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Fractional Calculus and Applied Analysis

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A survey on impulsive fractional differential equations

JinRong Wang / Michal Fečkan
  • Department of Mathematical Analysis and Numerical Mathematics Comenius University in Bratislava Mlynská dolina, 842 48 Bratislava, SLOVAKIA and Mathematical Institute of Slovak Academy of Sciences Šanikova 49, 814 73 Bratislava, Slovakia
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/ Yong Zhou
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/fca-2016-0044

Abstract

Recently, in series of papers we have proposed different concepts of solutions of impulsive fractional differential equations (IFDE). This paper is a survey of our main results about IFDE. We present several types of such equations with various boundary value conditions as well. Concept of solutions, existence results and examples are presented. Proofs are only sketched.

MSC 2010: Primary 26A33; Secondary 33E12; 34A08; 34K37; 35R11

Key Words and Phrases: fractional calculus; Mittag-Leffler type functions; fractional ordinary and partial differential equations; nonlocal impulsive fractional switched systems; noninstantaneous impulsive fractional differential equations

References

  • [1]

    R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta. Appl. Math. 109 (2010), 973–1033.Google Scholar

  • [2]

    R.P. Agarwal, S. Hristova, D. O'Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19, No 2 (2016), 290–318; DOI: 10.1515/fca-2016-0017; http://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml http://www.degruyter.com/.Web of ScienceCrossref

  • [3]

    B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3 (2009), 251–258.Google Scholar

  • [4]

    B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 4 (2010), 134–141.Google Scholar

  • [5]

    C. Atkinson, A. Osseiran, Rational solutions for the time-fractional diffusion equation. SIAM J. Appl. Math. 71 (2011), 92–106.Google Scholar

  • [6]

    K. Balachandran, S. Kiruthika, Existence of solutions of abstract fractional impulsive semilinear evolution equations. Electron. J. Qual. Theory Differ. Equ. 2010, No 4 (2010), 1–12.Google Scholar

  • [7]

    M. Benchohra, D. Seba, Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2009, No 8 (2009), 1–14.Google Scholar

  • [8]

    G. Bonanno, R. Rodríguez-López, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17, No 3 (2014), 717–744; DOI: 10.2478/s13540-014-0196-y; http://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml http://www.degruyter.com/.Web of ScienceCrossref

  • [9]

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

  • [10]

    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006).Google Scholar

  • [11]

    G.M. Mophou, Existence and uniqueness of mild solution to impulsive fractional differetial equations. Nonlinear Anal. 72 (2010), 1604–1615.Google Scholar

  • [12]

    N. Nyamoradi, Multiplicity of nontrivial solutions for boundary value problem for impulsive fractional differential inclusions via nonsmooth critical point theory. Fract. Calc. Appl. Anal. 18, No 6 (2015), 1470–1491; % DOI: 10.1515/fca-2015-0085; % http://www.degruyter.com/view/j/fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml http://www.degruyter.com/.CrossrefWeb of Science

  • [13]

    J.R. Wang, M. Fečkan, Y. Zhou, On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3050–3060.Google Scholar

  • [14]

    J.R. Wang, M. Fečkan, Y. Zhou, Presentation of solutions of impulsive fractional Langevin equations and existence results. Impulsive fractional Langevin equations. Eur. Phys. J. Spec. Top. 222 (2013), 1857–1874.Google Scholar

  • [15]

    J.R. Wang, Y. Zhou, M. Fečkan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl. 64 (2012), 3008–3020.Google Scholar

  • [16]

    J.R. Wang, M. Fečkan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8 (2011), 345–362.Google Scholar

  • [17]

    J.R. Wang, M. Fečkan, Y. Zhou, Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions. Submitted. %%% Update ????????Google Scholar

  • [18]

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

  • [19]

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

  • [20]

    W. Wei, X. Xiang, Y. Peng, Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization 55 (2006), 141–156.Google Scholar

  • [21]

    H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328 (2007), 1075–1081.Google Scholar

  • [22]

    Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59 (2010), 1063–1077.Google Scholar

About the article

Received: 2016-04-02

Published Online: 2016-08-31

Published in Print: 2016-08-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 4, Pages 806–831, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0044.

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