Skip to main content
SpringerLink
Log in

A Method of L-Quasi-upper and Lower Solutions for Boundary Value Problems of Impulsive Differential Equation in Banach Spaces

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

Under certain monotonicity conditions and the non-compactness measure conditions, by using the monotone iteration scheme with L-quasi-upper and lower solutions, the existence of minimal and maximal L-quasi-solutions of the boundary problems

$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle u'(t)=f(t,u(t),u(t)),&{} \quad t\in J, \; t\ne t_{k},\\ \triangle u|_{t=t_{k}}=I_{k}(u(t_{k})),&{} \quad k=1,2,3,\ldots ,m,\\ u(0)=u(1),\end{array}\right. \end{aligned}$$

is derived and the existence of solution for the problems between them is proved. Besides, the conditions that the problem has unique solution and the error estimate of iterative sequences of the unique solution are also given.

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

References

  1. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Carl, S., Heikkila, S.: On discontinuous implicit and explicit abstract impulsive boundary value problems. Nonlinear Anal. 41, 701–723 (2000)

    Article  MathSciNet  Google Scholar 

  4. He, Z., Yu, J.: Periodic boundary value problem for first-order impulsive ordinary differential equations. J. Comput. Appl. Math. 272, 67–78 (2002)

    MathSciNet  MATH  Google Scholar 

  5. McRae, F.A.: Monotone iterative technique and existence results for fractional differential equations. Nonlinear Anal. 71, 6093–6096 (2009)

    Article  MathSciNet  Google Scholar 

  6. Wang, G., Agarwal, R.P., Cabada, A.: Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 25, 1019–1024 (2012)

    Article  MathSciNet  Google Scholar 

  7. Ahmad, B., Sivasundaram, S.: Existence results and monotone iterative technique for impulsive hybrid functional differential systems with anticipation and retardation. Appl. Math. Comput. 197, 515–524 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Jankowski, T.: Monotone iterative method for first-order differential equations at resonance. Appl. Math. Comput. 23, 20–28 (2014)

    MathSciNet  MATH  Google Scholar 

  9. He, Z., He, X.: Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions. Comput. Math. Appl. 48, 73–84 (2004)

    Article  MathSciNet  Google Scholar 

  10. Jiang, D., Wei, J.: Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations. Nonlinear Anal. 50, 885–898 (2002)

    Article  MathSciNet  Google Scholar 

  11. Ding, W., Xing, Y., Han, M.: Antiperiodic boundary value problems for first order impulsive functional differential equations. Appl. Math. Comput. 186, 45–53 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Lakshmikantham, V., Leela, S., Vatsala, A.S.: Method of quasi-upper and lower solutions in abstract cones. Nonlinear Anal. 6, 833–838 (1982)

    Article  MathSciNet  Google Scholar 

  14. Guo, D., Lakshmikantham, V.: Coupled fixed points of nonlinear operator with applications. Nonlinear Anal. 11, 623–632 (1987)

    Article  MathSciNet  Google Scholar 

  15. Li, Y.: A method of quasi-upper and lower solutions for ordinary differential equations in Banach spaces. J. Northwest Normal Univ. (Nat. Sci.) 37, 6–11 (2001)

    MATH  Google Scholar 

  16. Guo, D.: Nonlinear Functional Analysis. Shandong Science and Technology, Jinan (2001). (in Chinese)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Li, Y.: Existence of solutions of initial value problems for abstract semilinear evolution equations. Acta Math. Sin. 48(6), 1089–1094 (2005). (in Chinese)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Guofeng Che & Haibo Chen

Authors
  1. Guofeng Che
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haibo Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haibo Chen.

Additional information

Research supported by Natural Science Foundation of China (11271372) and Hunan Provincial Natural Science Foundation of China (12JJ2004).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Che, G., Chen, H. A Method of L-Quasi-upper and Lower Solutions for Boundary Value Problems of Impulsive Differential Equation in Banach Spaces. Differ Equ Dyn Syst 26, 393–403 (2018). https://doi.org/10.1007/s12591-015-0266-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-015-0266-6

Keywords

Search

Navigation