Fractional infinite time-delay evolution equations with non-instantaneous impulsive

Ahmed Salem; Kholoud N. Alharbi; Ahmed Salem; Kholoud N. Alharbi

doi:10.3934/math.2023652

AIMS Mathematics

2023, Volume 8, Issue 6: 12943-12963. doi: 10.3934/math.2023652

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Fractional infinite time-delay evolution equations with non-instantaneous impulsive

Ahmed Salem ^{1
,
,},
Kholoud N. Alharbi ^1,2

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80203, Jeddah, 21589, Saudi Arabia
2.
Department of Mathematics, College of science and Arts in Uglat Asugour, Qassim university, Buraydah, Kingdom of Saudi Arabia

Received: 12 October 2022 Revised: 14 February 2023 Accepted: 17 March 2023 Published: 31 March 2023
MSC : 34A08, 34A12, 34A60, 34G99, 34K99

This dissertation is regarded to investigate the system of infinite time-delay and non-instantaneous impulsive to fractional evolution equations containing an infinitesimal generator operator. It turns out that its mild solution is existed and is unique. Our model is built using a fractional Caputo approach of order lies between 1 and 2. To get the mild solution, the families associated with cosine and sine which are linear strongly continuous bounded operators, are provided. It is common to use Krasnoselskii's theorem and the Banach contraction mapping principle to prove the existence and uniqueness of the mild solution. To confirm that our results are applicable, an illustrative example is introduced.
- Caputo fractional derivative,
- infinite time-delay,
- mild solution,
- non-instantaneous impulsive,
- Krasnoselskii's theorem
Citation: Ahmed Salem, Kholoud N. Alharbi. Fractional infinite time-delay evolution equations with non-instantaneous impulsive[J]. AIMS Mathematics, 2023, 8(6): 12943-12963. doi: 10.3934/math.2023652

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Abstract

This dissertation is regarded to investigate the system of infinite time-delay and non-instantaneous impulsive to fractional evolution equations containing an infinitesimal generator operator. It turns out that its mild solution is existed and is unique. Our model is built using a fractional Caputo approach of order lies between 1 and 2. To get the mild solution, the families associated with cosine and sine which are linear strongly continuous bounded operators, are provided. It is common to use Krasnoselskii's theorem and the Banach contraction mapping principle to prove the existence and uniqueness of the mild solution. To confirm that our results are applicable, an illustrative example is introduced.

References

[1]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Soliton. Fract., 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
[2]	B. Ghanabri, S. Djilali, Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population, Chaos Soliton. Fract., 138 (2020), 109960. https://doi.org/10.1016/j.chaos.2020.109960 doi: 10.1016/j.chaos.2020.109960
[3]	A. Salem, Existence results of solutions for ant-periodic fractional Langevin equation, J. Appl. Anal. Comput., 10 (2020), 2557–2574. https://doi.org/10.11948/20190419 doi: 10.11948/20190419
[4]	A. Salem, N. Mshary, On the existence and uniqueness of solution to fractional-order Langevin equation, Adv. Math. Phys., 2020 (2020), 8890575. https://doi.org/10.1155/2020/8890575 doi: 10.1155/2020/8890575
[5]	T. V. An, N. D. Phu, N. V. Hoa, A survey on non-instantaneous impulsive fuzzy differential equations involving the generalized Caputo fractional derivative in the short memory case, Fuzzy Set. Syst., 443 (2022), 160–197. https://doi.org/10.1016/j.fss.2021.10.008 doi: 10.1016/j.fss.2021.10.008
[6]	A. Salem, R. Babusail, Finite-time stability in nonhomogeneous delay differential equations of fractional Hilfer type, Mathematics, 10 (2022), 1520. https://doi.org/10.3390/math10091520 doi: 10.3390/math10091520
[7]	A. Salem, A. Al-dosari, Positive solvability for conjugate fractional differential inclusion of $(k, n-k)$ type without continuity and compactness, Axioms, 10 (2021), 170. https://doi.org/10.3390/axioms10030170 doi: 10.3390/axioms10030170
[8]	A. Salem, L. Almaghamsi, F. Alzahrani, An infinite system of fractional order with $p$-Laplacian operator in a tempered sequence space via measure of noncompactness technique, Fractal Fract., 5 (2021), 182. https://doi.org/10.3390/fractalfract5040182 doi: 10.3390/fractalfract5040182
[9]	R. Agarwal, R. Almeida, S. Hristova, D. O'Regan, Non-instantaneous impulsive fractional differential equations with state dependent delay and practical stability, Acta Math. Sci., 41 (2021), 1699–1718. https://doi.org/10.1007/s10473-021-0518-1 doi: 10.1007/s10473-021-0518-1
[10]	A. Salem, S. Abdullah, Non-instantaneous impulsive BVPs involving generalized Liouville-Caputo derivative, Mathematics, 10 (2022), 291. https://doi.org/10.3390/math10030291 doi: 10.3390/math10030291
[11]	S. Asawasamrit, Y. Thadang, S. K. Ntouyas, J. Tariboon, Non-instantaneous impulsive boundary value problems containing Caputo fractional derivative of a function with respect to another function and Riemann-Stieltjes fractional integral boundary conditions, Axioms, 10 (2021), 130. https://doi.org/10.3390/axioms10030130 doi: 10.3390/axioms10030130
[12]	A. Salem, A. Al-Dosari, Hybrid differential inclusion involving two multi-valued operators with nonlocal multi-valued integral condition, Fractal Fract., 6 (2022), 109. https://doi.org/10.3390/fractalfract6020109 doi: 10.3390/fractalfract6020109
[13]	S. Zhaoa, M. Song, Stochastic impulsive fractional differential evolution equations with infinite delay, Filomat, 31 (2017), 4261–4274. https://doi.org/10.2298/FIL1713261Z doi: 10.2298/FIL1713261Z
[14]	A. Salem, B. Alghamdi, Multi-strip and multi-point boundary conditions for fractional Langevin equation, Fractal Fract., 4 (2020), 18. https://doi.org/10.3390/fractalfract4020018 doi: 10.3390/fractalfract4020018
[15]	A. Salem, N. Mshary, Coupled fixed point theorem for the generalized Langevin equation with four-point and Strip conditions, Adv. Math. Phys., 2022 (2022), 1724221. https://doi.org/10.1155/2022/1724221 doi: 10.1155/2022/1724221
[16]	R. Saadati, E. Pourhadi, B. Samet, On the $\mathcal PC$-mild solutions of abstract fractional evolution equations with non-instantaneous impulses via the measure of noncompactness, Bound. Value Probl., 2019 (2019), 19. https://doi.org/10.1186/s13661-019-1137-9 doi: 10.1186/s13661-019-1137-9
[17]	A. El. Mfadel, S. Melliani, A. Kassidi, M. Elomari, Existence of mild solutions for nonlocal $\Psi$-Caputo-type fractional evolution equations with nondense domain, Nonautonomous Dyn. Syst., 9 (2022), 272–289. https://doi.org/10.1515/msds-2022-0157 doi: 10.1515/msds-2022-0157
[18]	A. Salem, L. Almaghamsi, Existence solution for coupled system of Langevin fractional differential equations of caputo type with Riemann-Stieltjes integral boundary conditions, Symmetry, 13 (2021), 2123. https://doi.org/10.3390/sym13112123 doi: 10.3390/sym13112123
[19]	A. Kumar, D. N. Pandey, Existence of mild solution of Atangana-Baleanu fractional differential equations with non-instantaneous impulses and with non-local conditions, Chaos Soliton. Fract., 132 (2020), 109551. https://doi.org/10.1016/j.chaos.2019.109551 doi: 10.1016/j.chaos.2019.109551
[20]	V. Kavitha, M. M. Arjunan, D. Baleanu, Non-instantaneous impulsive fractional-order delay differential systems with Mittag-Leffler kernel, AIMS Math., 7 (2022) 9353–9372. https://doi.org/10.3934/math.2022519 doi: 10.3934/math.2022519
[21]	M. Sova, Cosine operator functions, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1966.
[22]	I. Podlubny, Fractional differential equations, 1999.
[23]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006. https://doi.org/10.1016/s0304-0208(06)x8001-5
[24]	C. C. Travis, G. F. Webb, Cosine families and abstractnonlinear second order differential equations, Acta Math. Acad. Sci. Hung., 32 (1978), 75–96. https://doi.org/10.1007/BF01902205 doi: 10.1007/BF01902205
[25]	X. Zhang, X. Huanga, Z. Liu, The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Anal. Hybri., 4 (2010), 775–781. https://doi.org/10.1016/j.nahs.2010.05.007 doi: 10.1016/j.nahs.2010.05.007
[26]	M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahaba, 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
[27]	J. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 1978.
[28]	A. Salem, K. N. Alharbi, H. M. Alshehri, Fractional evolution equations with infinite time delay in abstract phase space, Mathematics, 10 (2022), 1332. https://doi.org/10.3390/math10081332 doi: 10.3390/math10081332

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