Non-instantaneous impulsive fractional-order delay differential systems with Mittag-Leffler kernel

Velusamy Kavitha; Mani Mallika Arjunan; Dumitru Baleanu; Velusamy Kavitha; Mani Mallika Arjunan; Dumitru Baleanu

doi:10.3934/math.2022519

AIMS Mathematics

2022, Volume 7, Issue 5: 9353-9372. doi: 10.3934/math.2022519

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Research article

Non-instantaneous impulsive fractional-order delay differential systems with Mittag-Leffler kernel

1.
Department of Mathematics, School of Sciences, Arts, Media & Management, Karunya Institute of Technology and Sciences, Karunya Nagar, Coimbatore-641114, Tamil Nadu, India
2.
Department of Mathematics, School of Arts, Science and Humanities, SASTRA Deemed to be University, Thanjavur-613401, Tamil Nadu, India
3.
Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, 06530 Ankara, Turkey
4.
Institute of Space Sciences, Magurele-Bucharest, Romania
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 25 August 2021 Revised: 22 February 2022 Accepted: 02 March 2022 Published: 11 March 2022
MSC : 26A33, 34A08, 34K37

The existence of fractional-order functional differential equations with non-instantaneous impulses within the Mittag-Leffler kernel is examined in this manuscript. Non-instantaneous impulses are involved in such equations and the solution semigroup is not compact in Banach spaces. We suppose that the nonlinear term fulfills a non-compactness measure criterion and a local growth constraint. We further assume that non-instantaneous impulsive functions satisfy specific Lipschitz criteria. Finally, an example is given to justify the theoretical results.
- Atangana-Baleanu fractional derivative,
- non-instantaneous impulses,
- Mittag-Leffler kernel,
- fixed point theorem
Citation: Velusamy Kavitha, Mani Mallika Arjunan, Dumitru Baleanu. Non-instantaneous impulsive fractional-order delay differential systems with Mittag-Leffler kernel[J]. AIMS Mathematics, 2022, 7(5): 9353-9372. doi: 10.3934/math.2022519

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Abstract

The existence of fractional-order functional differential equations with non-instantaneous impulses within the Mittag-Leffler kernel is examined in this manuscript. Non-instantaneous impulses are involved in such equations and the solution semigroup is not compact in Banach spaces. We suppose that the nonlinear term fulfills a non-compactness measure criterion and a local growth constraint. We further assume that non-instantaneous impulsive functions satisfy specific Lipschitz criteria. Finally, an example is given to justify the theoretical results.

References

[1]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[2]	A. Anguraj, M. Mallika Arjunan, E. Hernandez, Existence results for an impulsive neutral functional differential equation with state-dependent delay, Appl. Anal., 86 (2007), 861–872. https://doi.org/10.1080/00036810701354995 doi: 10.1080/00036810701354995
[3]	D. Aimene, D. Baleanu, D. Seba, Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delay, Chaos Soliton. Fract., 128 (2019), 51–57. https://doi.org/10.1016/j.chaos.2019.07.027 doi: 10.1016/j.chaos.2019.07.027
[4]	N. Al-Salti, E. Karimov, K. Sadarangani, On a differential equation with Caputo- Fabrizio fractional derivative of order $1 < \beta > 2$ and application to mass-spring-damper system, Progr. Fract. Differ. Appl., 2 (2016), 257–263. https://dx.doi.org/10.18576/pfda/020403 doi: 10.18576/pfda/020403
[5]	S. Abbas, M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput., 257 (2015), 190–198. https://doi.org/10.1016/j.amc.2014.06.073 doi: 10.1016/j.amc.2014.06.073
[6]	E. Bas, R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 116 (2018), 121–125. https://doi.org/10.1016/j.chaos.2018.09.019 doi: 10.1016/j.chaos.2018.09.019
[7]	G. Bahaa, A. Hamiaz, Optimality conditions for fractional differential inclusions with nonsingular Mittag-Leffler kernel, Adv. Differ. Equ., 2018 (2018), 257. https://doi.org/10.1186/s13662-018-1706-8 doi: 10.1186/s13662-018-1706-8
[8]	J. Banas, K. Goebel, Measures of noncompactness in Banach spaces, In: Lecture notes in pure and applied mathematics, New York: Marcel Dekker, 1980.
[9]	M. Benchohra, S. Litimein, Juan J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses, J. Fix. Point Theory A., 21 (2019), 21. https://doi.org/10.1007/s11784-019-0660-8 doi: 10.1007/s11784-019-0660-8
[10]	M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential equations and inclusions, New York: Hindawi Publishing Corporation, 2006.
[11]	N. Bastos, Calculus of variations involving Caputo-Fabrizio fractional differentiation, Stat. Optim. Inform. Comput., 6 (2018), 12–21. https://doi.org/10.19139/soic.v6i1.466 doi: 10.19139/soic.v6i1.466
[12]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. https://dx.doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[13]	P. Chen, X. Zhang, Y. Li, Existence of mild solutions to partial differential equations with non-instantaneous impulses, Electron. J. Differ. Equ., 2016 (2016), 1–11.
[14]	P. Chen, Y. Li, H. Yang, Perturbation method for nonlocal impulsive evolution equations, Nonlinear Anal. Hybri., 8 (2013), 22–30. https://doi.org/10.1016/j.nahs.2012.08.002 doi: 10.1016/j.nahs.2012.08.002
[15]	Y. K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Anal. Hybr., 2 (2008), 209–218. https://doi.org/10.1016/j.nahs.2007.10.001 doi: 10.1016/j.nahs.2007.10.001
[16]	K. Deimling, Nonlinear functional analysis, New York: Springer-Verlag, 1985.
[17]	Z. Fan, G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709–1727. https://doi.org/10.1016/j.jfa.2009.10.023 doi: 10.1016/j.jfa.2009.10.023
[18]	D. Guo, Existence of positive solutions for $n$th-order nonlinear impulsive singular integro-differential equations in Banach spaces, Nonlinear Anal. Theor., 68 (2008), 2727–2740. https://doi.org/10.1016/j.na.2007.02.019 doi: 10.1016/j.na.2007.02.019
[19]	G. R. Gautam, J. Dabas, Mild solutions for a class of neutral fractional functional differential equations with not instantaneous impulses, Appl. Math. Comput., 259 (2015), 480–489. https://doi.org/10.1016/j.amc.2015.02.069 doi: 10.1016/j.amc.2015.02.069
[20]	E. Hernández, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641–1649. https://doi.org/10.1090/S0002-9939-2012-11613-2 doi: 10.1090/S0002-9939-2012-11613-2
[21]	J. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11–41.
[22]	J. K. Hale, Theory of functional differential equations, New York: Springer-Verlag, 1977.
[23]	M. Haase, The functional calculus for sectorial operators, Springer Science & Business Media, 2006.
[24]	Y. Hino, S. Murakami, T. Naito, Functional differential equations with unbounded delay, Springer, 1991.
[25]	S. Liu, J. Wang, D. O'Regan, Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness, Topol. Method. Nonl. An., 58 (2021), 19–49. https://doi.org/10.12775/TMNA.2020.069 doi: 10.12775/TMNA.2020.069
[26]	V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Singapore: World Scientific, 1989.
[27]	A. Kumar, Dwijendra 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
[28]	M. Mallika Arjunan, T. Abdeljawad, V. Kavitha, A. Yousef, On a new class of Atangana-Baleanu fractional Volterra-Fredholm integro-differential inclusions with non-instantaneous impulses, Chaos Soliton. Fract., 148 (2021), 111075. https://doi.org/10.1016/j.chaos.2021.111075 doi: 10.1016/j.chaos.2021.111075
[29]	M. Mallika Arjunan, A. Hamiaz, V. Kavitha, Existence results for Atangana-Baleanu fractional neutral integro-differential systems with infinite delay through sectorial operators, Chaos Soliton. Fract., 149 (2021), 111042. https://doi.org/10.1016/j.chaos.2021.111042 doi: 10.1016/j.chaos.2021.111042
[30]	M. Mallika Arjunan, V. Kavitha, Existence results for Atangana-Baleanu fractional integro-differential systems with non-instantaneous impulses, Nonlinear Stud., 28 (2021), 865–877.
[31]	H. Monch, 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
[32]	I. Podlubny, Fractional differential equations, San Diego California: Academic Press, 1999.
[33]	W. Qiu, J. Wang, Iterative learning control for multi-agent systems with non-instantaneous impulsive consensus tracking, Int. J. Robust Nonlin., 31 (2021), 6507–6524. https://doi.org/10.1002/rnc.5627 doi: 10.1002/rnc.5627
[34]	S. Suganya, D. Baleanu, P. Kalamani, M. Mallika Arjunan, On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses, Adv. Differ. Equ., 2015 (2015), 372. https://doi.org/10.1186/s13662-015-0709-y doi: 10.1186/s13662-015-0709-y
[35]	X. B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal. Theor., 74 (2011), 2003–2011. https://doi.org/10.1016/j.na.2010.11.007 doi: 10.1016/j.na.2010.11.007
[36]	H. Zhang, Y. Miaolin, Y. Renyu, Jinde Cao, Synchronization stability of Riemann-Liouville fractional delay-coupled complex neural networks, Physica. A, 508 (2018), 155–165. https://doi.org/10.1016/j.physa.2018.05.060 doi: 10.1016/j.physa.2018.05.060
[37]	H. Zhang, Y. Renyu, Jinde Cao, A. Alsaedi, Delay-independent stability of Riemann-Liouville fractional neutral-type delayed neural networks, Neural Process Lett., 47 (2018), 427–442. https://doi.org/10.1007/s11063-017-9658-7 doi: 10.1007/s11063-017-9658-7
[38]	W. Zhang, H. Zhang, Jinde Cao, Fuad E. Alsaadi, D. Chen, Synchronization in uncertain fractional-order memristive complex-valued neural networks with multiple time delays, Neural Netw., 110 (2019), 186–198. https://doi.org/10.1016/j.neunet.2018.12.004 doi: 10.1016/j.neunet.2018.12.004

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