Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions

Najla Alghamdi; Bashir Ahmad; Esraa Abed Alharbi; Wafa Shammakh; Najla Alghamdi; Bashir Ahmad; Esraa Abed Alharbi; Wafa Shammakh

doi:10.3934/math.2024632

AIMS Mathematics

2024, Volume 9, Issue 5: 12964-12981. doi: 10.3934/math.2024632

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Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions

1.
Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 29 December 2023 Revised: 04 March 2024 Accepted: 25 March 2024 Published: 07 April 2024
MSC : 34B10, 34D20, 43A08

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order $ \beta_{\xi} $ instead of $ \beta_{1} $; for details, see Lemma 2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations.
- delay differential equations,
- stability criteria,
- nonlocal integral boundary conditions,
- Caputo fractional drivative
Citation: Najla Alghamdi, Bashir Ahmad, Esraa Abed Alharbi, Wafa Shammakh. Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions[J]. AIMS Mathematics, 2024, 9(5): 12964-12981. doi: 10.3934/math.2024632

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Abstract

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order $ \beta_{\xi} $ instead of $ \beta_{1} $; for details, see Lemma 2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations.

References

[1]	J. Sabatier, O. P. Agarwal, J. A. Ttenreiro Machado, Advances in fractional calculus, theoretical developments and applications in physics and engineering, New York: Springer, 2007. https://doi.org/10.1007/978-1-4020-6042-7
[2]	H. A. Fallahgoul, S. M. Focardi, F. J. Fabozzi, Calculus and fractional processes with applications to financial economics, theory and application, London: Academic Press, 2017.
[3]	D. Kusnezov, A. Bulgac, G. Dang, Quantum Levy processes and fractional kinetics, Phys. Rev. Lett., 82 (1999), 1136–11399. https://doi.org/10.1103/PhysRevLett.82.1136 doi: 10.1103/PhysRevLett.82.1136
[4]	F. Zhang, G. Chen, C. Li, J. Kurths, Chaos synchronization in fractional differential systems, Phil. Trans. R. Soc. A, 371 (2013), 20120155. https://doi.org/10.1098/rsta.2012.0155 doi: 10.1098/rsta.2012.0155
[5]	V. M. Bulavatsky, Mathematical models and problems of fractional-differential dynamics of some relaxation filtration processes, Cybern. Syst. Anal., 54 (2018), 727–736. https://doi.org/10.1007/s10559-018-0074-4 doi: 10.1007/s10559-018-0074-4
[6]	G. Alotta, M. Di Paola, F. P. Pinnola, M. Zingales, A fractional nonlocal approach to nonlinear blood flow in small-lumen arterial vessels, Meccanica, 55 (2020), 891–906. https://doi.org/10.1007/s11012-020-01144-y doi: 10.1007/s11012-020-01144-y
[7]	A. N. Chatterjee, B. Ahmad, A fractional-order differential equation model of COVID-19 infection of epithelial cells, Chaos Soliton Fract., 147 (2021), 110952. https://doi.org/10.1016/j.chaos.2021.110952 doi: 10.1016/j.chaos.2021.110952
[8]	C. Xu, Z. Liu, P. Li, J. Yan, L. Yao, Bifurcation mechanism for fractional-order three-triangle multi-delayed neural networks, Neural Proces. Lett., 55 (2023), 6125–6151. https://doi.org/10.1007/s11063-022-11130-y doi: 10.1007/s11063-022-11130-y
[9]	P. Li, R. Gao, C. Xu, Y. Li, A. Akgül, D. Baleanu, Dynamics exploration for a fractional-order delayed zooplankton-phytoplankton system, Chaos Soliton Fract., 166 (2023), 112975. https://doi.org/10.1016/j.chaos.2022.112975 doi: 10.1016/j.chaos.2022.112975
[10]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[11]	B. Ahmad, S. K. Ntouyas, Nonlocal nonlinear fractional-order boundary value problems, Hackensack: World Scientific, 2021. https://doi.org/10.1142/12102
[12]	R. P. Agarwal, V. Lupulescu, D. O'Regan, G. Rahman, Multi-term fractional differential equations in a nonreflexive Banach space, Adv. Differ. Equ., 2013 (2013), 302. https://doi.org/10.1186/1687-1847-2013-302 doi: 10.1186/1687-1847-2013-302
[13]	B. Ahmad, N. Alghamdi, A. Alsaedi, S. K. Ntouyas, A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions, Fract. Calc. Appl. Anal., 22 (2019), 601–618. https://doi.org/10.1515/FCA-2019-0034 doi: 10.1515/FCA-2019-0034
[14]	M. Delkhosh, K. Parand, A new computational method based on fractional Lagrange functions to solve multi-term fractional differential equations, Numer. Algorithms, 88 (2021), 729–766. https://doi.org/10.1007/s11075-020-01055-9 doi: 10.1007/s11075-020-01055-9
[15]	B. Ahmad, M. Alblewi, S. K. Ntouyas, A. Alsaedi, Existence results for a coupled system of nonlinear multi-term fractional differential equations with anti-periodic type coupled nonlocal boundary conditions, Math. Methods Appl. Sci., 44 (2021), 8739–8758. https://doi.org/10.1002/mma.7301 doi: 10.1002/mma.7301
[16]	B. Ahmad, A. Alsaedi, N. Alghamdi, S. K. Ntouyas, Existence theorems for a coupled system of nonlinear multi-term fractional differential equations with nonlocal boundary conditions, Kragujevac J. Math., 46 (2022), 317–331. https://doi.org/10.46793/KgJMat2202.317A doi: 10.46793/KgJMat2202.317A
[17]	A. Diop, Existence of mild solutions for multi-term time fractional measure differential equations, J. Anal., 30 (2022), 1609–1623. https://doi.org/10.1007/s41478-022-00420-2 doi: 10.1007/s41478-022-00420-2
[18]	H. Gou, On the $S$-asymptotically $\omega$-periodic mild solutions for multi-term time fractional measure differential equations, Topol. Methods Nonlinear Anal., 62 (2023), 569–590. https://doi.org/10.1080/17442508.2023.2300290 doi: 10.1080/17442508.2023.2300290
[19]	C. Chen, L. Liu, Q. Dong, Existence and Hyers-Ulam stability for boundary value problems of multi-term Caputo fractional differential equations, Filomat, 37 (2023), 9679–9692. https://doi.org/10.1186/s13662-018-1903-5 doi: 10.1186/s13662-018-1903-5
[20]	Y. S. Kang, S. H. Jo, Spectral collocation method for solving multi-term fractional integro-differential equations with nonlinear integral, Math. Sci., 18 (2024), 91–106. https://doi.org/10.1007/s40096-022-00487-9 doi: 10.1007/s40096-022-00487-9
[21]	M. Dieye, E. H. Lakhel, M. A. McKibben, Controllability of fractional neutral functional differential equations with infinite delay driven by fractional Brownian motion, IMA J. Math. Control Inform., 38 (2021), 929–956. https://doi.org/10.1093/imamci/dnab020 doi: 10.1093/imamci/dnab020
[22]	R. Chaudhary, V. Singh, D. N. Pandey, Controllability of multi-term time-fractional differential systems with state-dependent delay, J. Appl. Anal., 26 (2020), 241–255. https://doi.org/10.1515/jaa-2020-2016 doi: 10.1515/jaa-2020-2016
[23]	H. Zhao, J. Zhang, J. Lu, J. Hu, Approximate controllability and optimal control in fractional differential equations with multiple delay controls, fractional Brownian motion with Hurst parameter in $0 < H < \frac{1}{2}$, and Poisson jumps, Commun. Nonlinear Sci. Numer. Simul., 128 (2024), 107636. https://doi.org/10.1016/j.cnsns.2023.107636 doi: 10.1016/j.cnsns.2023.107636
[24]	H. Boulares, A. Ardjouni, Y. Laskri, Existence and uniqueness of solutions to fractional order nonlinear neutral differential equations, Fract. Differ. Calc., 7 (2017), 247–263. https://doi.org/10.7153/fdc-2017-07-10 doi: 10.7153/fdc-2017-07-10
[25]	Y. Guo, X. B. Shu, Y. Li, F. Xu, The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order $1 < \beta < 2$, Bound. Value Probl., 2019 (2019), 59. https://doi.org/10.1186/s13661-019-1172-6 doi: 10.1186/s13661-019-1172-6
[26]	X. Wang, D. Luo, Q. Zhu, Ulam-Hyers stability of Caputo type fuzzy fractional differential equations with time-delays, Chaos Soliton Fract., 156 (2022), 111822. https://doi.org/10.1016/j.chaos.2022.111822 doi: 10.1016/j.chaos.2022.111822
[27]	R. Chaharpashlou, A. M. Lopes, Hyers-Ulam-Rassias stability of a nonlinear stochastic fractional Volterra integro-differential equation, J. Appl. Anal. Comput., 13 (2023), 2799–2808. https://doi.org/10.11948/20230005 doi: 10.11948/20230005
[28]	C. Chen, L. Liu, Q. Dong, Existence and Hyers-Ulam stability for boundary value problems of multi-term Caputo fractional differential equations, Filomat, 37 (2023), 9679–9692. https://doi.org/10.2298/FIL2328679C doi: 10.2298/FIL2328679C
[29]	G. Rahman, R. P. Agarwal, D. Ahmad, Existence and stability analysis of $n$th order fractional delay differential equation, Chaos Soliton Fract., 155 (2022), 111709. https://doi.org/10.1016/j.chaos.2021.111709 doi: 10.1016/j.chaos.2021.111709
[30]	P. J. Torvik, R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294–298. https://doi.org/10.1115/1.3167615 doi: 10.1115/1.3167615
[31]	F. Mainardi, P. Pironi, F. Tampieri, On a generalization of the Basset problem via fractional calculus, In: Proceedings 15th Canadian congress of applied mechanics, 2 (1995), 836–837.
[32]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8

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