On implicit φ-Hilfer fractional differential equations with the p-Laplacian operator

Walid Benhadda; Ali El Mfadel; Abderrazak Kassidi; M'hamed Elomari

doi:10.24193/subbmath.2025.4.05

Authors

Walid Benhadda Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, Beni Mellal, Morocco
Ali El Mfadel Sultan Moulay Slimane University, Superior School of Technology, Khenifra, Morocco
Abderrazak Kassidi Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, Beni Mellal, Morocco
M'hamed Elomari Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, Beni Mellal, Morocco.

DOI:

https://doi.org/10.24193/subbmath.2025.4.05

Keywords:

φ-Hilfer fractional derivative, Topological degree, pLaplacian operator, Ulam-Hyers stability.

Abstract

In this paper, we establish the existence and uniqueness of solutions for a new class of nonlocal boundary implicit φ-Hilfer fractional differential equations involving the p-Laplacian operator. The existence results are derived using the topological degree method for condensing maps and the Banach contraction principle. Moreover, we investigate the Ulam-Hyers and generalized Ulam-Hyers stability of our main problem. To illustrate the applicability of our theoretical results, we provide an example.

References

Agarwal, R.P., Ntouyas, S.K., Ahmad, B., Alzahrani, A.K., Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments, Adv. Difference Equ., 1 (2006), 1-15.

Almalahi, M.A., Abdo, M.S., Panchal, S.K., Existence and Ulam-Hyers-Mittag-Leffler stability results of φ-Hilfer nonlocal Cauchy problem, Rend. Circ. Mat. Palermo (2)., 70 (2021), 57-77. DOI: https://doi.org/10.1007/s12215-020-00484-8

Almalahi, M.A., Abdo, M.S., Panchal, S.K., φ-Hilfer fractional functional differential equation by Picard operator method, J. Appl. Nonlinear Dyn., 9 (2020), 685-702. DOI: https://doi.org/10.5890/JAND.2020.12.011

Alsaedi, A., Alghanmi, M., Ahmad, B., Alharbi, B., Uniqueness of solutions for a φ-Hilfer fractional integral boundary value problem with the p-Laplacian operator, Demonstr. Math., 56 (2023), 11pp. DOI: https://doi.org/10.1515/dema-2022-0195

Bai, C., Existence and uniqueness of solutions for fractional boundary value problems with p-Laplacian operator, Adv. Difference Equ., 2018 (2018), 4pp. DOI: https://doi.org/10.1186/s13662-017-1460-3

Benhadda, W., El Mfadel, A., Kassidi, A., Existence of mild solutions for non-instantaneous impulsive φ-Caputo fractional integro-differential equations, Proyecciones-Journal of Mathematics, 43 (2024). DOI: https://doi.org/10.22199/issn.0717-6279-6282

Benhadda, W., Elomari, M., Kassidi, A., El Mfadel, A., Existence of Anti-Periodic Solutions for ψ-Caputo Fractional p-Laplacian Problems via Topological Degree Methods, Asia Pac. J. Math., 10 (2023), 10-13.

Benhadda, W., Kassidi, A., El Mfadel, A., Existence Results for an Implicit Coupled System Involving φ-Caputo and p-Laplacian Operators, Sahand Commun. Math. Anal., (2024).

Deimling, K., Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. DOI: https://doi.org/10.1007/978-3-662-00547-7

Green, J.W., Valentine, F.A., On the arzela-ascoli theorem, Math. Mag., 34 (1961), 199-202. DOI: https://doi.org/10.1080/0025570X.1961.11975217

Hilfer, R., Applications of Fractional Calculus in Physics, Singapore, 2000. DOI: https://doi.org/10.1142/3779

Isaia, F., On a nonlinear integral equation without compactness, Acta Math. Univ. Comen., New Ser., 75 (2006), 233-240.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and applications of fractional differential equations, Elsevier, 2006. DOI: https://doi.org/10.3182/20060719-3-PT-4902.00008

Liu, X., Jia, M., Xiang, X., On the solvability of a fractional differential equation model involving the p-Laplacian operator, Comput. Math. Appl., 10 (2012), 3267-3275. DOI: https://doi.org/10.1016/j.camwa.2012.03.001

Mali, A.D., Kucche, K.D., Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations, Math. Methods Appl. Sci., 15 (2020), 8608-8631. DOI: https://doi.org/10.1002/mma.6521

Rus, I.A., Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103–107.

Samko, S.G., Kilbas, A.A.and Marichev, O.I., Fractional integrals and derivatives: Theory and applications, 1993.

Sousa, J.V.C., Oliveira, E.C., On the φ- fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. DOI: https://doi.org/10.1016/j.cnsns.2018.01.005

Subashini,R., Jothimani, K., Nisar, K.S. Ravichandran, C., New results on nonlocal functional integro-differential equations via Hilfer fractional derivative, Alex. Eng. J., 59 (2020), 2891-2899. DOI: https://doi.org/10.1016/j.aej.2020.01.055

Vellappandi, M., Govindaraj V., Sousa, C., Fractional optimal reachability problems with φ-Hilfer fractional derivative, Math. Methods Appl. Sci., 45 (2022), 6255-6267. DOI: https://doi.org/10.1002/mma.8168

Yan, H., Qiao, Y., Duan, L., Miao, J., Synchronization of fractional-order gene regulatory networks mediated by miRNA with time delays and unknown parameters, J. Franklin Inst., 359 (2022), 2176-2191. DOI: https://doi.org/10.1016/j.jfranklin.2022.01.028

Zaslavsky, G.M., Hamiltonian chaos and fractional dynamics, Oxford University Press, 2005. DOI: https://doi.org/10.1093/oso/9780198526049.001.0001