Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi


IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

Online
ISSN
2191-0294
See all formats and pricing
More options …
Volume 19, Issue 7-8

Issues

Stability Analysis of Multi-point Boundary Value Problem for Sequential Fractional Differential Equations with Non-instantaneous Impulses

Akbar Zada / Sartaj Ali
Published Online: 2018-10-26 | DOI: https://doi.org/10.1515/ijnsns-2018-0040

Abstract

This paper deals with a new class of non-linear impulsive sequential fractional differential equations with multi-point boundary conditions using Caputo fractional derivative, where impulses are non instantaneous. We develop some sufficient conditions for existence, uniqueness and different types of Ulam stability, namely Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability and generalized Hyers–Ulam–Rassias stability for the given problem. The required conditions are obtained using fixed point approach. The validity of our main results is shown with the aid of few examples.

Keywords: sequential fractional differential equation; Caputo fractional derivative; fractional integral; non-instantaneous impulses; Ulam’s type stability; fixed point theorem

MSC 2010: 26A33; 34A08; 34B27

References

  • [1]

    S. Abbas, M. Benchohra and G. M. N’Guerekata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.Google Scholar

  • [2]

    S. Abbas, M. Benchohra and G. M. N’Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2014.Google Scholar

  • [3]

    A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006.Google Scholar

  • [4]

    V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.Google Scholar

  • [5]

    R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.Google Scholar

  • [6]

    M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, Springer, Dordrecht, 2011.Google Scholar

  • [7]

    M. H. Aqlan, A. Alsaedi, B. Ahmad and J. J. Nieto, Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions, Open Math. 14 (2016), 723–735.Google Scholar

  • [8]

    A. Alsaedi, B. Ahmad and M. H. Aqlan, Sequential fractional differential equations and unification of anti-periodic and multi-point boundary conditions, J. Nonlinear Sci. Appl. 10 (2017), 71–83.Google Scholar

  • [9]

    B. Ahmad and J. J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 64 (2012), 3046–3052.Google Scholar

  • [10]

    D. H. Hyers, G. Isac and Th. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser Boston, 1998.Google Scholar

  • [11]

    I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar

  • [12]

    M. S. Alhothuali, A. Alsaedi, B. Ahmad and M. H. Aqlan, Nonlinear sequential fractional differential equations with boundary conditions involving lower case fractional derivatives, Adv. Differ. Equ. 2017 (2017), 16.Google Scholar

  • [13]

    K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electron. J. Qual. Theory Differ. Equ. 2010 (2010), 1–12.Google Scholar

  • [14]

    M. Benchohra and D. Seba, Impulsive fractional differential equations in Banach spaces, Electron. J. Qual. Theory Differ. Equ. 2009 (2009), 1–14.Google Scholar

  • [15]

    R. W. Ibrahim, Stability of sequential fractional differential equation, Appl. Comput. Math. 14 (2015), 9.Google Scholar

  • [16]

    J. Jiang and L. Liu, Existence of solutions for a sequential fractional system with coupled boundary conditions, Adv. Differ. Equ. 2016 (2016), 15.Google Scholar

  • [17]

    N. Kosmatov, Initial value problems of fractional order with fractional impulsive conditions, Results Math. 63 (2013), 1289–1310.Google Scholar

  • [18]

    M. J. Mardanov, N. I. Mahmudov and Y. A. Sharifov, Existence and uniqueness theorems for impulsive fractional differential equations with two-point and integral boundary conditions, Sci. World J. 2014 (2014), 8.Google Scholar

  • [19]

    B. Sambandham and A. S. Vatsala, Basic results for sequential Caputo fractional differential equations, Mathematics 3 (2015), 76–91.Google Scholar

  • [20]

    J. R. Wang, Y. Zhou and Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput. 242 (2014), 649–657.Web of ScienceGoogle Scholar

  • [21]

    G. Wang, L. Zhang and G. Song, Systems of first order impulsive fractional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear Anal.: TMA 74 (2011), 974–982.Google Scholar

  • [22]

    A. Zada, S. Ali and Y. Li, Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equ. 317 (2017), 1–26.Google Scholar

  • [23]

    A. Bitsadze and A. Samarskii, On some simple generalizations of linear elliptic boundary problems, Soviet Math. Dokl. 10 (1969), 398–400.Google Scholar

  • [24]

    S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968.Google Scholar

  • [25]

    D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U. S. A. 27 (1941), 222–224.CrossrefGoogle Scholar

  • [26]

    T. Li and A. Zada, Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Differ. Equ. 153 (2016), 2070–2075.Google Scholar

  • [27]

    T. Li, A. Zada and S. Faisal, Hyers–Ulam stability of $n$th order linear differential equations, J. Nonlinear Sci. Appl. 9 (2016), 2070–2075.Google Scholar

  • [28]

    Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300.CrossrefGoogle Scholar

  • [29]

    S. Tang, A. Zada, S. Faisal, M. M. A. El-Sheikh and T. Li, Stability of higher-order nonlinear impulsive differential equations, J. Nonlinear, Sci. Appl. 9 (2016), 4713–4721.Google Scholar

  • [30]

    A. Zada, W. Ali and S. Farina, Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Method Appl. Sci. 40 (2017), 5502–5514.Google Scholar

  • [31]

    A. Zada, S. Faisal and Y. Li, On the Hyers–Ulam stability of first-order impulsive delay differential equations, J. Funct. Spaces 2016 (2016), 6.Google Scholar

  • [32]

    A. Zada, S. Faisal and Y. Li, Hyers–Ulam–Rassias stability of non-linear delay differential equations, J. Nonlinear Sci. Appl. 10 (2017), 504–510.Google Scholar

  • [33]

    A. Zada, S. O. Shah and R. Shah, Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problem, Appl. Math. Comput. 271 (2015), 512–518.Google Scholar

  • [34]

    K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.Google Scholar

  • [35]

    C. Bai, Impulsive periodic boundary value problems for fractional differential equation involving Riemann–Liouville sequential fractional derivative, J. Math. Anal. Appl. 384 (2011), 211–231.Google Scholar

  • [36]

    D. Baleanu, O.G. Mustafa and R.P. Agarwal, On Lp-solutions for a class of sequential fractional differential equations, Appl. Math. Comput. 218 (2011), 2074–2081.Google Scholar

  • [37]

    M. Klimek, Sequential fractional differential equations with Hadamard derivative, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 4689–4697.Google Scholar

  • [38]

    Z. Wei and W. Dong, Periodic boundary value problems for Riemann-Liouville sequential fractional differential equations, Electron. J. Qual. Theory Differ. Equ. 87 (2011), 1–13.Google Scholar

  • [39]

    Z. Wei, Q. Li and J. Che, Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative, J. Math. Anal. Appl. 367 (2010), 260–272.Google Scholar

  • [40]

    I. A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babeş Bolyai Math. 54 (2009), 125–133.Google Scholar

  • [41]

    I. A. Rus, Ulam stability of ordinary differential equations in a Banach spaces, Carpathian J. Math. 26 (2010), 103–107.Google Scholar

  • [42]

    J. R. Wang, M. Fečkan and Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (2012), 258–264.Google Scholar

  • [43]

    C. J. Jung, On generalized complete metric spaces, September, 1968.Google Scholar

  • [44]

    J. B. Diaz and B. Margolis, A fixed point theorem of alternative, for contractions on a generalized complete metric space, Bull. Am. Math. Soc. 74 (1968), 305–309.Google Scholar

About the article

Received: 2018-02-17

Accepted: 2018-10-05

Published Online: 2018-10-26

Published in Print: 2018-12-19


Competing interest: The authors declare that they have no competing interest regarding this research work.


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 7-8, Pages 763–774, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2018-0040.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in