Skip to main content
SpringerLink
Log in

Application of characteristic equation of first order neutral impulsive difference equations

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

In this work, the existence of oscillatory solution of first order nonlinear neutral impulsive difference equations of form:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta [x(n)-p(n)f(x(n-\tau ))] + q(n)h(x(n-\sigma ))=0,\, n\ne m_j\\ {\underline{\Delta }}[x(m_j-1)-p(m_j-1)f(x(m_j-\tau -1))] +r(m_j-1)h(x(m_j -\sigma -1))=0,\, j\in \mathbb {N} \end{array}\right. } \end{aligned}$$

is discussed for the various ranges of the neutral coefficient p(n). The technique employed here is due to the linearizaton method by using Banach contraction principle and Knaster-Tarski fixed point theorem. Some examples are given to show the feasibility and effectiveness 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

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. Agarwal, R.P. 2000. Difference equations and inequalities. New York: Marcel Dekker.

    Book  Google Scholar 

  2. Berezansky, L., and E. Braverman. 2003. Linearized oscillation theory for a nonlinear delay impulsive equation. Journal of Computational and Applied Mathematics 161: 477–495.

    Article  MathSciNet  Google Scholar 

  3. Berezansky, L., and E. Braverman. 2008. Linearized oscillation theory for a nonlinear equation with a distributed delay. Mathematical and Computer Modelling 48: 287–304.

    Article  MathSciNet  Google Scholar 

  4. Duan, Y., W. Feng, and J. Yan. 2002. Linearized oscillation of nonlinear impulsive delay differential equations. Computers and Mathematics with Applications 44: 1267–1274.

    Article  MathSciNet  Google Scholar 

  5. Gyori, I., and G. Ladas. 1991. Oscillation theory of delay differential equations with applications. New York: Claredon Press.

    MATH  Google Scholar 

  6. Lakshmikantham, V., D.D. Bainov, and P.V. Simieonov. 1989. Oscillation theory of impulsive differential equations. Singapore: World Scientific Press.

    Book  Google Scholar 

  7. Li, J., and J. Shen. 2006. Positive solutions for first order difference equations with impulses. International Journal of Difference Equations 10: 225–239.

    MathSciNet  MATH  Google Scholar 

  8. Lu, W., W.G. Ge, and Z.H. Zhao. 2010. Oscillatory criteria for third-order nonlinear difference equations with impulses. Journal of Computational and Applied Mathematics 234: 3366–3372.

    Article  MathSciNet  Google Scholar 

  9. Parhi, N., and A.K. Tripathy. 2003. Oscillation of forced nonlinear neutral delay difference equations of first order. Czechoslovak Mathematical Journal 53: 83–101.

    Article  MathSciNet  Google Scholar 

  10. Peng, D.H., M.A. Han, and H.Y. Wang. 2003. Linearized oscillations of first order nonlinear neutral delay difference equations. Computers and Mathematics with Applications 45: 1785–1796.

    Article  MathSciNet  Google Scholar 

  11. Samoilenko, A.M., and N.A. Perestynk. 1987. Differential equations with impulse effect. Kiev: Visca Skola.

    Google Scholar 

  12. Tripathy, A.K., and G.N. Chhatria. 2018. Oscillation criteria for forced first order nonlinear neutral impulsive differenc system. Tatra Mountains Mathematical Publications 71: 175–193.

    Article  MathSciNet  Google Scholar 

  13. Tripathy, A.K., and G.N. Chhatria. 2019. On oscillatory first order neutral impulsive difference equations. Mathematica Bohemica. https://doi.org/10.21136/MB.2019.0002-18.

    Article  MATH  Google Scholar 

  14. Tripathy, A.K., and G.N. Chhatria. 2019. Oscillation criteria for first order neutral impulsive difference equations with constant coefficients. Differential Equations and Dynamical Systems. https://doi.org/10.1007/s12591-019-00495-7.

    Article  MATH  Google Scholar 

  15. Wei, G.P. 2005. The persistance of nonoscillatory solutions of difference equation under impulsive perturbations. Computers and Mathematics with Applications 50: 1579–1586.

    Article  MathSciNet  Google Scholar 

  16. Xiao, Y., S. Tang, and L. Chen. 2002. A Linearized oscillation result for odd order neutral difference equations. The Indian Journal of Pure and Applied Mathematics 33: 277–286.

    MathSciNet  MATH  Google Scholar 

  17. Zhou, Z., and J.S. Yu. 1997. Linearized oscillations for difference equations of neutral type. Mathematical Science Research Hot-Line 1: 1–8.

    MathSciNet  MATH  Google Scholar 

  18. Zhou, Z., and J.S. Yu. 1999. Linearized oscillation theorems for neutral difference equations. Israel Journal of Mathematics 114: 149–156.

    Article  MathSciNet  Google Scholar 

  19. Zhou, Z., and Z. Lin. 2003. Some results on the linearized oscillation of the odd order neutral difference equations. Applicable Analysis 82: 401–409.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The author thanks the editor and two anonymous referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper.

Funding

This work is supported by Rajiv Gandhi National fellowship(UGC), New Delhi, India, through the Letter No. F1-17.1/2017-18/RGNF-2017-18-SC-ORI-35849, dated. 11th july, 2017.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sambalpur University, Sambalpur, 768019, India

    Gokula Nanda Chhatria

Authors
  1. Gokula Nanda Chhatria
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gokula Nanda Chhatria.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animal performed by any of the authors.

Additional information

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

Chhatria, G.N. Application of characteristic equation of first order neutral impulsive difference equations. J Anal 29, 191–206 (2021). https://doi.org/10.1007/s41478-020-00255-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-020-00255-9

Keywords

Mathematics Subject Classification

Search

Navigation