Existence and controllability of Hilfer fractional neutral differential equations with time delay via sequence method

Krishnan Kavitha; Velusamy Vijayakumar; Kottakkaran Sooppy Nisar; Anurag Shukla; Wedad Albalawi; Abdel-Haleem Abdel-Aty; Krishnan Kavitha; Velusamy Vijayakumar; Kottakkaran Sooppy Nisar; Anurag Shukla; Wedad Albalawi; Abdel-Haleem Abdel-Aty

doi:10.3934/math.2022706

AIMS Mathematics

2022, Volume 7, Issue 7: 12760-12780. doi: 10.3934/math.2022706

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Research article

Existence and controllability of Hilfer fractional neutral differential equations with time delay via sequence method

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632 014, Tamil Nadu, India
2.
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia
3.
Department of Applied Science, Rajkiya Engineering College Kannauj, Kannauj-209732, India
4.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
5.
Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia
6.
Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

Received: 09 March 2022 Revised: 20 April 2022 Accepted: 22 April 2022 Published: 05 May 2022
MSC : 26A33, 34A08, 47D09, 47H10, 93C43

This paper deals with the existence and approximate controllability outcomes for Hilfer fractional neutral evolution equations. To begin, we explore existence outcomes using fractional computations and Banach contraction fixed point theorem. In addition, we illustrate that a neutral system with a time delay exists. Further, we prove the considered fractional time-delay system is approximately controllable using the sequence approach. Finally, an illustration of our main findings is offered.
- hilfer fractional derivative,
- approximate controllability,
- neutral system,
- time delay,
- mild solutions,
- sequence approach,
- nonlocal conditions
Citation: Krishnan Kavitha, Velusamy Vijayakumar, Kottakkaran Sooppy Nisar, Anurag Shukla, Wedad Albalawi, Abdel-Haleem Abdel-Aty. Existence and controllability of Hilfer fractional neutral differential equations with time delay via sequence method[J]. AIMS Mathematics, 2022, 7(7): 12760-12780. doi: 10.3934/math.2022706

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Abstract

This paper deals with the existence and approximate controllability outcomes for Hilfer fractional neutral evolution equations. To begin, we explore existence outcomes using fractional computations and Banach contraction fixed point theorem. In addition, we illustrate that a neutral system with a time delay exists. Further, we prove the considered fractional time-delay system is approximately controllable using the sequence approach. Finally, an illustration of our main findings is offered.

References

[1]	D. D. Dai, T. T. Ban, Y. L. Wang, W. Zhang, The piecewise reproducing kernel method for the time variable fractional order advection-reaction-diffusion equations, Therm. Sci., 25 (2021), 1261–1268. https://doi.org/10.2298/TSCI200302021D doi: 10.2298/TSCI200302021D
[2]	A. Debbouche, V. Antonov, Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Hilbert spaces, Chaos Soliton Fract., 102 (2017), 140–148. https://doi.org/10.1016/j.chaos.2017.03.023 doi: 10.1016/j.chaos.2017.03.023
[3]	K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[4]	C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K. S. Nisar, A. Shukla, A note on the approximate controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order $1 < r < 2$, Math. Comput. Simulat., 190 (2021), 1003–1026. https://doi.org/10.1016/j.matcom.2021.06.026 doi: 10.1016/j.matcom.2021.06.026
[5]	C. Dineshkumar, K. S. Nisar, R. Udhayakumar, V. Vijayakumar, New discussion about the approximate controllability of fractional stochastic differential inclusions with order $1 < r < 2$, Asian J. Control, 2021. https://doi.org/10.1002/asjc.2663
[6]	C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, A. Shukla, K. S. Nisar, A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order with delay, Chaos Soliton Fract., 153 (2021), 111565. https://doi.org/10.1016/j.chaos.2021.111565 doi: 10.1016/j.chaos.2021.111565
[7]	K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616–1626. https://doi.org/10.1016/j.camwa.2012.01.009 doi: 10.1016/j.camwa.2012.01.009
[8]	F. D. Ge, H. C. Zhou, C. H. Kou, Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique, Appl. Math. Comput., 275 (2016), 107–120. https://doi.org/10.1016/j.amc.2015.11.056 doi: 10.1016/j.amc.2015.11.056
[9]	H. Gu, J. J. Trujillo, Existence of integral solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344–354. https://doi.org/10.1016/j.amc.2014.10.083 doi: 10.1016/j.amc.2014.10.083
[10]	C. H. He, C. Liu, J. H. He, H. M. Sedighi, A. Shokri, K. A. Gepreel, A fractal model for the internal temperature response of a porous concrete, Appl. Math. Comput., 22 (2021), 71–77.
[11]	Y. T. Zuo, C. Liu, H. J. He, Fractal approach to mechanical and electrical properties of graphene/sic composites, Facta Univ. Ser.: Mech. Eng., 19 (2021), 271–284. https://doi.org/10.22190/FUME201212003Z doi: 10.22190/FUME201212003Z
[12]	J. H. He, G. M. Moatimid, M. H. Zekry, Forced nonlinear oscillator in a fractal space, Facta Univ. Ser.: Mech. Eng., 20 (2022), 1–20. https://doi.org/10.22190/FUME220118004H doi: 10.22190/FUME220118004H
[13]	J. H. He, C. Liu, A modified frequency-amplitude formulation for fractal vibration systems, Fractals, 2022. https://doi.org/10.1142/S0218348X22500463
[14]	J. H. He, F. Y. Ji, Two-scale mathematics and fractional calculus for thermodynamics, Therm. sci., 23 (2019), 2131–2133. https://doi.org/10.2298/TSCI1904131H doi: 10.2298/TSCI1904131H
[15]	S. Ji, Approximate controllability of semilinear nonlocal fractional differential systems via an approximating method, Appl. Math. Comput., 236 (2014), 43–53. https://doi.org/10.1016/j.amc.2014.03.027 doi: 10.1016/j.amc.2014.03.027
[16]	J. W. He, Y. Liang, B. Ahmad, Y. Zhou, Nonlocal fractional evolution inclusions of order $\alpha \in (1, 2)$, Mathematics, 2019 (2019), 209. https://doi.org/10.3390/math7020209 doi: 10.3390/math7020209
[17]	R. Hilfer, Application of fractional calculus in physics, Singapore: World Scientific, 2000. https://doi.org/10.1142/3779
[18]	A. Haq, N. Sukavanam, Existence and approximate controllability of Riemann-Liouville fractional integrodifferential systems with damping, Chaos Soliton Fract., 139 (2020), 110043. https://doi.org/10.1016/j.chaos.2020.110043 doi: 10.1016/j.chaos.2020.110043
[19]	A Haq, Partial-approximate controllability of semi-linear systems involving two Riemann-Liouville fractional derivatives, Chaos Soliton Fract., 157 (2022), 111923. https://doi.org/10.1016/j.chaos.2022.111923 doi: 10.1016/j.chaos.2022.111923
[20]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, N. Sakthivel, K. S. Nisar, A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay, Math. Methods Appl. Sci., 44 (2021), 4428–4447. https://doi.org/10.1002/mma.7040 doi: 10.1002/mma.7040
[21]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control, 2021. https://doi.org/10.1002/asjc.2549
[22]	K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar, R. Udhayakumar, Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type, Chaos Soliton Fract., 151 (2021), 111264. https://doi.org/10.1016/j.chaos.2021.111264 doi: 10.1016/j.chaos.2021.111264
[23]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[24]	V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
[25]	J. Liang, H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions, Appl. Math. Comput., 254 (2015), 20–29. https://doi.org/10.1016/j.amc.2014.12.145 doi: 10.1016/j.amc.2014.12.145
[26]	X. Li, Z. Liu, C. C. Tisdell, Approximate controllability of fractional control systems with time delay using the sequence method, Electron. J. Differ. Equ., 272 (2017), 1–11.
[27]	N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal., 68 (2008), 536–46. https://doi.org/10.1016/j.na.2006.11.018 doi: 10.1016/j.na.2006.11.018
[28]	M. Mohan Raja, V. Vijayakumar, R. Udhayakumar, Results on the existence and controllability of fractional integro-differential system of order $1 < r < 2$ via measure of noncompactness, Chaos Soliton Fract., 139 (2020), 110299. https://doi.org/10.1016/j.chaos.2020.110299 doi: 10.1016/j.chaos.2020.110299
[29]	K. S. Nisar, V. Vijayakumar, Results concerning to approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral delay differential system, Math. Methods Appl. Sci., 44 (2021), 13615–13632. https://doi.org/ 10.1002/mma.7647 doi: 10.1002/mma.7647
[30]	K. S. Nisar, K. Jothimani, K. Kaliraj, C, Ravichandran An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos Soliton Fract., 146 (2021), 110915. https://doi.org/10.1016/j.chaos.2021.110915 doi: 10.1016/j.chaos.2021.110915
[31]	R. Patel, A. Shukla, S. S. Jadon, Existence and optimal control problem for semilinear fractional order $(1, 2]$ control system, Math. Methods Appl. Sci., 2020. https://doi.org/10.1002/mma.6662
[32]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, New York Springer, 1983. https://doi.org/10.1007/978-1-4612-5561-1
[33]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to method of their solution and some of their applications, San Diego: Academic Press, 1999.
[34]	R. Sakthivel, R. Ganesh, S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput., 225 (2013), 708–717. https://doi.org/10.1016/j.amc.2013.09.068 doi: 10.1016/j.amc.2013.09.068
[35]	A. Shukla, N. Sukavanam, D. N. Pandey, Controllability of semilinear stochastic control system with finite delay, IMA J. Math. Control Inf., 35 (2018), 427–449. https://doi.org/10.1093/imamci/dnw059 doi: 10.1093/imamci/dnw059
[36]	A. Shukla, N. Sukavanam, D. N. Pandey Complete controllability of semi-linear stochastic system with delay, Rend. Circ. Mat. Palermo, 64 (2015), 209–220. https://doi.org/10.1007/s12215-015-0191-0 doi: 10.1007/s12215-015-0191-0
[37]	A. Shukla, N. Sukavanam, D. N. Pandey, Approximate controllability of semilinear fractional stochastic control system, Asian-Eur. J. Math., 11 (2018), 1850088. https://doi.org/10.1142/S1793557118500882 doi: 10.1142/S1793557118500882
[38]	A. Shukla, N. Sukavanam, D. N. Pandey, Controllability of semilinear stochastic system with multiple delays in control, IFAC Proc. Vol., 47 (2014), 306–312. https://doi.org/10.3182/20140313-3-IN-3024.00107 doi: 10.3182/20140313-3-IN-3024.00107
[39]	A. Shukla, R. Patel, Existence and optimal control results for second-order semilinear system in Hilbert spaces, Circuits Syst. Signal Proccess., 40 (2021), 4246–4258. https://doi.org/10.1007/s00034-021-01680-2 doi: 10.1007/s00034-021-01680-2
[40]	A. Shukla, R. Patel, Controllability results for fractional semilinear delay control systems, J. Appl. Math. Comput., 65 (2021), 861–875. https://doi.org/10.1007/s12190-020-01418-4 doi: 10.1007/s12190-020-01418-4
[41]	A. Shukla, N. Sukavanam, D. N. Pandey, Approximate controllability of semilinear fractional control systems of order $\alpha\in (1, 2)$, 2015 Proceedings of the Conference on Control and its Applications, 2015,175–180. https://doi.org/10.1137/1.9781611974072.25
[42]	V. Vijayakumar, C. Ravichandran, K. S. Nisar, K. D. Kucche, New discussion on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order $1 < r < 2$, Numer. Methods Partial Differ. Equ., 2021. https://doi.org/10.1002/num.22772
[43]	R. Subashini, K. Jothimani, K. S. Nisar, C. Ravichandran, New results on nonlocal functional integro-differential equations via Hilfer fractional derivative, Alex. Eng. J., 5 (2020), 2891–2899. https://doi.org/10.1016/j.aej.2020.01.055 doi: 10.1016/j.aej.2020.01.055
[44]	V. Vijayakumar, S. K. Panda, K. S. Nisar, H. M. Baskonus, Results on approximate controllability results for second-order Sobolev-type impulsive neutral differential evolution inclusions with infinite delay, Numer. Methods Partial Differ. Equ., 37 (2021), 1200–1221. https://doi.org/10.1002/num.22573 doi: 10.1002/num.22573
[45]	V. Vijayakumar, R. Udhayakumar, C. Dineshkumar, Approximate controllability of second order nonlocal neutral differential evolution inclusions, IMA J. Math. Control Inf., 38 (2021), 192–210. https://doi.org/10.1093/imamci/dnaa001 doi: 10.1093/imamci/dnaa001
[46]	V. Vijayakumar, R. Murugesu, Controllability for a class of second order evolution differential inclusions without compactness, Appl. Anal., 98 (2019), 1367–1385. https://doi.org/10.1080/00036811.2017.1422727 doi: 10.1080/00036811.2017.1422727
[47]	V. Vijayakumar, R. Murugesu, M. Tamil Selvan, Controllability for a class of second order functional evolution differential equations without uniqueness, IMA J. Math. Control Inf., 36 (2019), 225–246. https://doi.org/10.1093/imamci/dnx048 doi: 10.1093/imamci/dnx048
[48]	V. Vijayakumar, R. Udhayakumar, S. K. Panda, K. S. Nisar, Results on approximate controllability of Sobolev type fractional stochastic evolution hemivariational inequalities, Numer. Methods Partial Differ. Equ., 2020. https://doi.org/10.1002/num.22690
[49]	K. L. Wang, S. W. Yao, He's fractional derivative for the evolution equation, Therm. Sci., 24 (2020), 2507–2513. https://doi.org/10.2298/TSCI2004507W doi: 10.2298/TSCI2004507W
[50]	C. Han, Y. L. Wang, Z. Y. Li, Numerical solutions of space fractional variable-coefficient kdv-modified kdv equation by fourier spectral method, Fractals, 29 (2021), 2150246. https://doi.org/10.1142/S0218348X21502467 doi: 10.1142/S0218348X21502467
[51]	J. R. Wang, Y. R. Zhang, Nonlocal initial value problems for differential equation with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850–859. https://doi.org/10.1016/j.amc.2015.05.144 doi: 10.1016/j.amc.2015.05.144
[52]	W. K. Williams, V. Vijayakumar, R. Udhayakumar, K. S. Nisar, A new study on existence and uniqueness of nonlocal fractional delay differential systems of order $1 < r < 2$ in Hilbert spaces, Numer. Methods Partial Differ. Equ., 37 (2021), 949–961. https://doi.org/10.1002/num.22560 doi: 10.1002/num.22560
[53]	F. Xianlong, L. Xingbo, Controllability of non-densely defined neutral functional differential systems in abstract space, Chin. Ann. Math. Ser. B, 28 (2007), 243–252. https://doi.org/10.1002/num.22560 doi: 10.1002/num.22560
[54]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014. https://doi.org/10.1142/10238
[55]	Y. Zhou, Fractional evolution equations and inclusions, Analysis and Control, Elsevier, 2015. https://doi.org/10.1016/C2015-0-00813-9
[56]	Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., 11 (2010), 4465–4475. https://doi.org/10.1016/j.nonrwa.2010.05.029 doi: 10.1016/j.nonrwa.2010.05.029
[57]	Y. Zhou, L. Zhang, X. H. Shen, Existence of mild solutions for fractional evolution equations, J. Integral Equ. Appl., 25 (2013), 557–585. https://doi.org/10.1216/JIE-2013-25-4-557 doi: 10.1216/JIE-2013-25-4-557
[58]	S. Zahoor, S. Naseem, Design and implementation of an efficient FIR digital filter, Cogent Eng., 4 (2017), 1323373. https://doi.org/10.1080/23311916.2017.1323373 doi: 10.1080/23311916.2017.1323373

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