Skip to main content
SpringerLink
Log in

A spectrum relation of almost periodic solution of second order scalar functional differential equations with piecewise constant argument

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, the spectrum relation of almost periodic solution for the equation (x(t) + px(t − 1))″ = qx([t]) + f(t) is investigated. Although this has been discussed in an article, some counterexamples are constructed to show that some part of the spectrum inclusion in that article is not correct. The key point which causes such problem is found out. A new statement is formulated and proved.

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. Cooke, K. L., Wiener, J.: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 99, 265–297 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Shah, S. M., Wiener, J.: Advanced differential equations with piecewise constant argument deviations. Internat. J. Math. & Math. Sci., 6, 671–703 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Busenberg, S., Cooke, K. L.: Models of Vertically Transmitted Diseases with Sequential-continuous Dynamics. In: Nonlinear Phenomena in Mathematical Sciences (V. Lakshmikantham Ed.), Academic Press, New York, 1982, 179–187

    Google Scholar 

  4. Yuan, R.: On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument. Nonlinear Analysis: Theory, Methods & Applications, 59(8), 1189–1205 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Yuan, R.: On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument. J. Math. Anal. Appl., 303(1), 103–118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Seifert, G.: Second-order scalar functional differential equations with piecewise constant arguments. J. Difference Equations Appl., 8(5), 427–445 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Seifert, G.: Second-order neutral delay-differential equations with piecewise constant time dependence. J. Math. Anal. Appl., 281(1), 1–9 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Ait Dads, E., Lhachimi, L.: New approach for the existence of pesudo almost periodic solutions for some second order differential equation with piecewise constant. Nonlinear Analysis: Theory, Methods & Applications, 64(6), 1307–1324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yuan, R.: A new almost periodic type solution of first order neutral delay differential equations with piecewise constant argument. Acta Mathematica Sinica, English Series, 17(3), 403–412 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lin, G. J., Yuan, R.: Pseudo almost periodic solutions of a singularly perturbed differential equation with piecewise constant argument. Acta Mathematica Sinica, English Series, 23(3), 423–438 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang, L., Zhang, C. Y.: Almost periodic solutions of second-order neutral delay-differential equations with piecewise constant argument. Acta Mathematica Sinica, Chinese Series, 53(2), 227–232 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Corduneanu, C.: Almost Periodic Function, Wiley, New York, 1968

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, P. R. China

    Li Wang

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China

    Rong Yuan

  3. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P. R. China

    Chuan Yi Zhang

Authors
  1. Li Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rong Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chuan Yi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Wang.

Additional information

The first author is supported by NPU Foundation for Fundamental Research (NPU-FFR-JC20100220); the second author is supported by National Natural Science Foundation (Grant No. 11031002) and RFDP; the third author is supported by National Natural Science Foundation (Grant No. 11071048

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, L., Yuan, R. & Zhang, C.Y. A spectrum relation of almost periodic solution of second order scalar functional differential equations with piecewise constant argument. Acta. Math. Sin.-English Ser. 27, 2275–2284 (2011). https://doi.org/10.1007/s10114-011-8392-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-011-8392-8

Keywords

MR(2000) Subject Classification

Search

Navigation