Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions

Madeaha Alghanmi; Shahad Alqurayqiri; Madeaha Alghanmi; Shahad Alqurayqiri

doi:10.3934/math.2024729

AIMS Mathematics

2024, Volume 9, Issue 6: 15040-15059. doi: 10.3934/math.2024729

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Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions

Madeaha Alghanmi ^{1
,
,},
Shahad Alqurayqiri ²

1.
Department of Mathematics, College of Sciences & Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 02 February 2024 Revised: 15 April 2024 Accepted: 16 April 2024 Published: 25 April 2024
MSC : 26A33, 34K05

This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results.
- coupled system,
- infinite delay,
- Caputo fractional derivative,
- integral boundary conditions,
- existence
Citation: Madeaha Alghanmi, Shahad Alqurayqiri. Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions[J]. AIMS Mathematics, 2024, 9(6): 15040-15059. doi: 10.3934/math.2024729

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Abstract

This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results.

References

[1]	A. Carvalho, C. M. A. Pinto, A delay fractional order model for the co-infection of malaria and HIV/AIDS, Int. J. Dynam. Control, 5 (2017), 168–186. https://doi.org/10.1007/s40435-016-0224-3 doi: 10.1007/s40435-016-0224-3
[2]	N. H. Sweilam, S. M. Al-Mekhlafi, Z. N. Mohammed, D. Baleanu, Optimal control for variable order fractional HIV/AIDS and malaria mathematical models with multi-time delay, Alex. Eng. J., 59 (2020), 3149–3162. https://doi.org/10.1016/j.aej.2020.07.021 doi: 10.1016/j.aej.2020.07.021
[3]	C. Lu, J. Chen, X. Fan, L. Zhang, Dynamics and simulations of a stochastic predator-prey model with infinite delay and impulsive perturbations, J. Appl. Math. Comput., 57 (2018), 437–465. https://doi.org/10.1007/s12190-017-1114-3 doi: 10.1007/s12190-017-1114-3
[4]	S. Pati, J. R. Graef, S. Padhi, Positive periodic solutions to a system of nonlinear differential equations with applications to Lotka-Volterra-type ecological models with discrete and distributed delays, J. Fixed Point Theory Appl., 21 (2019), 80. https://doi.org/10.1007/s11784-019-0715-x doi: 10.1007/s11784-019-0715-x
[5]	Z. S. Aghayan, A. Alfi, A. M. Lopes, LMI-based delayed output feedback controller design for a class of fractional-order neutral-type delay systems using guaranteed cost control approach, Entropy, 24 (2022), 1496. https://doi.org/10.3390/e24101496 doi: 10.3390/e24101496
[6]	M. S. Ali, G. Narayanan, V. Shekher, A. Alsaedi, B. Ahmad, Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays, Commun. Nonlinear Sci. Numer. Simul., 83 (2020), 105088. https://doi.org/10.1016/j.cnsns.2019.105088 doi: 10.1016/j.cnsns.2019.105088
[7]	K. Cui, J. Lu, C. Li, Z. He, Y. M. Chu, Almost sure synchronization criteria of neutral-type neural networks with Lévy noise and sampled-data loss via event-triggered control, Neurocomputing, 325 (2019), 113–120. https://doi.org/10.1016/j.neucom.2018.10.013 doi: 10.1016/j.neucom.2018.10.013
[8]	B. Ghanbari, A fractional system of delay differential equation with nonsingular kernels in modeling hand-foot-mouth disease, Adv. Differ. Equ., 2020 (2020), 536. https://doi.org/10.1186/s13662-020-02993-3 doi: 10.1186/s13662-020-02993-3
[9]	L. Hu, S. Zhang, Existence results for a coupled system of fractional differential equations with $p$-Laplacian operator and infinite-point boundary conditions, Bound. Value Probl., 2017 (2017), 88. https://doi.org/10.1186/s13661-017-0819-4 doi: 10.1186/s13661-017-0819-4
[10]	W. Zhang, W. Liu, T. Xue, Existence and uniqueness results for the coupled systems of implicit fractional differential equations with periodic boundary conditions, Adv. Differ. Equ., 2018 (2018), 413. https://doi.org/10.1186/s13662-018-1867-5 doi: 10.1186/s13662-018-1867-5
[11]	A. M. A. El-Sayed, S. A. Abd El-Salam, Coupled system of a fractional order differential equations with weighted initial conditions, Open Math., 17 (2019), 1737–1749. https://doi.org/10.1515/math-2019-0120 doi: 10.1515/math-2019-0120
[12]	B. Ahmad, M. Alghanmi, A. Alsaedi, Existence results for a nonlinear coupled system involving both Caputo and Riemann-Liouville generalized fractional derivatives and coupled integral boundary conditions, Rocky Mountain J. Math., 50 (2020), 1901–1922. https://doi.org/10.1216/rmj.2020.50.1901 doi: 10.1216/rmj.2020.50.1901
[13]	S. Aljoudi, Existence and uniqueness results for coupled system of fractional differential equations with exponential kernel derivatives, AIMS Mathematics, 8 (2023), 590–606. https://doi.org/10.3934/math.2023027 doi: 10.3934/math.2023027
[14]	K. Zhao, Generalized UH-stability of a nonlinear fractional coupling $({p}_1, {p}_2)$-Laplacian system concerned with nonsingular Atangana-Baleanu fractional calculus, J. Inequal. Appl., 2023 (2023), 96. https://doi.org/10.1186/s13660-023-03010-3 doi: 10.1186/s13660-023-03010-3
[15]	M. Alghanmi, R. P. Agarwal, B. Ahmad, Existence of solutions for a coupled system of nonlinear implicit differential equations involving $\varrho$-fractional derivative with anti periodic boundary conditions, Qual. Theory Dyn. Syst., 23 (2024), 6. https://doi.org/10.1007/s12346-023-00861-5 doi: 10.1007/s12346-023-00861-5
[16]	K. Zhao, Study on the stability and its simulation algorithm of a nonlinear impulsive ABC-fractional coupled system with a Laplacian operator via F-contractive mapping, Adv. Cont. Discr. Mod., 2024 (2024), 5. https://doi.org/10.1186/s13662-024-03801-y doi: 10.1186/s13662-024-03801-y
[17]	J. K. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkc. Ekvacioj, 21 (1978), 11–41.
[18]	K. Schumacher, Existence and continuous dependence for functional-differential equations with unbounded delay, Arch. Rational Mech. Anal., 67 (1978), 315–335. https://doi.org/10.1007/BF00247662 doi: 10.1007/BF00247662
[19]	F. Kappel, W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differ. Equ., 37 (1980), 141–183. https://doi.org/10.1016/0022-0396(80)90093-5 doi: 10.1016/0022-0396(80)90093-5
[20]	J. K. Hale, Functional differential equations with infinite delays, J. Math. Anal. Appl., 48 (1974), 276–283. https://doi.org/10.1016/0022-247X(74)90233-9 doi: 10.1016/0022-247X(74)90233-9
[21]	C. Corduneanu, V. Lakshmikantham, Equations with unbounded delay: A survey, Nonlinear Anal., 4 (1980), 831–877. https://doi.org/10.1016/0362-546X(80)90001-2 doi: 10.1016/0362-546X(80)90001-2
[22]	Y. Hino, S. Murakami, T. Naito, Functional differential equations with infinite delay, Berlin, Heidelberg: Springer, 1991. https://doi.org/10.1007/BFb0084432
[23]	V. Lakshmikantham, L. Wen, B. Zhang, Theory of differential equations with unbounded delay, New York: Springer, 1994. https://doi.org/10.1007/978-1-4615-2606-3
[24]	M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340–1350. https://doi.org/10.1016/j.jmaa.2007.06.021 doi: 10.1016/j.jmaa.2007.06.021
[25]	H. Bao, J. Cao, Existence of solutions for fractional stochastic impulsive neutral functional differential equations with infinite delay, Adv. Differ. Equ., 2017 (2017), 66. https://doi.org/10.1186/s13662-017-1106-5 doi: 10.1186/s13662-017-1106-5
[26]	M. Benchohra, S. Litimein, J. J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses, J. Fixed Point Theory Appl., 21 (2019), 21. https://doi.org/10.1007/s11784-019-0660-8 doi: 10.1007/s11784-019-0660-8
[27]	Y. Li, Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differ. Equ., 266 (2019), 3514–3558. https://doi.org/10.1016/j.jde.2018.09.009 doi: 10.1016/j.jde.2018.09.009
[28]	B. Ahmad, M. Alghanmi, A. Alsaedi, R. P. Agarwal, Nonlinear impulsive multiorder Caputo-type generalized fractional differential equations with infinite delay, Mathematics, 7 (2019), 1108. https://doi.org/10.3390/math7111108 doi: 10.3390/math7111108
[29]	K. Zhao, Y. Ma, Study on the existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equation with infinite delay, Fractal Fract., 5 (2021), 52. https://doi.org/10.3390/fractalfract5020052 doi: 10.3390/fractalfract5020052
[30]	C. Chen, Q. Dong, Existence and Hyers-Ulam stability for a multi-term fractional differential equation with infinite delay, Mathematics, 10 (2022), 1013. https://doi.org/10.3390/math10071013 doi: 10.3390/math10071013
[31]	M. Alghanmi, S. Alqurayqiri, Existence results for fractional neutral functional differential equations with infinite delay and nonlocal boundary conditions, Adv. Cont. Discr. Mod., 2023 (2023), 36. https://doi.org/10.1186/s13662-023-03782-4 doi: 10.1186/s13662-023-03782-4
[32]	X. Yang, Y. Feng, K. F. C. Yiu, Q. Song, , F. E. Alsaadi, Synchronization of coupled neural networks with infinite time distributed delays via quantized intermittent pinning control, Nonlinear Dyn., 94 (2018), 2289–2303. https://doi.org/10.1007/s11071-018-4449-x doi: 10.1007/s11071-018-4449-x
[33]	J. Liu, K. Zhao, Existence of mild solution for a class of coupled systems of neutral fractional integro-differential equations with infinite delay in Banach space, Adv. Differ. Equ. 2019 (2019), 284. https://doi.org/10.1186/s13662-019-2232-z doi: 10.1186/s13662-019-2232-z
[34]	F. Z. Mokkedem, Approximate controllability for a class of linear neutral evolution systems with infinite delay, J. Dyn. Control Syst., 28 (2022), 917–943. https://doi.org/10.1007/s10883-021-09560-3 doi: 10.1007/s10883-021-09560-3
[35]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam, Boston: Elsevier, 2006.
[36]	V. Pata, Fixed point theorems and applications, Cham: Springer, 2019. https://doi.org/10.1007/978-3-030-19670-7

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