Existence of mild solutions for impulsive fractional evolution equations with periodic boundary conditions

Abstract

In this paper, we concern on the periodic boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space E. First, we establish the existence results of mild solutions for the associated linear periodic boundary value problem. Next, we obtain the existence results of mild solutions by using the monotone iterative technique with L-quasi-upper and lower solutions, the results are new and extend some previously known results. Finally, two examples are also given to illustrate the main results.

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

References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, P., Li, Y., Chen, Q., Feng, B.: On the initial value problem of fractional evolution equations with noncompact semigroup. Comput. Math. Appl. 67, 1108–1115 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Agarwal, P., Al-Mdallal, Q.M., Cho, Y.J., Jain, S.: Fractional differential equations for the generalized Mittag-Leffler function. Adv. Differ. Equ. 2018, 58 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Sitho, S., Ntouyas, S.K., Agarwal, P., Tariboon, J.: Noninstantaneous impulsive inequalities via conformable fractional calculus. J. Inequal. Appl. 2018, 261 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Phys. A: Stat. Mech. Appl. 500, 40–49 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Li, Y., Liu, Z.: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 66, 83–92 (2007)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mu, J., Li, Y.X.: Monotone iterative technique for impulsive fractional evolution equations. J. Inequal. Appl. 2011, 125 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Du, S., Lakshmikantham, V.: Monotone iterative technique for differential equtions in Banach spaces. J. Anal. Math. Anal. 87, 454–459 (1982)

    Article  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

    Chen, P., Mu, J.: Monotone iterative method for semilinear impulsive evolution equations of mixed type in Banach spaces. Electron. J. Differ. Equ. 2010(149), 1–13 (2010)

    MathSciNet  MATH  Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

    Li, B.L., Gou, H.D.: Monotone iterative method for the periodic boundary value problems of impulsive evolution equations in Banach spaces. Chaos Solitons Fractals 110, 209–215 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Wang, L., Wang, Z.: Monotone iterative technique for parameterized BVPs of abstract semilinear evolution equations. Comput. Math. Appl. 46, 1229–1243 (2003)

    MathSciNet  Article  Google Scholar 

  17. 17.

    EI-Gebeily, M.A., ORegan, D., Nieto, J.J.: A monotone iterative technique for stationary and time dependent problems in Banach spaces. J. Comput. Appl. Math. 233, 2359–2404 (2010)

    MathSciNet  Google Scholar 

  18. 18.

    Wang, J., Fec̆kan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8, 345–361 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Mu, J.: Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions. Bound. Value Probl. 2012, 71 (2012)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Li, Y.X.: The positive solutions of abstract semilinear evolution equations and their applications. Acta Math. Sin. 39(5), 666–672 (1996). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Guo, D.J., Sun, J.X.: Ordinary Differential Equations in Abstract Spaces. Shandong Science and Technology, Jinan (1989). (in Chinese)

    Google Scholar 

  22. 22.

    Deiling, K.: Nonlinear Functional Analysis. Springer, New York (1985)

    Google Scholar 

  23. 23.

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

    MathSciNet  Article  Google Scholar 

  24. 24.

    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Google Scholar 

  25. 25.

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

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Haide Gou.

Additional information

Publisher's Note

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

This work is supported by the National Natural Science Foundation of China (Grant No. 11661071).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gou, H., Li, Y. Existence of mild solutions for impulsive fractional evolution equations with periodic boundary conditions. J. Pseudo-Differ. Oper. Appl. 11, 425–445 (2020). https://doi.org/10.1007/s11868-019-00278-2

Download citation

Keywords

  • Monotone iterative technique
  • Coupled L-quasi-upper and lower solutions
  • Periodic boundary conditions
  • Analytic semigroup

Mathematics Subject Classification

  • 26A33
  • 34K30
  • 34K45
  • 35B10