Nonlocal impulsive differential equations and inclusions involving Atangana-Baleanu fractional derivative in infinite dimensional spaces

Muneerah Al Nuwairan; Ahmed Gamal Ibrahim; Muneerah Al Nuwairan; Ahmed Gamal Ibrahim

doi:10.3934/math.2023595

AIMS Mathematics

2023, Volume 8, Issue 5: 11752-11780. doi: 10.3934/math.2023595

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Research article

Nonlocal impulsive differential equations and inclusions involving Atangana-Baleanu fractional derivative in infinite dimensional spaces

Muneerah Al Nuwairan ^{1
,
,},
Ahmed Gamal Ibrahim ²

1.
Department of Mathematics, College of Sciences, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mathematics, College of Sciences, Cairo University, Egypt

Received: 03 January 2023 Revised: 03 March 2023 Accepted: 09 March 2023 Published: 17 March 2023
MSC : 34A08, 26A33

The aim of this paper is to derive conditions under which the solution set of a non-local impulsive differential inclusions involving Atangana-Baleanu fractional derivative is a nonempty compact set in an infinite dimensional Banach spaces. Existence results for solutions in the presence of instantaneous or non-instantaneous impulsive effect are given. We considered the case where the right hand side is either a single valued function, or a multifunction. This generalizes recent results to the case when there are impulses, the right hand side is a multifunction, and where the dimension of the space is infinite. Examples are given to illustrate the effectiveness of the established results.
Citation: Muneerah Al Nuwairan, Ahmed Gamal Ibrahim. Nonlocal impulsive differential equations and inclusions involving Atangana-Baleanu fractional derivative in infinite dimensional spaces[J]. AIMS Mathematics, 2023, 8(5): 11752-11780. doi: 10.3934/math.2023595

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Abstract

The aim of this paper is to derive conditions under which the solution set of a non-local impulsive differential inclusions involving Atangana-Baleanu fractional derivative is a nonempty compact set in an infinite dimensional Banach spaces. Existence results for solutions in the presence of instantaneous or non-instantaneous impulsive effect are given. We considered the case where the right hand side is either a single valued function, or a multifunction. This generalizes recent results to the case when there are impulses, the right hand side is a multifunction, and where the dimension of the space is infinite. Examples are given to illustrate the effectiveness of the established results.

References

[1]	Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson, Y. L. Danon, Pulse mass measlesvaccination across age cohorts, Proc. Natl. Acad. Sci. U. S. A., 90 (1993), 11698–11702. http://doi.org/10.1073/pnas.90.24.11698 doi: 10.1073/pnas.90.24.11698
[2]	X. Z. Liu, G. Ballinger, Boundedness for impulsive delaydifferential equations and applications in populations growth models, Nonlinear Anal., 53 (2003), 1041–1062. http://doi.org/10.1016/S0362-546X(03)00041-5 doi: 10.1016/S0362-546X(03)00041-5
[3]	M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential equations and inclusions, Hindawi Publishing Corporation, 2007.
[4]	J. R. Wang, M. Fečkan, Non-instantaneous impulsive differential equations: basic theory and computation, IOP Publishing Ltd, 2018.
[5]	R. Agarwal, S. Hristova, D. O'Regan, Non-instantaneous impulses in differential equations, Springer, 2017.
[6]	A. G. Ibrahim, Differential equations and inclusions of fractional order with impulse effect in Banach spaces, Bull. Malays. Math. Sci. Soc., 43 (2020), 69–109. http://doi.org/10.1007/s40840-018-0665-2 doi: 10.1007/s40840-018-0665-2
[7]	J. Wang, A. G. Ibrahim, D. O'Regan, Nonempties and compactness of the solution set for fractional evolution inclusions with non-instantaneous impulses, Electron. J. Differ. Equations, 2019 (2019), 1–17.
[8]	J. R. Wang, A. G. Ibrahim, D. O'Regan, A. A. Almandouh, Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order $\alpha \in (1, 2)$, Int. J. Nonlinear Sci. Numer. Simul., 22 (2021), 593–603. http://doi.org/10.1515/ijnsns-2019-0179 doi: 10.1515/ijnsns-2019-0179
[9]	R. Agarwal, S. Hristova, D. O'Regan, Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions, J. Franklin Inst., 354 (2017), 3097–3119. http://doi.org/10.1016/j.jfranklin.2017.02.002 doi: 10.1016/j.jfranklin.2017.02.002
[10]	K. Liu, Stability analysis for $(w, c)$-periodic non-instantaneous impulsive differential equations, AIMS Math., 7 (2021), 1758–1774. http://doi.org/10.3934/math.2022101 doi: 10.3934/math.2022101
[11]	I. N. Kavallaris, T. Suzuki, Non-local partial differential equations for engineering and biology, Springer, 2018.
[12]	T. S. Hassan, R. G. Ahmed, A. M. A. El-Sayed, R. A. El-Nabulsi, O. Moaaz, M. B. Mesmouli, Solvability of a state-dependence functional integro-differential inclusion with delay nonlocal condition, Mathematics, 10 (2022), 2420. http://doi.org/10.3390/math10142420 doi: 10.3390/math10142420
[13]	X. P. Zhang, P. Y. Chen, A. Abdelmonem, Y. X. Li, Mild solutionsof stochastic partial differential equations with nonlocal conditions and non compact semigroups, J. Math. Slovaca, 69 (2019), 111–124. http://doi.org/10.1515/ms-2017-0207 doi: 10.1515/ms-2017-0207
[14]	B. F. Martínez-Salgado, R. Rosas-Sampayo, A. Torres-Hernandez, C. Fuentes, Application of fractional calculus to oil industry, Intech, 2017. http://doi.org/10.5772/intechopen.68571
[15]	H. Hardy, R. A. Beier, Fractals in reservoir engineering, World Scientific, 1994. http://doi.org/10.1142/2574
[16]	K. A. Lazopoulos, A. K. Lazopoulos, Fractional vector calculus and fluid mechanics, J. Mech. Behav. Mater., 26 (2017), 43–54. http://doi.org/10.1515/jmbm-2017-0012 doi: 10.1515/jmbm-2017-0012
[17]	G. U. Varieschi, Applications of fractional calculus to Newtonian mechanics, arXiv, 2018. https://doi.org/10.48550/arXiv.1712.03473
[18]	R. C. Velázquez, G. Fuentes-Cruz, M. Vásquez-Cruz, Decline-curve analysis of fractured reservoirs with fractal geometry, SPE Res. Eval. Eng., 11 (2008), 606–619. https://doi.org/10.2118/104009-PA doi: 10.2118/104009-PA
[19]	J. F. Douglas, Some applications of fractional calculus to polymer science, Adv. Chem. Phys., John Wiley Sons Inc., 2007. https://doi.org/10.1002/9780470141618.ch3
[20]	E. Reyes-Melo, J. Martinez-Vega, C. Guerrero-Salazar, U. Ortiz-Mendez, Modeling of relaxation phenomena in organic dielectric materials. Applications of differential and integral operators of fractional order, J. Optoelectron. Adv. Mater., 6 (2004), 1037–1043.
[21]	R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51 (1984), 299–307. https://doi.org/10.1115/1.3167616 doi: 10.1115/1.3167616
[22]	R. Herrmann, Fractional calculus: an introduction for physicists, World Scientific, 2011.
[23]	A. A. Kilbas, H. H. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Inc., 2006.
[24]	K. Diethelm, The analysis of fractional differential equations, Springer, 2010.
[25]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[26]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769.
[27]	K. A. Abro, A. Atangana, A comparative analysis of electromechanical model of piezoelectric actuator through Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, Math. Methods Appl. Sci., 43 (2020), 9681–9691. https://doi.org/10.1002/mma.6638 doi: 10.1002/mma.6638
[28]	B. Ghanbari, A. Atangana, A new application of fractional Atangana-Baleanu derivatives: designing ABC-fractional masks in image processing, Phys. A, 542 (2020), 123516. https://doi.org/10.1016/j.physa.2019.123516 doi: 10.1016/j.physa.2019.123516
[29]	M. A. Khan, A. Atangana, Modeling the dynamics of novelcoronavirus (2019-nCov) with fractional derivative, Alex. Eng. J., 59 (2020), 2379–2389. https://doi.org/10.1016/j.aej.2020.02.033 doi: 10.1016/j.aej.2020.02.033
[30]	T. Abdeljawad, D. Baleanu, Itegration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098–1107. https://doi.org/10.22436/jnsa.010.03.20 doi: 10.22436/jnsa.010.03.20
[31]	T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equations, 2016 (2016), 232. https://doi.org/10.1186/s13662-016-0949-5 doi: 10.1186/s13662-016-0949-5
[32]	M. S. Abdo, T. Abdeljawad, S. M. Ali, K. Shah, On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions, Adv. Differ. Equations, 2021 (2021), 37. https://doi.org/10.1186/s13662-020-03196-6 doi: 10.1186/s13662-020-03196-6
[33]	F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame ofAtangana-Baleanu fractional derivative, Chaos Solitons Fract., 117 (2018), 16–20. https://doi.org/10.1016/j.chaos.2018.10.006 doi: 10.1016/j.chaos.2018.10.006
[34]	T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 130. https://doi.org/10.1186/s13660-017-1400-5 doi: 10.1186/s13660-017-1400-5
[35]	Asma, S. Shabbir, K. Shah, T. Abdeljawad, Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative, Adv. Differ. Equations, 2021 (2021), 395. https://doi.org/10.1186/s13662-021-03551-1 doi: 10.1186/s13662-021-03551-1
[36]	A. Devi, A. Kumar, Existence and uniqueness results for integro fractional differential equations with Atangana-Baleanu fractional derivative, J. Math. Ext., 15 (2021), 1–24. https://doi.org/10.30495/JME.SI.2021.2128 doi: 10.30495/JME.SI.2021.2128
[37]	M. I. Syam, M. Al-Refai, Fraction differential equations with Atangana-Baleanu fractional derivative: analysis and applications, Chaos Solitions Fract., 2 (2019), 100013. https://doi.org/10.1016/j.csfx.2019.100013 doi: 10.1016/j.csfx.2019.100013
[38]	M. Hassouna, E. H. El Kinani, A. Ouhadan, Global existence and uniqueness of solution of Atangana-Baleanu Caputo fractional differential equation with nonlinear term and approximate solutions, Int. J. Differ. Equations, 2021 (2021), 5675789. https://doi.org/10.1155/2021/5675789 doi: 10.1155/2021/5675789
[39]	M. A. Almalahi, S. K. Panchal, M. S. Abdo, F. Jarad, On Atangana-Baleanu-type nonlocal boundary fractional differential equations, J. Funct. Spaces, 2022 (2022), 1812445. https://doi.org/10.1155/2022/1812445 doi: 10.1155/2022/1812445
[40]	S. T. Sutar, K. D. Kucche, Existence and data dependence results for fractional differential equations involving Atangana-Baleanu derivative, Rend. Circ. Mat. Palermo Ser. 2, 71 (2022), 647–663. https://doi.org/10.1007/s12215-021-00622-w doi: 10.1007/s12215-021-00622-w
[41]	R. Knapik, Impulsive differential equations with non-local conditions, Morehead Electron. J. Appl. Math., 2002.
[42]	K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630–637. https://doi.org/10.1006/jmaa.1993.1373 doi: 10.1006/jmaa.1993.1373
[43]	T. Cardinali, P. Rubbioni, Impulsive mild solution for semilinear differential inclusions with nonlocal conditions in Banach spaces, Nonlinear Anal., 75, (2012), 871–879. https://doi.org/10.1016/j.na.2011.09.023
[44]	D. Bothe, Multivalued perturbation of $m$-accerative differential inclusions, Isr. J. Math., 108 (1998), 109–138. https://doi.org/10.1007/BF02783044 doi: 10.1007/BF02783044
[45]	H. Ye, J. M. Gao, Y. S. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075–1081. https://doi.org/10.1016/j.jmaa.2006.05.061 doi: 10.1016/j.jmaa.2006.05.061
[46]	S. C. Hu, N. S. Papageorgiou, Handbook of multi-valued analysis, Springer, 1997.
[47]	M. Kamenskii, V. Obukhowskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, Walter de Gruyter, 2001. https://doi.org/10.1515/9783110870893
[48]	R. Almeida, S. Hristova, S. Dashkovskiy, Uniform bounded input bounded output stability of fractional-order delay nonlinear systems with input, Int. J. Robust Nonlinear Control, 31 (2021), 225–249. https://doi.org/10.1002/rnc.5273 doi: 10.1002/rnc.5273

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