Skip to main content
SpringerLink
Log in

Extremal Mild Solutions of Fractional Evolution Equation with Mixed Monotone Impulsive Conditions

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

The well established mixed monotone iterative technique is utilized explicitly to study the existence of the extremal solutions of a class of impulsive system with Hilfer fractional order in this paper. The procedure of finding mild L-quasi solutions of such impulsive evolution equation with non-compact semigroups involves measure of non-compactness and Sadovskii’s fixed point theorem as well. An example is provided to illustrate the main results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of data and materials

Not applicable.

References

  1. Ahmed, H. M., El-Borai, M. M., El-Owaidy, H. M., Ghanem, A. H.: Impulsive Hilfer fractional differential equations, Adv. Difference Equ. 2018, Paper No. 226 (2018)

  2. Chang, S.S., Guo, W.P.: On the existence and uniqueness theorems of solutions for the systems of mixed monotone operator equations with applications. Gaoxiao Yingyong Shuxue Xuebao Ser. B 8(1), 1–14 (1993)

    MathSciNet  MATH  Google Scholar 

  3. Chang, S.S., Ma, Y.H.: Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming. J. Math. Anal. Appl. 160(2), 468–479 (1991)

    Article  MathSciNet  Google Scholar 

  4. Chen, P.: Mixed monotone iterative technique for impulsive periodic boundary value problems in Banach spaces. Bound. Value Probl. 2011, Art. ID 421261 (2011)

  5. Chen, P., Li, Y.: Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces. Nonlinear Anal. 74(11), 3578–3588 (2011)

    Article  MathSciNet  Google Scholar 

  6. Chen, P., Li, Y.: Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions. Z. Angew. Math. Phys. 65(4), 711–728 (2014)

    Article  MathSciNet  Google Scholar 

  7. Debbouche, A., Antonov, V.: Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces. Chaos Solitons Fractals 102(3), 140–148 (2017)

    Article  MathSciNet  Google Scholar 

  8. Du, J., Jiang, W., Niazi, A.U.K.: Approximate controllability of impulsive Hilfer fractional differential inclusions. J. Nonlinear Sci. Appl. 10(2), 595–611 (2017)

    Article  MathSciNet  Google Scholar 

  9. Du, S.W., Lakshmikantham, V.: Monotone iterative technique for differential equations in a Banach space. J. Math. Anal. Appl. 87(2), 454–459 (1982)

    Article  MathSciNet  Google Scholar 

  10. Furati, K.M., Kassim, M.D., Tatar, N.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64(6), 1616–1626 (2012)

    Article  MathSciNet  Google Scholar 

  11. Gu, H., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Guo, D.J.: Fixed points of mixed monotone operators with applications. Appl. Anal. 31(3), 215–224 (1988)

    Article  MathSciNet  Google Scholar 

  13. Guo, D.J., Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11(5), 623–632 (1987)

    Article  MathSciNet  Google Scholar 

  14. Guo, D.J., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, vol. 5. Academic Press, Boston (1988)

    Google Scholar 

  15. Gou, H., Li, Y.: Upper and lower solution method for Hilfer fractional evolution equations with nonlocal conditions. Bound. Value Probl. 2019, Paper No. 187 (2019)

  16. Gou, H., Li, Y.: The method of lower and upper solutions for impulsive fractional evolution equations. Ann. Funct. Anal. 11(2), 350–369 (2020)

    Article  MathSciNet  Google Scholar 

  17. Gou, H., Li, Y., Li, Q.: Mixed monotone iterative technique for Hilfer fractional evolution equations with nonlocal conditions. J. Appl. Anal. Comput. 10(5), 1823–1847 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Guo, Y., Chen, M., Shu, X.-B., Xu, F.: The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm. Stoch. Anal. Appl. 39(4), 643–666 (2021)

    Article  MathSciNet  Google Scholar 

  19. Heinz, H.-P.: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal. 7(12), 1351–1371 (1983)

    Article  MathSciNet  Google Scholar 

  20. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  21. Jaiswal, A., Bahuguna, D.: Hilfer fractional differential equations with almost sectorial operators. Differ. Equ. Dyn. Syst. (2020)

  22. Li, Y., Gou, H.: Mixed monotone iterative technique for semilinear impulsive fractional evolution equations. J. Appl. Anal. Comput. 9(4), 1216–1241 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    MATH  Google Scholar 

  24. Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  25. Rzepecki, B.: Applications of Sadovskiĭ’s fixed point theorem to equations in Banach spaces, in Differential equations and optimal control (Żagań: 44–53. Higher College Engrg, Zielona Góra (1986)

  26. Shu, X.-B., Xu, F.: Upper and lower solution method for fractional evolution equations with order \(1<\alpha <2\). J. Korean Math. Soc. 51(6), 1123–1139 (2014)

    Article  MathSciNet  Google Scholar 

  27. Stamova, I.M., Stamov, T.G.: Functional and Impulsive Differential Equations of Fractional Order. CRC Press, Boca Raton (2017)

    Book  Google Scholar 

  28. Sun, J.X., Liu, L.S.: Iterative method for coupled quasi-solutions of mixed monotone operator equations. Appl. Math. Comput. 52(2–3), 301–308 (1992)

    MathSciNet  MATH  Google Scholar 

  29. Vanterler da Costa Sousa, J., Benchohra, M., N’Guérékata, G.M.: Attractivity for differential equations of fractional order and \(\psi \)-Hilfer type. Fract. Calc. Appl. Anal. 23(4), 1188–1207 (2020)

    Article  MathSciNet  Google Scholar 

  30. Vanterler da Costa Sousa, J., Capelas de Oliveira, E.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    Article  MathSciNet  Google Scholar 

  31. Vanterler da Costa Sousa, J., Capelas de Oliveira, E.: A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator. Differ. Equ. Appl. 11(1), 87–106 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Xu, S., Jia, B.: Fixed-point theorems of \(\phi \) concave-\((-\psi )\) convex mixed monotone operators and applications. J. Math. Anal. Appl. 295(2), 645–657 (2004)

    Article  MathSciNet  Google Scholar 

  33. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)

    Article  MathSciNet  Google Scholar 

  34. Zhang, Z.: New fixed point theorems of mixed monotone operators and applications. J. Math. Anal. Appl. 204(1), 307–319 (1996)

    Article  MathSciNet  Google Scholar 

  35. Zhao, J., Wang, R.: Mixed monotone iterative technique for fractional impulsive evolution equations. Miskolc Math. Notes 17(1), 683–696 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful for the referees for their valuable comments.

Funding

No funding available.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    Divya Raghavan & Sukavanam Nagarajan

Authors
  1. Divya Raghavan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sukavanam Nagarajan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both the authors contributed equally.

Corresponding author

Correspondence to Divya Raghavan.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Communicated by Pham Huu Anh Ngoc.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raghavan, D., Nagarajan, S. Extremal Mild Solutions of Fractional Evolution Equation with Mixed Monotone Impulsive Conditions. Bull. Malays. Math. Sci. Soc. 45, 1427–1452 (2022). https://doi.org/10.1007/s40840-022-01288-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-022-01288-y

Keywords

Mathematics Subject Classification

Search

Navigation