://doi.org/10.1016/j.chaos.2015.12.014

[23] W. Yang, Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions, J. Appl. Math. Comput., 49 (2015), 357-381. https://doi.org/10.1007/s12190-014-0843-9

[24] W. Yang, Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations, J. Nonlinear Sci. Appl., 8 (2015), 110-129. http://dx.doi.org/10.22436/jnsa.008.02.04

[25] A. Tudorache, R. Luca, Systems of Hilfer-Hadamard fractional differential equations with nonlocal coupled boundary conditions, Fractal Fract., 7 (2023), 816. https://doi.org/10.3390/fractalfract7110816

[26] A. Tudorache, R. Luca, Positive solutions to a system of coupled Hadamard fractional boundary value problems, Fractal Fract., 8 (2024), 543. https://doi.org/10.3390/fractalfract8090543

[27] C. Kiataramkul, W. Yukunthorn, S. K. Ntouyas, J. Tariboon, Sequential Riemann-Liouville and Hadamard-Caputo fractional differential systems with nonlocal coupled fractional integral boundary conditions, Axioms, 10 (2021), 174. https://doi.org/10.3390/axioms10030174

[28] N. Mehmood, A. Abbas, A. Akgül, T. Abdeljawad, M. A. Alqudah, Existence and stability results for coupled system of fractional differential equations involving AB-Caputo derivative, Fractals, 31 (2023), 2340023. https://doi.org/10.1142/S0218348X23400236

[29] A. Salim, S. Bouriah, M. Benchohra, J. E. Lazreg, E Karapinar, A study on k-generalized ψ-Hilfer fractional differential equations with periodic integral conditions , Math. Methods Appl. Sci., 47 (2024), 12044-12061. https://doi.org/10.1002/mma.9056

[30] W. Yang, Positive solutions for a class of nonlinear p-Laplacian Hadamard fractional differential systems with coupled nonlocal Riemann-Stieltjes integral boundary conditions , Filomat, 36 (2022), 6631-6654. https://doi.org/10.2298/FIL2219631Y

[31] S. Zibar, B. Tellab, A. Amara, H. Emadifar, A. Kumar, S. Widatalla, Existence, uniqueness and stability analysis of a nonlinear coupled system involving mixed φ-Riemann-Liouville and ψ-Caputo fractional derivatives, Bound. Value Probl., 2025 (2025), 8. https://doi.org/10.1186/s13661-025-01994-z

[32] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.

[33] B. Basti, Y. Arioua, N. Benhamidouche, Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations, J. Math. Appl., 42 (2019), 35-61. https://doi.org/10.7862/rf.2019.3

[34] B. Łupińska, E. Schmeidel, Analysis of some Katugampola fractional differential equations with fractional boundary conditions, Math. Biosci. Eng., 18 (2021), 7269-7279. https://doi.org/10.3934/mbe.2021359

[35] S. N. Srivastava, S. Pati, J. R. Graef, A. Domoshnitsky, S. Padhi, Existence of solution for a Katugampola fractional differential equation using coincidence degree theory, Mediterr. J. Math., 21 (2024), 123. https://doi.org/10.1007/s00009-024-02658-5

[36] L. Sadek, S. A. ldris, F. Jarad, The general Caputo-Katugampola fractional derivative and numerical approach for solving the fractional differential equations, Alex. Eng. J., 121 (2025), 539-557. https://doi.org/10.1016/j.aej.2025.02.065

[37] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. https://doi.org/10.1016/j.amc.2011.03.062

[38] B. Łupińska, T. Odzijewicz, A Lyapunov-type inequality with the Katugampola fractional derivative, Math. Methods Appl. Sci., 41 (2018), 8985-8996. https://doi.org/10.1002/mma.4782

[39] W. Yang, Eigenvalue problems for a coupled system of singular Katugampola fractional differential equations with four-point boundary conditions, Differ. Equations Appl., 17 (2025), 133-162. https://doi.org/10.7153/dea-2025-17-09

[40] R. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications , Cambridge University Press, 2001.

[41] M. A. Krasnoselskii, Positive Solutions of Operator Equations , Noordhoff Gronigen, Netherland, 1964.