Skip to main content
SpringerLink
Log in

Mild Solutions for Impulsive Functional Differential Equations of Order \(\alpha \in (1,2)\)

  • Conference paper
  • First Online:
Mathematical Analysis and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 143))

  • 1644 Accesses

Abstract

In this research paper, first we develop the definition of mild solutions for impulsive fractional differential equations of order \(\alpha \in (1,2).\) Second, we study the uniqueness result of mild solutions for impulsive fractional differential equation with state-dependent delay by applying fixed point theorem and solution operator. At last, we present an example to illustrate the uniqueness result using fractional partial derivatives.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abbas, S., Benchohra, M.: Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay. An Int. J. Theory Appl. (2010). ISSN 1311–0454

    Google Scholar 

  2. Agarwal, R.P., Andrade, B.D.: On fractional integro-differential equations with state-dependent delay. Comput. Math. Appl. 62, 1143–1149 (2011)

    Article  MathSciNet  Google Scholar 

  3. Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. TMA 69, 3692–3705 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benchohra, M., Litimein, S., N’Guerekata, G.: On fractional integro-differential inclusions with state-dependent delay in Banach spaces. Appl. Anal. 2011, 1–16 (2011)

    Google Scholar 

  5. Benchohra, M., Berhoun, F.: Impulsive fractional differential equations with state dependent delay. Commun. Appl. Anal. 14(2), 213–224 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Carvalho dos Santos, J.P., Arjunan, M.M.: Existence results for fractional neutral integro-differential equations with state-dependent delay. Comp. and Math. with Appl. 62, 1275–1283 (2011)

    Google Scholar 

  7. Chauhan, A., Dabas, J., Kumar, M.: Integral boundary-value problem for impulsive fractional functional integro-differential equation with infinite delay. Electr. J. Diff. Eqn. 2012(229), 1–13 (2012)

    Google Scholar 

  8. Chauhan, A., Dabas, J.: Existence of mild solutions for impulsive fractional order semilinear evolution equations with nonlocal conditions. Electr. J. Diff. Eqn. 2011(107), 1–10 (2011)

    Google Scholar 

  9. Chauhan, A., Dabas, J.: Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition. Commun. Nonlinear Sci. Num. Simul. 19, 821–829 (2014)

    Article  MathSciNet  Google Scholar 

  10. Dabas, J., Chauhan, A.: Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equations with infinity delay. Math. Comput. Mod.57, 754–763 (2013)

    Google Scholar 

  11. Dabas, J., Gautam, G.R.: Impulsive neutral fractional integro-differential equation with state dependent delay and integral boundary condition. Electr. J. Diff. Eqn. 2013(273), 1–13 (2013)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Fu, X., Huang, R.: Existence of solutions for neutral integro-differential equations with state-dependent delay. Appl. Math. Comput. 224, 743–759 (2013)

    Article  MathSciNet  Google Scholar 

  14. Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 21, 11–41 (1978)

    MathSciNet  MATH  Google Scholar 

  15. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies, Elsevier Science, Amsterdam (2006)

    Google Scholar 

  16. Lakshmikantham, V., Leela, V.S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    Google Scholar 

  17. Mainardi, F., Gorenflo, R.: On Mittag-Leffer-type functions in fractional evolution processes. J. Comput.Appl. Math. 118, 283–299 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  19. Podlubny, I.: Fractional Differential Equation. Academic Press, San Diego (1999)

    Google Scholar 

  20. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  21. Shu, X., Wang, Q.: The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order \(1 < \alpha < 2.\) Comput. Math. Appl. 64, 2100–2110 (2012)

    Google Scholar 

  22. Wang, J., Fekckan, M., Zhou, Y.: On the new concept of solutions and existence results for mpulsive fractional evolution equations. Dyn. PDE 8(4), 345–361 (2011)

    MATH  Google Scholar 

  23. Wang, J., Li, X., Wei, W.: On the natural solution of an impulsive fractional differential equation of order \(q \in (1, 2)\). Commun. Nonlinear. Sci. Numer. Simul. 17, 4384–4394 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Science and Engineering, IIT Roorkee, Saharanpur Campus, 247001, Saharanpur, India

    Ganga Ram Gautam & Jaydev Dabas

Authors
  1. Ganga Ram Gautam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaydev Dabas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ganga Ram Gautam .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    P. N. Agrawal

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    R. N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Uaday Singh

  4. Dept. of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H. M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Gautam, G.R., Dabas, J. (2015). Mild Solutions for Impulsive Functional Differential Equations of Order \(\alpha \in (1,2)\) . In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol 143. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2485-3_23

Download citation

Publish with us

Policies and ethics

Search

Navigation