Unique positive solution for a nonlinear p-Laplacian Hadamard fractional differential boundary value problem

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This article was retracted on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

The purpose of this article is to investigate a class of nonlinear p-Laplacian Hadamard fractional differential equations. By employing two fixed–point theorems of the sum operator, we get the uniqueness of a positive solution for such p-Laplacian equations. In addition, an iterative sequence can be constructed to approximate the unique positive solution. The validity of the results is demonstrated through two illustrative numerical examples in the final of this paper.

Keywords

fixed point theorem / uniqueness / fractional differential equations / p-Laplacian operator

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This article has been retracted.Please see the Retraction Notice for more detail: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] L. Guo, C. Li, J. Zhao, Existence of monotone positive solutions for caputo-hadamard nonlinear fractional differential equation with infinite-point boundary value conditions, Symmetry, 15 (2023), 970. https://doi.org/10.3390/sym15050970

[2] Y. Li, Y. Liu, Multiple solutions for a class of boundary value problems of fractional differential equations with generalized Caputo derivatives, AIMS Math., 6 (2021), 13119-13142. https://doi.org/10.3934/math.2021758

[3] X. Zuo, W. Wang, Existence of solutions for fractional differential equation with periodic boundary condition, AIMS Math., 7 (2022), 6619-6633. https://doi.org/10.3934/math.2022369

[4] C. Ciftci, F. Deren, Analysis of p-Laplacian Hadamard fractional boundary value problems with the derivative term involved in the nonlinear term, Math. Method Appl. Sci., 46 (2023), 8945-8955. https://doi.org/10.1002/mma.9028

[5] L. Guo, H. Liu, C. Li, J. Zhao, J. Xu, Existence of positive solutions for singular p-Laplacian hadamard fractional differential equations with the derivative term contained in the nonlinear term, Nonlinear Anal. Modell. Control, 28 (2023), 491-515. https://doi.org/10.15388/namc.2023.28.31728

[6] K. Zhang, J. Wang, W. Ma, Solutions for integral boundary value problems of nonlinear Hadamard fractional differential equations, J. Funct. Spaces, 2018 (2018), 1-10. https://doi.org/10.1155/2018/2193234

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

[8] J. Xu, J. Liu, D. O’Regan, Solvability for a Hadamard-type fractional integral boundary value problem, Nonlinear Anal. Model. Control, 28 (2023), 672-696. https://doi.org/10.15388/namc.2023.28.32130

[9] H. Lu, Z. Han, S. Sun, Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with p-Laplacian, Adv. Differ. Equations, 30 (2013). https://doi.org/10.1186/1687-1847-2013-30

[10] L. Guo, Y. Wang, H. Liu, C. Li, W. Wang, Y. Cui, et al., On iterative positive solutions for a class of singular infinite-point p-Laplacian fractional differential equation with singular source terms, J. Appl. Anal. Comput., 13 (2023), 2827-2842. https://doi.org/10.15388/NA.2018.2.3

[11] K. Jong, H. Choi, Y. Ri, Existence of positive solutions of a class of multi-point boundary value problems for p-Laplacian fractional differential equations with singular source terms, Commun. Nonlinear Sci. Numer. Simul., 72 (2019), 272-281. https://doi.org/10.1016/j.cnsns.2018.12.021

[12] J. Wu, X. Zhang, L. Liu, Y. Wu, Y. Cui, The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity, Bound Value Probl., 82 (2018). https://doi.org/10.1186/s13661-018-1003-1

[13] X. Zhang, C. Mao, L. Liu, Y. Wu, Exact iterative solution for an abstract fractional dynamic system model for bioprocess, Qual. Theory Dyn. Syst., 16 (2017), 205-222. https://doi.org/10.1007/s12346-015-0162-z

[14] X. Zhang, L. Liu, Y. Wu, Y. Cui, The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach, J. Math. Anal. Appl., 464 (2018), 1089-1106. https://doi.org/10.1016/j.jmaa.2018.04.040

[15] X. Zhang, P. Chen, Y. Wu, B. Wiwatanapataphee, A necessary and sufficient condition for the existence of entire large solutions to a k-Hessian system, Appl. Math. Lett., 145 (2023). https://doi.org/10.1016/j.aml.2023.108745

[16] Z. Zhang, X. Yang, S. Wang, The alternating direction implicit difference scheme and extrapolation method for a class of three dimensional hyperbolic equations with constant coefficients, Electron. Res. Arch., 33 (2025), 3348-3377. https://doi.org/10.3934/era.2025148

[17] J. Zhang, X. Yang, S. Wang, The ADI difference and extrapolation scheme for high-dimensional variable coefficient evolution equations, Electron. Res. Arch., 33 (2025), 3305-3327. https://doi.org/10.3934/era.2025146

[18] T. Liu, H. Zhang, X. Yang, The ADI compact difference scheme for three-dimensional integro-partial differential equation with three weakly singular kernels, J. Appl. Math. Comput., 71 (2025), 1-29. https://doi.org/10.1007/s12190-025-02386-3

[19] K. Liu, Z. He, H. Zhang, X. Yang, A Crank-Nicolson ADI compact difference scheme for the three-dimensional nonlocal evolution problem with a weakly singular kernel, Comput. Appl. Math., 44 (2025), 164. https://doi.org/10.1007/s40314-025-03125-x

[20] X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7

[21] J. Wang, X. Jiang, H. Zhang, X. Yang, A new fourth-order nonlinear difference scheme for the nonlinear fourth-order generalized Burgers-type equation, J. Appl. Math. Comput., (2025), 1-31. https://doi.org/10.1007/s12190-025-02467-3

[22] X. Yang, Z. Zhang, Superconvergence analysis of a robust orthogonal Gauss collocation method for 2D fourth-order subdiffusion equations, J. Sci. Comput., 100 (2024), 62. https://doi.org/10.1007/s10915-024-02616-z

[23] Y. Shi, X. Yang, The pointwise error estimate of a new energy-preserving nonlinear difference method for supergeneralized viscous Burgers’ equation, Comput. Appl. Math., 44 (2025), 257. https://doi.org/10.1007/s40314-025-03222-x

[24] L. Guo, W. Wang, C. Li, J. Zhao, D. Min, Existence results for a class of nonlinear singular p-Laplacian Hadamard fractional differential equations, Electron. Res. Arch., 32 (2024), 928-944. http://doi.org/10.3934/era.2024045

[25] C. Zhai, Y. Ma, H. Li, Unique positive solution for a p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral, AIMS Math., 5 (2020), 4754-4769. https://doi.org/10.3934/math.2020304

[26] B. Ahmad, A. Alsaedi, S. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities , Springer: Cham, Switzerland, 2017. http://dx.doi.org/10.1007/978-3-319-52141-1

[27] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations , Elsevier: Amsterdam, 2003. http://doi.org/10.1016/S0304-0208(06)80001-0

[28] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones. Academic Press , Academic Press, Boston, 1998. https://doi.org/10.1016/c2013-0-10750-7

[29] C. Zhai, D. Anderson, A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations, J. Math. Anal. Appl., 375 (2011), 388-400. https://doi.org/10.1016/j.jmaa.2010.09.017

[30] C. Yang, C. Zhai, M. Hao, Uniqueness of positive solutions for several classes of sum operator equations and applications, J. Inequal. Appl., 2014 (2014), 58. https://doi.org/10.1186/1029-242X-2014-58