Search Advanced

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 7020-7041. doi: 10.3934/mbe.2023303
Previous Article Next Article
Theory article Special Issues

Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse

  • 1.
    Department of System Science and Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
  • 2.
    Key Laboratory of Applied Statistics and Data Analysis of Department of Education of Yunnan Province, Kunming, Yunnan 650500, China
  • 3.
    School of Mathematical and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

  • Academic Editor: Feliz Minhós
  • Received: 27 November 2022 Revised: 15 January 2023 Accepted: 20 January 2023 Published: 09 February 2023

  • This paper considers the stability of a fractional differential equation with multi-point boundary conditions and non-instantaneous integral impulse. Some sufficient conditions for the existence, uniqueness and at least one solution of the aforementioned equation are studied by using the Diaz-Margolis fixed point theorem. Secondly, the Ulam stability of the equation is also discussed. Lastly, we give one example to support our main results. It is worth pointing out that these two non-instantaneous integral impulse and multi-point boundary conditions factors are simultaneously considered in the fractional differential equations studied for the first time.

    Citation: Guodong Li, Ying Zhang, Yajuan Guan, Wenjie Li. Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 7020-7041. doi: 10.3934/mbe.2023303

    Related Papers:

  • This paper considers the stability of a fractional differential equation with multi-point boundary conditions and non-instantaneous integral impulse. Some sufficient conditions for the existence, uniqueness and at least one solution of the aforementioned equation are studied by using the Diaz-Margolis fixed point theorem. Secondly, the Ulam stability of the equation is also discussed. Lastly, we give one example to support our main results. It is worth pointing out that these two non-instantaneous integral impulse and multi-point boundary conditions factors are simultaneously considered in the fractional differential equations studied for the first time.


    加载中


    [1] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1998.
    [2] J. Klafter, S. Lim, R. Metzler, Fractional Dynamics in Physics, World Scientific, Sinapore, 2011.
    [3] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific, Singapore, 3 (2012). https://doi.org/10.1142/10044
    [4] F. Mainardi, P. Pironi, The fractional Langevin equation: Brownian motion revisited, Extr. Math., 10 (1996), 140–154. https://doi.org/10.48550/arXiv.0806.1010 doi: 10.48550/arXiv.0806.1010
    [5] K. M. Saad, D. Baleanu, A. Atangana, New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger's equations, Comput. Appl. Math., 37 (2018), 5203–5216. https://doi.org/10.1007/s40314-018-0627-1 doi: 10.1007/s40314-018-0627-1
    [6] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-14003-7
    [7] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Nort-Holand Mathematics Studies, Amsterdam, 204 (2006), 1–540.
    [8] K. Zhao, Multiple positive solutions of integral BVPs for high–order nonlinear fractional differential equations with impulses and distributed delays, Dyn. Syst., 30 (2015), 208–223. https://doi.org/10.1080/14689367.2014.995595 doi: 10.1080/14689367.2014.995595
    [9] K. Zhao, Impulsive integral boundary value problems of the higher–order fractional differential equation with eigenvalue arguments, Adv. Differ. Equations, 2015 (2015), 1–16. https://doi.org/10.1186/s13662-015-0725-y doi: 10.1186/s13662-015-0725-y
    [10] Y. Tian, Z. Bai, Impulsive boundary value problem for differential equations with fractional order, Differ. Equations Dyn. Syst., 21 (2013), 253–260. https://doi.org/10.1007/s12591-012-0150-6 doi: 10.1007/s12591-012-0150-6
    [11] J. Wang, F. Michal, Y. Zhou, Presentation of solutions of impulsive fractional Langevin equations and existence results, Eur. Phys. J. Spec. Top., 222 (2013), 1857–1874. https://doi.org/10.1140/epjst/e2013-01969-9 doi: 10.1140/epjst/e2013-01969-9
    [12] J. Wang, Z. Yong, L. Zeng, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649–657. https://doi.org/10.1016/j.amc.2014.06.002 doi: 10.1016/j.amc.2014.06.002
    [13] S. Ulam, A Collection of Mathematical Problems, New York: Interscience Publishers, 1960.
    [14] D. H. Hyers, On the stability of the linear functional equation, PNAS, 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
    [15] T. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
    [16] H. Khan, J. F. Gómez-Aguilar, A. Khan, T. S. Khan, Stability analysis for fractional order advection–reaction diffusion system, Physica A, 521 (2019), 737–751. https://doi.org/10.1016/j.physa.2019.01.102 doi: 10.1016/j.physa.2019.01.102
    [17] A. Khan, J. F. Gómez-Aguilar, T. S. Khan, H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos, Solitons Fractals, 122 (2019), 119–128. https://doi.org/10.1016/j.chaos.2019.03.022 doi: 10.1016/j.chaos.2019.03.022
    [18] R. Rizwan, A. Zada, X. Wang, Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses, Adv. Differ. Equations, 2019 (2019), 1–31. https://doi.org/10.1186/s13662-019-1955-1 doi: 10.1186/s13662-019-1955-1
    [19] I. Rus, Ulam stability of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107. Available form: http://www.jstor.org/stable/43999438.
    [20] J. Wang, A. Zada, W. Ali, Ulam's-type stability of first–order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 553–560. https://doi.org/10.1515/ijnsns-2017-0245 doi: 10.1515/ijnsns-2017-0245
    [21] J. Wang, K. Shah, A. Ali, Existence and Hyers–Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., 41 (2018), 2392–2402. https://doi.org/10.1002/mma.4748 doi: 10.1002/mma.4748
    [22] K. Zhao, P. Gong, Positive solutions of m-point multi–term fractional integral BVP involving time–delay for fractional differential equations, Boundary Value Probl., 2015 (2015), 1–19. https://doi.org/10.1186/s13661-014-0280-6 doi: 10.1186/s13661-014-0280-6
    [23] A. Zada, S. Ali, Y. Li, Ulam–type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equations, 2017 (2017), 1–26. https://doi.org/10.1186/s13662-017-1376-y doi: 10.1186/s13662-017-1376-y
    [24] A. Zada, S. Ali, Stability analysis of multi–point boundary value problem for sequential fractional differential equations with non-instantaneous impulses, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 763–774. https://doi.org/10.1515/ijnsns-2018-0040 doi: 10.1515/ijnsns-2018-0040
    [25] J. D. Stein, On generalized complete metric spaces, Bull. Amer. Math. Soc., 75 (1969), 113–116. https://doi.org/10.1090/S0002-9904-1969-12210-X doi: 10.1090/S0002-9904-1969-12210-X
    [26] J. Diaz, B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Am. Math. Soc., 74 (1968), 305–309. https://doi.org/10.1090/S0002-9904-1968-11933-0 doi: 10.1090/S0002-9904-1968-11933-0
    [27] W. Li, J. Ji, L. Huang, Global dynamics analysis of a water hyacinth fish ecological system under impulsive control, J. Franklin Inst., 359 (2022), 10628–10652. https://doi.org/10.1016/j.jfranklin.2022.09.030 doi: 10.1016/j.jfranklin.2022.09.030
    [28] W. Li, J. Ji, L. Huang, L. Zhang, Global dynamics and control of malicious signal transmission in wireless sensor networks, Nonlinear Anal. Hybrid Syst., 48 (2023), 101324. https://doi.org/10.1016/j.nahs.2022.101324 doi: 10.1016/j.nahs.2022.101324
    [29] Z. Cai, L. Huang, Generalized Lyapunov approach for functional differential inclusions, Automatica, 113 (2020), 108740. https://doi.org/10.1016/j.automatica.2019.108740 doi: 10.1016/j.automatica.2019.108740
    [30] W. Li, J. Ji, L. Hunag, Y. Zhang, Complex dynamics and impulsive control of a chemostat model under the ratio threshold policy, Chaos, Solitons Fractals, 167 (2023), 113077. https://doi.org/10.1016/j.chaos.2022.113077 doi: 10.1016/j.chaos.2022.113077
    [31] Q. Zhu, H. Wang, Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function, Automatica, 87 (2018), 166–175. https://doi.org/10.1016/j.automatica.2017.10.004 doi: 10.1016/j.automatica.2017.10.004
    [32] B. Wang, Q. Zhu, Stability analysis of discrete time semi-markov jump linear systems, IEEE Trans. Autom. Control, 65 (2020), 5415–5421. https://doi.org/10.1109/TAC.2020.2977939 doi: 10.1109/TAC.2020.2977939
    [33] H. Wang, Q. Zhu, Global stabilization of a class of stochastic nonlinear time-delay systems with SISS inverse dynamics, IEEE Trans. Autom. Control, 65 (2020), 4448–4455. https://doi.org/10.1109/TAC.2020.3005149 doi: 10.1109/TAC.2020.3005149
    [34] K. Ding, Q. Zhu, Extended dissipative anti-disturbance control for delayed switched singular semi-Markovian jump systems with multi-disturbance via disturbance observer, Automatica, 18 (2021), 109556. https://doi.org/10.1016/j.automatica.2021.109556 doi: 10.1016/j.automatica.2021.109556
    [35] 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
    [36] G. Wang, B. Ahmad, L. Zhang, Impulsive anti–periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal. Theory Methods Appl., 74 (2011), 792–804. https://doi.org/10.1016/j.na.2010.09.030 doi: 10.1016/j.na.2010.09.030
    [37] D. Luo, M. Tian, Q. Zhu, Some results on finite-time stability of stochastic fractional–order delay differential equations, Chaos, Solitons Fractals, 158 (2022), 111996. https://doi.org/10.1016/j.chaos.2022.111996 doi: 10.1016/j.chaos.2022.111996
    [38] X. Wang, D. Luo, Q. Zhu, Ulam–Hyers stability of caputo type fuzzy fractional differential equations with time-delays, Chaos, Solitons Fractals, 156 (2022), 111822. https://doi.org/10.1016/j.chaos.2022.111822 doi: 10.1016/j.chaos.2022.111822
    [39] 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. https://doi.org/10.1016/j.aml.2021.107549 doi: 10.1016/j.aml.2021.107549
    [40] I. Rus, Ulam stability of ordinary differential equations, Studia Universitatis Babes Bolyai Mathematica, 54 (2009), 125–133. Available form: https://www.cs.ubbcluj.ro/studia-m/2009-4/rus-final.pdf.
    [41] S. O. Shah, A. Zada, Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales, Appl. Math. Comput., 359 (2019), 202–213. https://doi.org/10.1016/j.amc.2019.04.044 doi: 10.1016/j.amc.2019.04.044
    [42] F. Haq, K. Shah, G. ur Rahman, M. Shahzad, Hyers–Ulam stability to a class of fractional differential equations with boundary conditions, Int. J. Appl. Comput. Math., 3 (2017), 1135–1147. https://doi.org/10.1007/s40819-017-0406-5 doi: 10.1007/s40819-017-0406-5
    [43] W. Li, Y. Zhang, L. Huang, Dynamics analysis of a predator–prey model with nonmonotonic functional response and impulsive control, Math. Comput. Simul., 204 (2023), 529–555. https://doi.org/10.1016/j.matcom.2022.09.002 doi: 10.1016/j.matcom.2022.09.002
    [44] W. Li, J. Ji, L. Huang, Z. Guo, Global dynamics of a controlled discontinuous diffusive SIR epidemic system, Appl. Math. Lett., 121 (2021), 107420. https://doi.org/10.1016/j.aml.2021.107420 doi: 10.1016/j.aml.2021.107420
  • 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
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(1675) PDF downloads(131) Cited by(51)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog