Research article Special Issues

Discussion on iterative process of nonlocal controllability exploration for Hilfer neutral impulsive fractional integro-differential equation

  • Received: 12 February 2023 Revised: 14 April 2023 Accepted: 27 April 2023 Published: 15 May 2023
  • MSC : 34A08, 34A12, 34A37, 45J05, 93B05

  • This manuscript primarily focuses on the nonlocal controllability results of Hilfer neutral impulsive fractional integro-differential equations of order $ 0\leq w\leq1 $ and $ 0 < g < 1 $ in a Banach space. The outcomes are derived from the strongly continuous operator, Wright function, linear operator, and bounded operator. First, we explore the existence and uniqueness of the results of the mild solution of Hilfer's neutral impulsive fractional integro-differential equations using Schauder's fixed point theorem and an iterative process. In order to determine nonlocal controllability, the Banach fixed point technique is used. We employed some specific numerical computations and applications to examine the effectiveness of the results.

    Citation: Kanagaraj Muthuselvan, Baskar Sundaravadivoo, Kottakkaran Sooppy Nisar, Suliman Alsaeed. Discussion on iterative process of nonlocal controllability exploration for Hilfer neutral impulsive fractional integro-differential equation[J]. AIMS Mathematics, 2023, 8(7): 16846-16863. doi: 10.3934/math.2023861

    Related Papers:

  • This manuscript primarily focuses on the nonlocal controllability results of Hilfer neutral impulsive fractional integro-differential equations of order $ 0\leq w\leq1 $ and $ 0 < g < 1 $ in a Banach space. The outcomes are derived from the strongly continuous operator, Wright function, linear operator, and bounded operator. First, we explore the existence and uniqueness of the results of the mild solution of Hilfer's neutral impulsive fractional integro-differential equations using Schauder's fixed point theorem and an iterative process. In order to determine nonlocal controllability, the Banach fixed point technique is used. We employed some specific numerical computations and applications to examine the effectiveness of the results.



    加载中


    [1] A. Akg$\ddot{u}$l, S. H. A. Khoshnaw, Application of fractional derivative on non-linear biochemical reaction models, Int. J. Intell. Netw., 1 (2020), 52–58. https://doi.org/10.1016/j.ijin.2020.05.001 doi: 10.1016/j.ijin.2020.05.001
    [2] R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201–210. https://doi.org/10.1122/1.549724 doi: 10.1122/1.549724
    [3] K. Diethelm, The analysis of fractional differential equations, In: Lecture Notes in Mathematics, 2010. https://doi.org/10.1007/978-3-642-14574-2
    [4] C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, J. H. T. Bates, The role of fractional calculus in modelling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141–159. https://doi.org/10.1016/j.cnsns.2017.04.001 doi: 10.1016/j.cnsns.2017.04.001
    [5] R. L. Magin, Fractional Calculus in Bioengineering, Chicago: University of Illinois-Chicago, 2006.
    [6] I. Podlubny, Fractional Differential Equation, Academic Press, 1998.
    [7] Y. Cao, Y. Kao, J. H. Park, H. Bao, Global Mittag–Leffler stability of the delayed fractional-coupled reaction-diffusion system on networks without strong connectedness, IEEE Trans. Neur. Net. Lear. Syst., 33 (2021), 6473–6483. https://doi.org/10.1109/TNNLS.2021.3080830 doi: 10.1109/TNNLS.2021.3080830
    [8] Y. Kao, Y. Li, J. H. Park, X. Chen, Mittag–Leffler synchronization of delayed fractional memristor neural networks via adaptive control, IEEE Trans. Neur. Net. Lear. Syst., 32 (2021), 2279–2284. https://doi.org/10.1109/TNNLS.2020.2995718 doi: 10.1109/TNNLS.2020.2995718
    [9] Y. Kao, Y. Cao, X. Chen, Global Mittag-Leffler synchronization of coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks via sliding mode control, Chaos, 32 (2022), 113123. https://doi.org/10.1063/5.0102787 doi: 10.1063/5.0102787
    [10] G. Li, Y. Zhang, Y. Guan, W. Li, Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse, Math. Biosci. Eng., 20 (2023), 7020–7041. https://doi.org/10.3934/mbe.2023303 doi: 10.3934/mbe.2023303
    [11] R. Rao, Z. Lin, X. Ai, J. Wu, Synchronization of epidemic systems with Neumann boundary value under delayed impulse, Mathematics, 10 (2022), 2064. https://doi.org/10.3390/math10122064 doi: 10.3390/math10122064
    [12] Y. Zhao, L. Wang, Practical exponential stability of impulsive stochastic food chain system with time-varying delays, Mathematics, 11 (2023), 147. https://doi.org/10.3390/math11010147 doi: 10.3390/math11010147
    [13] R. Agarwal, S. Hristova, D. O'Regan, Non-Instantaneous impulses in Caputo fractional differential equations, Fract. Calc. Appl. Anal., 20 (2017), 595–622. https://doi.org/10.1515/fca-2017-0032 doi: 10.1515/fca-2017-0032
    [14] H. M. Ahmed, M. M. El-Borai, H. M. El-Owaidy, A. S. Ghanem, Impulsive Hilfer fractional differential equations, Adv. Differ. Equ., 2018 (2018), 226. https://doi.org/10.1186/s13662-018-1679-7 doi: 10.1186/s13662-018-1679-7
    [15] D. D. Bainov, P. S. Simeonov, Oscillation Theory of Impulsive Differential Equations, Orlando: International Publications, 1998.
    [16] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, Singapore: World Scientific Publishing, 1989.
    [17] 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
    [18] K. Muthuselvan, B. S. Vadivoo, Analyze existence, uniqueness and controllability of impulsive fractional functional differential equations, Adv. Stud.: Euro-Tbil. Math. J., 10 (2022), 171–190.
    [19] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, Singapore: World Scientific Publishing Co. Pte. Ltd., 14 (1995). https://doi.org/10.1142/2892
    [20] B. S. Vadivoo, R. Raja, J. Cao, H. Zhang, X. Li, Controllability analysis of nonlinear neutral-type fractional-order differential systems with state delay and impulsive effects, Int. J. Control Autom. Syst., 16 (2018), 659–669. http://doi.org/10.1007/s12555-017-0281-1 doi: 10.1007/s12555-017-0281-1
    [21] X. Fu, X. Liu, Controllability of non-densely defined on neutral functional differential systems in abstract space, Chin. Ann.Math. Ser. B, 28 (2007), 243–252. http://doi.org/10.1007/s11401-005-0028-9 doi: 10.1007/s11401-005-0028-9
    [22] K. Jothimani, K. Kaliraj, S. Kumari Panda, K. S. Nisar, C. Ravichandran, Results on controllability of non-densely characterized neutral fractional delay differential system, Evol. Equ. Control The., 10 (2021), 619–631. https://doi.org/10.3934/eect.2020083 doi: 10.3934/eect.2020083
    [23] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 204 (2006).
    [24] P. Bedi, A. Kumar, T. Abdeljawad, A. Khan, Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations, Adv. Differ. Equ., 2020 (2020), 155. https://doi.org/10.1186/s13662-020-02615-y doi: 10.1186/s13662-020-02615-y
    [25] J. Du, W. Jiang, D. Pang, A. U. Niazi, Exact controllability for Hilfer fractional differential inclusion involving nonlocal initial conditions, Complexity, 2018 (2018), 9472847. https://doi.org/10.1155/2018/9472847 doi: 10.1155/2018/9472847
    [26] X. Liu, Y. Li, G. Xu, On the finite approximate controllability for Hilfer fractional evolution systems, Adv. Differ. Equ., 2020 (2020), 22. https://doi.org/10.1186/s13662-019-2478-5 doi: 10.1186/s13662-019-2478-5
    [27] D. Luo, Q. Zhu, Z. Luo, A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients, Appl. Math. Lett., 122 (2021), 107549. http://dx.doi.org/10.1016/j.aml.2021.107549 doi: 10.1016/j.aml.2021.107549
    [28] K. S. Nisar, K. Jothimani, C. Ravichandran, D. Baleanu, D. Kumar, New approach on controllability of Hilfer fractional derivatives with nondense domain, AIMS Mathematics, 7 (2022), 10079–10095. https://doi.org/10.3934/math.2022561 doi: 10.3934/math.2022561
    [29] Y. Zhou, J. W. He, A Cauchy problem for fractional evolution equations with Hilfer's fractional derivative on semi-infinite interval, Fract. Calc. Appl. Anal., 25 (2022), 924–961. https://doi.org/10.1007/s13540-022-00057-9 doi: 10.1007/s13540-022-00057-9
    [30] M. Zhou, C. Li, Y. Zhou, Existence of mild solutions for Hilfer fractional evolution equations with almost sectorial operators, Axioms, 11 (2022), 144. https://doi.org/10.3390/axioms11040144 doi: 10.3390/axioms11040144
    [31] 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
    [32] H. Gu, J. J. Trujillo, Existence of mild 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
    [33] A. Kumar, D. N. Pandey, Controllability results for non-densely defined impulsive fractional differential equations in abstract space, Differ. Equ. Dyn. Syst., 29 (2021), 227–237. https://doi.org/10.1007/s12591-019-00471-1 doi: 10.1007/s12591-019-00471-1
    [34] C. Ravichandran, N. Valliammal, J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin Inst., 356 (2019), 1535–1565. https://doi.org/10.1016/j.jfranklin.2018.12.001 doi: 10.1016/j.jfranklin.2018.12.001
    [35] Y. Zhou, J. R. Wang, L. Zhang, Basic Theory of Fractional Differential Equation, Singapore: World Scientific Publishing, 2016.
    [36] R. Hilfer, Applications of Fractional Calculus in Physics, Singapore: World Scientific Publishing, 2000.
    [37] Y. Cao, J. Sun, Controllability of measure driven evolution systems with nonlocal conditions, Appl. Math. Comput., 299 (2017), 119–126. https://doi.org/10.1016/j.amc.2016.11.037 doi: 10.1016/j.amc.2016.11.037
    [38] K. Balachandran, J. P. Dauer, Elements of Control Theory, Narosa Publishing House, 1999.
    [39] 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
    [40] Y. K. Ma, K. Kavitha, W. Albalawi, A. Shukla, K. S. Nisar, V. Vijayakumar, An analysis on the approximate controllability of Hilfer fractional neutral differential systems in Hilbert spaces, Alex. Eng. J., 61 (2022), 7291–7302. https://doi.org/10.1016/j.aej.2021.12.067 doi: 10.1016/j.aej.2021.12.067
    [41] V. Vijayakumar, R. Udhayakumar, A new exploration on existence of Sobolev-type Hilfer fractional neutral integro-differential equations with infinite delay, Numer. Meth. Part. Differ. Equ., 37 (2021), 750–766. https://doi.org/10.1002/num.22550 doi: 10.1002/num.22550
    [42] 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
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(986) PDF downloads(52) Cited by(3)

Article outline

Figures and Tables

Figures(4)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog