Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Controllability of Impulsive Fractional Integro-Differential Evolution Equations

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper, we are concerned with the controllability for a class of impulsive fractional integro-differential evolution equation in a Banach space. Sufficient conditions of the existence of mild solutions and approximate controllability for the concern problem are presented by considering the term \(u'(\cdot )\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_{b}\) and \(u'(b)=u'_{b}\). The discussions are based on Mönch fixed point theorem as well as the theory of fractional calculus and \((\alpha ,\beta )\)-resolvent operator. Finally, an example is given to illustrate the feasibility of our results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

References

  1. 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  MATH  Google Scholar 

  2. 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)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang, Y.K., Pereira, A., Ponce, R.: Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract. Calc. Appl. Anal. 20(4), 963–987 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhu, B., Liu, L., Wu, Y.: Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations. Fract. Calc. Appl. Anal. 20, 1338–1355 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Adda, F.B., Cresson, J.: Fractional differential equations and the Schrodinger equation. Appl. Math. Comput. 161, 323–345 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Agarwal, R.P., Benchohra, M., Slimani, B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Bahuguna, D., Srvastavai, S.K.: Semilinear integro-differential equations with compact semigroup. J. Appl. Math. Stoch. Anal. 11, 507–517 (1998)

    Article  MathSciNet  Google Scholar 

  9. Baleanu, D., Mustafa, O.G., Agarwal, R.P.: On the solution set for a class of sequential fractional differential equations. J. Phys. A, Math. Theor. 43, 385209 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Baleanu, D., Mustafa, O.G., Agarwal, R.P.: An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 23, 1129–1132 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)

    Book  MATH  Google Scholar 

  12. Heard, M.L., Rankin, S.M.: A semilinear parabolic Volterra integro-differential equation. J. Differ. Equ. 71, 201–233 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shu, X.B., Wang, Q.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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Mophou, G.M., N’Guérékata, G.M.: On some classes of almost automorphic functions and applications to fractional differential equations. Comput. Math. Appl. 59, 1310–1317 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kalman, R.E.: Controllability of linear dynamical systems. Contrib. Differ. Equ. 1(1), 189–213 (1963)

    MathSciNet  Google Scholar 

  17. Tai, Z., Wang, X.: Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces. Appl. Math. Lett. 22, 1760–1765 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Subalakshmi, R., Balachandran, K.: Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in Hilbert spaces. Chaos Solitons Fractals 42, 2035–2046 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Balachandran, K., Park, J.Y.: Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Anal. Hybrid Syst. 3, 363–367 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Balachandran, K., Sakthivel, R.: Controllability of integrodifferential systems in Banach spaces. Appl. Math. Comput. 118, 63–71 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Sakthivel, R., Choi, Q.H., Anthoni, S.M.: Controllability result for nonlinear evolution integrodifferential systems. Appl. Math. Lett. 17, 1015–1023 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sakthivel, R., Anthoni, S.M., Kim, J.H.: Existence and controllability result for semilinear evolution integrodifferential Systems. Math. Comput. Model. 41, 1005–1011 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Abada, N., Benchohra, M., Hammouche, H.: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ. 246, 3834–3863 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Benchohra, M., Gorniewicz, L., Ntouyas, S.K., Ouahab, A.: Controllability results for impulsive functional differential inclusions. Rep. Math. Phys. 54, 211–228 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sakthivel, R., Anandhi, E.R.: Approximate controllability of impulsive differential equations with state-dependent delay. Int. J. Control 83(2), 387–393 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jeong, J.M., Kim, J.R., Roh, H.H.: Controllability for semilinear retarded control systems in Hilbert spaces. J. Dyn. Control Syst. 13(4), 577–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Arthi, G., Park, J.: On controllability of second-order impulsive neutral integro-differential systems with infinite delay. IMA J. Math. Control Inf. 32, 639–657 (2014)

    Article  MATH  Google Scholar 

  30. Arthi, G., Balachandran, K.: Controllability results for damped second-order impulsive neutral integro-differential systems with nonlocal conditions. J. Control Theory Appl. 11, 186–192 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Arthi, G., Balachandran, K.: Controllability of damped second-order neutral functional differential systems with impulses. Taiwan. J. Math. 16, 89–106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hernández, E., Regan, D.O.: Controllability of Volterra-Fredholm type systems in Banach spaces. J. Franklin Inst. 346, 95–101 (2009)

    Article  MathSciNet  Google Scholar 

  33. Ji, S., Li, G., Wang, M.: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput. 217, 6981–6989 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Kumar, S., Sukavanam, N.: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equ. 252, 6163–6174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mahmudov, N.I., Zorlu, S.: On the approximate controllability of fractional evolution equations with compact analysis semigroup. J. Comput. Appl. Math. 259, 194–204 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liang, J., Yang, H.: Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl. Math. Comput. 254, 20–29 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Sakthivel, R., Mahmudov, N.I., Nieto, J.J.: Controllability for a class of fractional-order neutral evolution control systems. Appl. Math. Comput. 218, 10334–10340 (2012)

    MathSciNet  MATH  Google Scholar 

  38. Debbouchea, A., Baleanu, D.: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442–1450 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lian, T., Fan, Z., Li, G.: Approximate controllability of semilinear fractional differential systems of order \(1 < \alpha < 2\) via resolvent operators. Filomat 31(18), 5769–5781 (2017)

    Article  MathSciNet  Google Scholar 

  40. Singh, V., Pandey, D.N.: Controllability of second-order Sobolev type impulsive delay differential systems. Math. Methods Appl. Sci. 42(5), 1377–1388 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Bajlekova, E.G.: Fractional evolution equations in Banach spaces. PhD Thesis. Department of Mathematics, Eindhoven University of Technology (2001)

  42. Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60, p. 97. Dekker, New York (1980)

    MATH  Google Scholar 

  43. Li, Y.: 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 

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

    Google Scholar 

  45. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)

    MATH  Google Scholar 

  46. Shu, X.B., Shi, Y.: A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. 273, 465–476 (2016)

    MathSciNet  MATH  Google Scholar 

  47. Mönch, H.: Boundary value problems for linear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)

    Article  MathSciNet  MATH  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

Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Haide Gou & Yongxiang Li

Authors
  1. Haide Gou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongxiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

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.

Supported by the National Natural Science Foundation of China (Grant No. 12061062, 11661071).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gou, H., Li, Y. Controllability of Impulsive Fractional Integro-Differential Evolution Equations. Acta Appl Math 175, 5 (2021). https://doi.org/10.1007/s10440-021-00433-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-021-00433-2

Keywords

Mathematics Subject Classification (2010)

Search

Navigation