A survey on impulsive fractional differential equations

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.

   Article  MathSciNet  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, 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.

   Article  MathSciNet  Google Scholar 

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.

   Article  MathSciNet  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.

   Article  MathSciNet  Google Scholar 

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

   Article  MathSciNet  Google Scholar 

6. K. Balachandran, S. Kiruthika, Existence of solutions of abstract fractional impulsive semilinear evolution equations. Electron. J. Qual. Theory Differ. Equ. 2010, 4 (2010), 1–12.

   Article  MathSciNet  Google Scholar 

7. M. Benchohra, D. Seba, Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2009, 8 (2009), 1–14.

   Article  MathSciNet  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, 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.

   Article  MathSciNet  Google Scholar 

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.

   Article  MathSciNet  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).

    MATH  Google Scholar 

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

    Article  MathSciNet  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, 6 (2015), 1470–1491; 10.1515/fca-2015-0085; http://www.degruyter.com/view/j/fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml.

    Article  MathSciNet  Google Scholar 

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.

    Article  MathSciNet  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.

    Article  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.

    Article  MathSciNet  Google Scholar 

16. Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8 (2011), 345–362.

    Article  MathSciNet  Google Scholar 

17. J.R Wang, M. Fečkan, Y. Zhou, Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions. Submitted

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

    MathSciNet  MATH  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.

    Article  MathSciNet  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.

    Article  MathSciNet  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.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references