Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


Uniqueness and Stability Results for Non-local Impulsive Implicit Hadamard Fractional Differential Equations

Journal of Applied Nonlinear Dynamics 9(1) (2020) 23--29 | DOI:10.5890/JAND.2020.03.002

P. Karthikeyan, R. Arul

Department of Mathematics, Sri Vasavi college, Erode, Tamilnadu, India-638316

Download Full Text PDF

 

Abstract

We analyze the uniqueness and Ulam stability results for implicit impulsive fractional differential equations connecting nonlocal form of the Hadamard derivative of fractional order ϑ. The main results are studied by using the Banach contraction principle and Ulam stability. The finding of the result evoluted by the example.

References

  1. [1]  Abbas, S., Benchohra, M., and N'Guerekata, G.M. (2012), Topics in fractional differential equations, New York, Springer.
  2. [2]  Granas, A. and Dugundji, J. (2003), Fixed Point Theory, Springer, New York.
  3. [3]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Elsevier, Amsterdam.
  4. [4]  Oldham, K.B. and Spanier, J. (1974), The Fractional Calculus, Academic Press, New York, London.
  5. [5]  Podlubny, I. (1999), Fractional differential equation, Academic Press, San Diego.
  6. [6]  Zhou, Y. (2014), Basic theory of fractional differential equations, World Scientific(1st ed.), Singapore.
  7. [7]  Abbas, S., Benchohra, M., Lazreg, J.E., and Zhou, Y. (2017), A survey on Hadamard and Hilfer fractional dif- ferential equations: Analysis and stability, Nonlinear Science, and Nonequilibrium and Complex Phenomena, 1-25.
  8. [8]  Benchohra, M. and Lazreg, J.E. (2017), Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative, Studia Universitatis Babe-Bolyai Mathematica, 62, 27-38.
  9. [9]  Kilbas, A.A. (2001), Hadamard-type fractional calculus, Journal of the Korean Mathematical Society, 38(6), 1191-1204.
  10. [10]  Li, C. and Zhang, F. (2011), A survey on the stability of fractional differential equations, The European Physical Journal Special Topics, 193(1), 27-47.
  11. [11]  Wang, J., Feckan, M., and Zhou, Y. (2012), Ulam's type stability of impulsive ordinary differential equations, Journal of Mathematical Analysis and Applications, 395, 258-264.
  12. [12]  Wang, J.R. and Li, X. (2016), A uniform method to Ulam-Hyers stability for some linear fractional equations, Mediterranean Journal of Mathematics, 13, 625-635.
  13. [13]  Benchohra, M. and Lazreg, J.E. (2014), Existence results for nonlinear implicit fractional differential equa- tions, Surveys in mathematics and its applications, 9, 79-92.
  14. [14]  Anguraj, A., Kasthuri, M., and Karthikeyan, P., (2017), Existence of solutions for impulsive fractional differential equations with anti-periodic and integral jump conditions, Nonlinear Studies, 22(4), 501-510.
  15. [15]  Benavides, T.D. (1978), An existence theorem for implicit differential equations in a Banach space, Annali di Matematica Pura ed Applicata, 4, 119-130.
  16. [16]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of impulsive differential equations, Worlds Scientific, Singapore.
  17. [17]  Nieto, J.J., Ouahab, A., and Venktesh, V. (2015), Implicit fractional differential equations via the Liouville- Caputo derivative, Mathematics, 3, 398-411.
  18. [18]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive differential equations, World Scientific, Singapore.
  19. [19]  Wang, J., Feckan, M., and Zhou, Y. (2012), Nonlinear impulsive problems for fractional differential equations and Ulam stability, Computers and Mathematics with Applications, 64, 3389-3405.
  20. [20]  Tidke, H.L. and Mahajan, R.P. (2017), Existence and uniqueness of nonlinear implicit fractional differential equations with Riemann-Liouville derivative, American Journal of computational and applied mathematics, 7(2), 46-50.
  21. [21]  Abbas, S., Albarakati, W., Benchohra, M., and Trujillo, J.J. (2016), Ulam stabilities for partial hadamard fractional integral equations, Arabian Journal of Mathematics, 5(1), 1-7.
  22. [22]  Anuradha, A., Baleanu, D., Suganya, S., and Arjunan, M.M. (2017), On some impulsive fractional neutral differential systems with nonlocal condition through fractional operators, Nonlinear Studies, 24(3), 575-590.
  23. [23]  Karthikeyan, P. and Arul, R., (2018), Existence of solutions for hadamard fractional hybrid differential equations with impulsive and nonlocal conditions, Journal of Fractional Calculus And Applications, 9(1), 232-240.
  24. [24]  Karthikeyan, P. and Arul, R. (2018), Stability for impulsive implicit hadamard fractional differential equa- tions, Malaya Journal of Matematik, 6(1), 28-33.
  25. [25]  Benchohra, M. and Bouriah, S. (2016), Existence and Stability results for nonlinear implicit fractional differential equations with impluses, Memoirs on differential Equations and Mathematical Physics, 69, 15-31.