Existence results by Mönch's fixed point theorem for a tripled system of sequential fractional differential equations

Abeer Al Elaiw; Murugesan Manigandan; Muath Awadalla; Kinda Abuasbeh; Abeer Al Elaiw; Murugesan Manigandan; Muath Awadalla; Kinda Abuasbeh

doi:10.3934/math.2023199

AIMS Mathematics

2023, Volume 8, Issue 2: 3969-3996. doi: 10.3934/math.2023199

Previous Article Next Article

Research article Special Issues

Existence results by Mönch's fixed point theorem for a tripled system of sequential fractional differential equations

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf, Al Ahsa, 31982, Saudi Arabia
2.
Sri Ramakrishna Mission vidyalaya college of arts and science, Coimbatore-641020, Tamil Nadu, India

Received: 17 August 2022 Revised: 12 November 2022 Accepted: 24 November 2022 Published: 30 November 2022
MSC : 26A33, 34B15, 34B18

In this paper, we study the existence of the solutions for a tripled system of Caputo sequential fractional differential equations. The main results are established with the aid of Mönch's fixed point theorem. The stability of the tripled system is also investigated via the Ulam-Hyer technique. In addition, an applied example with graphs of the behaviour of the system solutions with different fractional orders are provided to support the theoretical results obtained in this study.
- Caputo fractional derivative,
- tripled system,
- existence,
- mixed boundary conditions,
- H-U stability
Citation: Abeer Al Elaiw, Murugesan Manigandan, Muath Awadalla, Kinda Abuasbeh. Existence results by Mönch's fixed point theorem for a tripled system of sequential fractional differential equations[J]. AIMS Mathematics, 2023, 8(2): 3969-3996. doi: 10.3934/math.2023199

Related Papers:

Abstract

In this paper, we study the existence of the solutions for a tripled system of Caputo sequential fractional differential equations. The main results are established with the aid of Mönch's fixed point theorem. The stability of the tripled system is also investigated via the Ulam-Hyer technique. In addition, an applied example with graphs of the behaviour of the system solutions with different fractional orders are provided to support the theoretical results obtained in this study.

References

[1]	D. Hinton, Handbook of differential equations (Daniel Zwillinger), SIAM Review, 36 (1994), 126–127. https://doi.org/10.1137/1036029 doi: 10.1137/1036029
[2]	K. Oldham, J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
[3]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
[4]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Switzerland: Gordon and breach science, 1993.
[5]	N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 45 (2006), 765–771. https://doi.org/10.1007/s00397-005-0043-5 doi: 10.1007/s00397-005-0043-5
[6]	M. Awadalla, Y. Y. Y. Noupoue, K. A. Asbeh, N. Ghiloufi, Modeling drug concentration level in blood using fractional differential equation based on Psi-Caputo derivative, J. Math., 2022 (2022), 9006361. https://doi.org/10.1155/2022/9006361 doi: 10.1155/2022/9006361
[7]	Y. Y. Y. Noupoue, Y. Tandoğdu, M. Awadalla, On numerical techniques for solving the fractional logistic differential equation, Adv. Differ. Equ., 2019 (2019), 108. https://doi.org/10.1186/s13662-019-2055-y doi: 10.1186/s13662-019-2055-y
[8]	M. Manigandan, S. Muthaiah, T. Nandhagopal, R. Vadivel, B. Unyong, N. Gunasekaran, Existence results for coupled system of nonlinear differential equations and inclusions involving sequential derivatives of fractional order, AIMS Math., 7 (2022), 723–755. https://doi.org/10.3934/math.2022045 doi: 10.3934/math.2022045
[9]	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
[10]	M. M. Matar, I. A. Amra, J. Alzabut, Existence of solutions for tripled system of fractional differential equations involving cyclic permutation boundary conditions, Bound. Value Probl., 2020 (2020), 140. https://doi.org/10.1186/s13661-020-01437-x doi: 10.1186/s13661-020-01437-x
[11]	X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2021), 64–69. https://doi.org/10.1016/j.aml.2008.03.001 doi: 10.1016/j.aml.2008.03.001
[12]	M. Subramanian, M. Manigandan, C. Tung, T. N. Gopal, J. Alzabut, On 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
[13]	A. Hamoud, Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integrodifferential equations, Adv. Theor. Nonlinear Anal. Appl., 4 (2020), 321–331. https://doi.org/10.31197/atnaa.799854 doi: 10.31197/atnaa.799854
[14]	H. Khan, C. Tunc, W. Chen, A. Khan, Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator, J. Appl. Anal. Comput., 8 (2018), 1211–1226. https://doi.org/10.11948/2018.1211 doi: 10.11948/2018.1211
[15]	S. Ferraoun, Z. Dahmani, Existence and stability of solutions of a class of hybrid fractional differential equations involving RL-operator, J. Interdiscip. Math., 23 (2020), 885–903. https://doi.org/10.1080/09720502.2020.1727617 doi: 10.1080/09720502.2020.1727617
[16]	A. Al Elaiw, M. M. Awadalla, M. Manigandan, K. Abuasbeh, A novel implementation of Mönch's fixed point theorem to a system of nonlinear Hadamard fractional differential equations, Fractal Fract., 6 (2022), 586. https://doi.org/10.3390/fractalfract6100586 doi: 10.3390/fractalfract6100586
[17]	W. Al-Sadi, Z. Y. Huang, A. Alkhazzan, Existence and stability of a positive solution for nonlinear hybrid fractional differential equations with singularity, J. Taibah Univ. Sci., 13 (2019), 951–960. https://doi.org/10.1080/16583655.2019.1663783 doi: 10.1080/16583655.2019.1663783
[18]	M. Subramanian, M. Manigandan, T. N. Gopal, Fractional differential equations involving Hadamard fractional derivatives with nonlocal multi-point boundary conditions, Discontinuity, Nonlinearity, and Complexity, 9 (2020), 421–431. https://doi.org/10.5890/DNC.2020.09.006 doi: 10.5890/DNC.2020.09.006
[19]	M. Awadalla, K. Abuasbeh, M. Subramanian, M. Manigandan, On a system of $\psi$-Caputo hybrid fractional differential equations with Dirichlet boundary conditions, Mathematics, 10 (2022), 1681. https://doi.org/10.3390/math10101681 doi: 10.3390/math10101681
[20]	A. Al-khateeb, H. Zureigat, O. Ala'Zyed, 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
[21]	I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107.
[22]	M. Manigandan, M. Subramanian, T. N. Gopal, B. Unyong, Existence and stability results for a tripled system of the Caputo type with multi-point and integral boundary conditions, Fractal Fract., 6 (2022), 285. https://doi.org/10.3390/fractalfract6060285 doi: 10.3390/fractalfract6060285
[23]	V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. Theor., 74 (2011), 4889–4897. https://doi.org/10.1016/j.na.2011.03.032 doi: 10.1016/j.na.2011.03.032
[24]	K. Karakaya, N. E. Bouzara, K. DoLan, Y. Atalan, Existence of tripled fixed points for a class of condensing operators in Banach spaces, Sci. World J., 2014 (2014), 140. https://doi.org/10.1155/2014/541862 doi: 10.1155/2014/541862
[25]	B. Ahmad, S. Hamdan, A. Alsaedi, S. K. Ntouyas, A study of a nonlinear coupled system of three fractional differential equations with nonlocal coupled boundary conditions, Adv. Differ. Equ., 2021 (2021), 278. https://doi.org/10.1186/s13662-021-03440-7 doi: 10.1186/s13662-021-03440-7
[26]	M. M. Matar, I. A. Amra, J. Alzabut, Existence of solutions for tripled system of fractional differential equations involving cyclic permutation boundary conditions, Bound. Value Probl., 2020 (2020), 140. https://doi.org/10.1186/s13661-020-01437-x doi: 10.1186/s13661-020-01437-x
[27]	Y. Alruwaily, B. Ahmad, S. K. Ntouyas, A. S. Alzaidi, Existence results for coupled nonlinear sequential fractional differential equations with coupled Riemann-Stieltjes integro-multipoint boundary conditions, Fractal Fract., 6 (2022), 123. https://doi.org/10.3390/fractalfract6020123 doi: 10.3390/fractalfract6020123
[28]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[29]	M. Benchohra, J. Henderson, D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Comm. Appl. Anal., 12 (2008), 419–428.
[30]	D. J. Guo, V. Lakshmikantham, X. Z. Liu, Nonlinear integral equations in abstract spaces, Kluwer Academic Publishers, 1996.
[31]	E. Zeidler, Nonlinear functional analysis and its applications: Part 2 B: Nonlinear monotone operators, Springer, 1989.
[32]	H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. Theor., 4 (1980), 985–999. https://doi.org/10.1016/0362-546X(80)90010-3 doi: 10.1016/0362-546X(80)90010-3

Reader Comments

Your name:*

Email:*
© 2023 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)