Research article

Study of time fractional order problems with proportional delay and controllability term via fixed point approach

  • Received: 17 January 2021 Accepted: 08 March 2021 Published: 12 March 2021
  • MSC : 26A33, 34A08

  • In the current manuscript, we are tying to study one of the important class of differential equations known is evolution equations. Here, we considered the problem under controllability term and with proportional delay. Before going to numerical or analytical solution it is important to check the existence and uniqueness of the solution. So, we will consider our problem for qualitative theory using fixed point theorems of Banach's and Krasnoselskii's type. For numerical solution the stability is important, hence the problem is also studied for Ulam-Hyer's type stability. At the end an example is constructed to ensure the establish results.

    Citation: Muhammad Sher, Kamal Shah, Zareen A. Khan. Study of time fractional order problems with proportional delay and controllability term via fixed point approach[J]. AIMS Mathematics, 2021, 6(5): 5387-5396. doi: 10.3934/math.2021317

    Related Papers:

  • In the current manuscript, we are tying to study one of the important class of differential equations known is evolution equations. Here, we considered the problem under controllability term and with proportional delay. Before going to numerical or analytical solution it is important to check the existence and uniqueness of the solution. So, we will consider our problem for qualitative theory using fixed point theorems of Banach's and Krasnoselskii's type. For numerical solution the stability is important, hence the problem is also studied for Ulam-Hyer's type stability. At the end an example is constructed to ensure the establish results.



    加载中


    [1] F. Liu, K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl., 62 (2011), 822-833. doi: 10.1016/j.camwa.2011.03.002
    [2] R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000.
    [3] V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems, Cambridge: Cambridge Academic Publishers, 2009.
    [4] M. ur Rehman, R. A. Khan, Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett., 23 (2010), 1038-1044. doi: 10.1016/j.aml.2010.04.033
    [5] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland mathematics studies, New York: Elsevier Science Inc., 204 (2006).
    [6] Y. He, Q. G. Wang, C. Lin, An improved $H_{\alpha}$ filter design for systems with time-Varying interval delay, IEEE T. Circuits-II, 53 (2006), 1235-1239.
    [7] D. H. He, L. G. Xu, Exponential stability of impulsive fractional switched systems with time delays, IEEE T. Circuits-II, 2020, 1-1. DOI: 10.1109/TCSII.2020.3037654.
    [8] L. G. Xu, X. Y. Chu, H. X. Hu, Exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses, Appl. Math. Lett., 99 (2020), 106000. doi: 10.1016/j.aml.2019.106000
    [9] L. G. Xu, J. K. Li, S. S. Ge, Impulsive stabilization of fractional differential systems, ISA T., 70 (2017), 125-131. doi: 10.1016/j.isatra.2017.06.009
    [10] M. Sher, K. Shah, M. Fečkan, R. A. Khan, Qualitative analysis of multi-terms fractional order delay differential equations via the topological degree theory, Mathematics, 8 (2020), 218. doi: 10.3390/math8020218
    [11] M. Sher, K. Shah, J. Rassias, On qualitative theory of fractional order delay evolution equation via the prior estimate method, Math. Method. Appl. Sci., 43 (2020), 6464-6475. doi: 10.1002/mma.6390
    [12] S. M. Ullam, Problems in modern mathematics (Chapter VI), Science Editors, New York: Wiley, 1940.
    [13] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U. S. A., 27 (1941), 222-224. doi: 10.1073/pnas.27.4.222
    [14] T. M. Rassias, On the stability of the linear mapping in Banach spaces, P. Am. Math. Soc., 72 (1978), 297-300. doi: 10.1090/S0002-9939-1978-0507327-1
    [15] M. Sher, K. Shah, Y. M. Chu, R. A. Khan, Applicability of topological degree theory to evolution equation with proportional delay, Fractals, 28 (2020), 2040028. doi: 10.1142/S0218348X20400289
    [16] K. Shah, M. Sher, T. Abdeljawad, Study of evolution problem under Mittag-Leffler type fractional order derivative, Alex. Eng. J., 59 (2020), 3945-3951. doi: 10.1016/j.aej.2020.06.050
    [17] M. Sher, K. Shah, Z. A. Khan, H. Khan, A. Khan, Computational and theoretical modeling of the transmission dynamics of novel COVID-19 under Mittag-Leffler power law, Alex. Eng. J., 59 (2020), 3133-3147. doi: 10.1016/j.aej.2020.07.014
    [18] J. R. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. London Math. Phy. Sci., 322 (1971), 447-468.
    [19] D. F. Li, C. J. Zhang, Long time numerical behaviors of fractional pantograph equations, Math. Comput. Simul., 172 (2020), 244-257. doi: 10.1016/j.matcom.2019.12.004
    [20] D. F. Li, W. W. Sun, C. M, Wu, A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions, Numer. Math. Theor, Meth Appl., 14 (2021), 355-376. doi: 10.4208/nmtma.OA-2020-0129
    [21] L. G. Xu, H. X. Hu, Boundedness analysis of stochastic pantograph differential systems, Appl. Math. Lett., 111 (2021), 106630. doi: 10.1016/j.aml.2020.106630
    [22] V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
    [23] I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103-107.
    [24] K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear. Sci., 16 (2011), 1970-1977. doi: 10.1016/j.cnsns.2010.08.005
    [25] T. A. Burton, T. Furumochi, Krasnoselskiis fixed point theorem and stability, Nonlinear. Anal.-Theory., 49 (2002), 445-454. doi: 10.1016/S0362-546X(01)00111-0
    [26] D. Jekel, The Heine-Borel and Arzelá-Ascoli theorems, John Nachbar Washington University Press, 2016, 1-13.
  • Reader Comments
  • © 2021 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(1619) PDF downloads(145) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog