Utilizing Schaefer's fixed point theorem in nonlinear Caputo sequential fractional differential equation systems

Muath Awadalla; Manigandan Murugesan; Manikandan Kannan; Jihan Alahmadi; Feryal AlAdsani; Muath Awadalla; Manigandan Murugesan; Manikandan Kannan; Jihan Alahmadi; Feryal AlAdsani

doi:10.3934/math.2024687

AIMS Mathematics

2024, Volume 9, Issue 6: 14130-14157. doi: 10.3934/math.2024687

Previous Article Next Article

Research article Special Issues

Utilizing Schaefer's fixed point theorem in nonlinear Caputo sequential fractional differential equation systems

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 31982, Al Ahsa, Saudi Arabia
2.
Center for Computational Modeling, Chennai Institute of Technology, Chennai, 600069, Tamil Nadu, India
3.
Department of Mathematics, College of Science and Humanity, Prince Sattam bin Abdulaziz University, Sulail, Al-Kharj 11942, Saudi Arabia

Received: 26 January 2024 Revised: 21 March 2024 Accepted: 29 March 2024 Published: 18 April 2024
MSC : 34A08, 34B15, 45G15

In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.
- Caputo fractional derivative,
- existence and uniqueness,
- sequential derivatives,
- fixed point theorem,
- boundary conditions
Citation: Muath Awadalla, Manigandan Murugesan, Manikandan Kannan, Jihan Alahmadi, Feryal AlAdsani. Utilizing Schaefer's fixed point theorem in nonlinear Caputo sequential fractional differential equation systems[J]. AIMS Mathematics, 2024, 9(6): 14130-14157. doi: 10.3934/math.2024687

Related Papers:

Abstract

In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.

References

[1]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, New York: Elsevier, 2006.
[3]	S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in fractional differential equations, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-4036-9
[4]	E. Shishkina, S. Sitnik, Transmutations, singular and fractional differential equations with applications to mathematical physics, London: Academic Press, 2020.
[5]	J. L. Zhou, S. Q. Zhang, Y. B. He, Existence and stability of solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 498 (2021), 124921. https://doi.org/10.1016/j.jmaa.2020.124921 doi: 10.1016/j.jmaa.2020.124921
[6]	S. Abbas, M. Benchohra, J. R. Graef, J. Henderson, Implicit fractional differential and integral equations: Existence and stability, Berlin, Boston: De Gruyter, 2018. https://doi.org/10.1515/9783110553819
[7]	D. R. Smart, Fixed point theorems, London: Cambridge University Press, 1974.
[8]	M. Awadalla, M. Manigandan, Existence results for Caputo tripled fractional differential inclusions with integral and multi-point boundary conditions, Fractal Fract., 7 (2023), 182. https://doi.org/10.3390/fractalfract7020182 doi: 10.3390/fractalfract7020182
[9]	A. Zada, M. Yar, T. Li, Existence and stability analysis of the nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions, Ann. Univ. Paedagog. Crac. Stud. Math., 17 (2018), 103–125. https://doi.org/10.2478/aupcsm-2018-0009 doi: 10.2478/aupcsm-2018-0009
[10]	N. I. Mahmudov, A. Al-Khateeb, Existence and Ulam-Hyers stability of coupled sequential fractional differential equations with integral boundary conditions, J. Inequal. Appl., 2019 (2019), 165. https://doi.org/10.1186/s13660-019-2115-6 doi: 10.1186/s13660-019-2115-6
[11]	A. Al Elaiw, M. Manigandan, M. Awadalla, K. Abuasbeh, Existence results by Mönch's fixed point theorem for a tripled system of sequential fractional differential equations, AIMS Mathematics, 8 (2023), 3969–3996. http://dx.doi.org/10.3934/math.2023199 doi: 10.3934/math.2023199
[12]	B. Ahmad, A. Alsaedi, F. M. Alotaibi, M. Alghanmi, Nonlinear coupled Liouville-Caputo fractional differential equations with a new class of nonlocal boundary conditions, Miskolc Math. Notes, 24 (2023), 31–46. http://dx.doi.org/10.18514/MMN.2023.3839 doi: 10.18514/MMN.2023.3839
[13]	A. Zada, M. Alam, U. Riaz, Analysis of q-fractional implicit boundary value problem having Stieltjes integral conditions, Math. Methods Appl. Sci., 44 (2021), 4381–4413. https://doi.org/10.1002/mma.7038 doi: 10.1002/mma.7038
[14]	K. K. Saha, N. Sukavanam, S. Pan, Existence and uniqueness of solutions to fractional differential equations with fractional boundary conditions, Alex. Eng. J., 72 (2023), 147–155. https://doi.org/10.1016/j.aej.2023.03.076 doi: 10.1016/j.aej.2023.03.076
[15]	A. Al-khateeb, H. Zureigat, O. Ala'yed, S. Bawaneh, Ulam-Hyers stability and uniqueness for nonlinear sequential fractional differential equations involving integral boundary conditions, Fractal Fract., 5 (2021), 235. https://doi.org/10.3390/fractalfract5040235 doi: 10.3390/fractalfract5040235
[16]	A. Thakur, J. Ali, R. Rodríguez-López, Existence of solutions to a class of nonlinear arbitrary order differential equations subject to integral boundary conditions, Fractal Fract., 5 (2021), 220. https://doi.org/10.3390/fractalfract5040220 doi: 10.3390/fractalfract5040220
[17]	S. K. Ntouyas, H. H. Al-Sulami, A study of coupled systems of mixed order fractional differential equations and inclusions with coupled integral fractional boundary conditions, Adv. Differ. Equ., 2020 (2020), 73. https://doi.org/10.1186/s13662-020-2539-9 doi: 10.1186/s13662-020-2539-9
[18]	B. Ahmad, M. Alghanmi, A. Alsaedi, J. J. Nieto, Existence and uniqueness results for a nonlinear coupled system involving Caputo fractional derivatives with a new kind of coupled boundary conditions, Appl. Math. Lett., 116 (2021), 107018. https://doi.org/10.1016/j.aml.2021.107018 doi: 10.1016/j.aml.2021.107018
[19]	H. A. Hammad, H. Aydi, H. Işık, M. De la Sen, Existence and stability results for a coupled system of impulsive fractional differential equations with Hadamard fractional derivatives, AIMS Mathematics, 8 (2023), 6913–6941. https://doi.org/10.3934/math.2023350 doi: 10.3934/math.2023350
[20]	H. A. Hammad, R. A. Rashwan, A. Nafea, M. E. Samei, S. Noeiaghdam, Stability analysis for a tripled system of fractional pantograph differential equations with nonlocal conditions, J. Vib. Control, 30 (2024), 632–647. https://doi.org/10.1177/10775463221149232 doi: 10.1177/10775463221149232
[21]	Humaira, H. A. Hammad, M. Sarwar, M. De la Sen, Existence theorem for a unique solution to a coupled system of impulsive fractional differential equations in complex-valued fuzzy metric spaces, Adv. Differ. Equ., 2021 (2021), 242. https://doi.org/10.1186/s13662-021-03401-0 doi: 10.1186/s13662-021-03401-0
[22]	S. M. Ulam, A collection of mathematical problems, New York: Interscience, 1960.
[23]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[24]	T. M. Rassias, On the stability of the linear mappings in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297–300. https://doi.org/10.2307/2042795 doi: 10.2307/2042795
[25]	Z. Ali, P. Kumam, K. Shah, A. Zada, Investigation of Ulam stability results of a coupled system of nonlinear implicit fractional differential equations, Mathematics, 7 (2019), 341. https://doi.org/10.3390/math7040341 doi: 10.3390/math7040341
[26]	J. Wang, K. Shah, A. Ali, Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., 41 (2018), 2392–2402. https://doi.org/10.1002/mma.4748 doi: 10.1002/mma.4748
[27]	S. Abbas, M. Benchohra, J. E. Lagreg, A. Alsaedi, Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Differ. Equ., 2017 (2017), 180. https://doi.org/10.1186/s13662-017-1231-1 doi: 10.1186/s13662-017-1231-1
[28]	D. Chalishajar, A. Kumar, Existence, uniqueness and Ulam's stability of solutions for a coupled system of fractional differential equations with integral boundary conditions, Mathematics, 6 (2018), 96. https://doi.org/10.3390/math6060096 doi: 10.3390/math6060096
[29]	M. Awadalla, 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 doi: 10.3390/sym14112273
[30]	J. V. C. Sousa, E. C. de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integral-differential equation, Appl. Math. Lett., 81 (2018), 50–56. https://doi.org/10.1016/j.aml.2018.01.016 doi: 10.1016/j.aml.2018.01.016
[31]	M. Murugesan, S. Muthaiah, J. Alzabut, T. N. Gopal, Existence and H-U stability of a tripled system of sequential fractional differential equations with multipoint boundary conditions, Bound. Value Probl., 2023 (2023), 56. https://doi.org/10.1186/s13661-023-01744-z doi: 10.1186/s13661-023-01744-z
[32]	E. Fadhal, K. Abuasbeh, M. Manigandan, M. Awadalla, Applicability of Mónch's fixed point theorem on a system of $(k, \psi)$-Hilfer type fractional differential equations, Symmetry, 14 (2022), 2572. https://doi.org/10.3390/sym14122572 doi: 10.3390/sym14122572
[33]	M. Houas, Sequential fractional pantograph differential equations with nonlocal boundary conditions: Uniqueness and Ulam-Hyers-Rassias stability, Results Nonlinear Anal., 5 (2022), 29–41. https://doi.org/10.53006/rna.928654 doi: 10.53006/rna.928654
[34]	M. Subramanian, M. Manigandan, C. Tunç, T. N. Gopal, J. Alzabut, On the system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order, J. Taibah Univ. Sci., 16 (2022), 1–23. https://doi.org/10.1080/16583655.2021.2010984 doi: 10.1080/16583655.2021.2010984
[35]	M. Manigandan, S. Muthaiah, T. Nandhagopal, R. Vadivel, B. Unyong, N. Gunasekaran, Existence results for a coupled system of nonlinear differential equations and inclusions involving sequential derivatives of fractional order, AIMS Mathematics, 7 (2022), 723–755. https://doi.org/10.3934/math.2022045 doi: 10.3934/math.2022045
[36]	B. Ahmad, J. J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 64 (2012), 3046–3052. https://doi.org/10.1016/j.camwa.2012.02.036 doi: 10.1016/j.camwa.2012.02.036
[37]	S. Rezapour, B. Tellab, C. T. Deressa, S. Etemad, K. Nonlaopon, H-U-type stability and numerical solutions for a nonlinear model of the coupled systems of Navier BVPs via the generalized differential transform method, Fractal Fract., 5 (2021), 166. https://doi.org/10.3390/fractalfract5040166 doi: 10.3390/fractalfract5040166
[38]	J. Jiang, L. Liu, Existence of solutions for a sequential fractional differential system with coupled boundary conditions, Bound. Value Probl., 2016 (2016), 159. https://doi.org/10.1186/s13661-016-0666-8 doi: 10.1186/s13661-016-0666-8
[39]	K. Deimling, Nonlinear functional analysis, Berlin, Heidelberg: Springer, 1985. https://doi.org/10.1007/978-3-662-00547-7

Reader Comments

Your name:*

Email:*
© 2024 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)