Skip to main content
SpringerLink
Log in

Meromorphic Solutions of Some Complex Non-Linear Difference Equations

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

This paper is devoted to investigating the growth and zeros of meromorphic solutions of the generalized Fermat functional equation

$${f^2}\left({z + 1} \right) + {A_1}\left(z \right)f\left({z + 1} \right)f\left({z + 1} \right)f\left(z \right) + {A_2}{f^2}\left(z \right) = {A_3}\left(z \right),$$

where A1(z), A2(z), A3(z) are polynomials with A3(z) ≢ 0. The corresponding homogeneous equation of the above equation is studied as well.

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. Z. X. Chen and K. H. Shon, On zeros and fixed points of difference of meromorphic functions, J. Math. Anal. Appl., 344 (2008), 373–383.

    Article  MathSciNet  Google Scholar 

  2. Z. X. Chen, On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations, Sci. China Math., 54 (2011), 2123–2133.

    Article  MathSciNet  Google Scholar 

  3. Z. X. Chen and K. H. Shon, On growth of meromorphic solutions for linear difference equations, Abstract Appl. Anal. (2013), ID 619296, 6 pp.

  4. Z. X. Chen, Complex Differences and Difference Equations, Science Press (Beijing, 2014).

    Book  Google Scholar 

  5. W. Chen, Q. Han and J. B. Liu, On Fermat Diophantine functional equations, little Picard theorem and beyond, Aequationes Math., 93 (2019), 425–432.

    Article  MathSciNet  Google Scholar 

  6. Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane, Ramanujan J., 16 (2008), 105–129.

    Article  MathSciNet  Google Scholar 

  7. W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc., 142 (2007), 133–147.

    Article  MathSciNet  Google Scholar 

  8. R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487.

    Article  MathSciNet  Google Scholar 

  9. R. G. Halburd and R. J. Korhonen, Finite order meromorphic solutions and the discrete Painleveequations, Proc. Lond. Math. Soc., 94 (2007), 443–474.

    Article  MathSciNet  Google Scholar 

  10. W. K. Hayman, Slowly growing integral and subharmonic functions, Comment Math. Helv., 34 (1960), 75–84.

    Article  MathSciNet  Google Scholar 

  11. W. K. Hayman, Meromorphic Functions, Clarendon Press (Oxford, 1964).

    MATH  Google Scholar 

  12. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, Complex difference equations of Malmquist type, Comput. Methods Funct. Theory, 1 (2001), 27–39.

    Article  MathSciNet  Google Scholar 

  13. P. C. Hu and Q. Wang, On meromorphic solutions of functional equations of Fermat type, Bull. Malays. Math. Sci. Soc., 42 (2019), 2497–2515.

    Article  MathSciNet  Google Scholar 

  14. G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser (Basel-Boston, 1985).

    Book  Google Scholar 

  15. K. Ishizaki and N. Yanagihara, Wiman-Valiron method for difference equations, Nagoya Math. J., 175 (2004), 75–102.

    Article  MathSciNet  Google Scholar 

  16. K. Ishizaki and R. J. Korhonen, Meromorphic solutions of algebraic difference equations, Constr. Approx., 48 (2018), 371–384.

    Article  MathSciNet  Google Scholar 

  17. I. Laine, J. Rieppo and H. Silvennoinen, Remarks on complex difference equation, Comput. Methods Funct. Theory, 5 (2005), 77–88.

    Article  MathSciNet  Google Scholar 

  18. K. Liu and L. Z. Yang, A note on meromorphic solutions of Fermat types equations, An. Stiint. Univ. Al. I. Cuza Iaşi Mat. (NS), 1 (2016), 317–325.

    MathSciNet  MATH  Google Scholar 

  19. E. G. Saleeby, On complex analytic solutions of certain trinomial functional and partial differential equations, Math., 85 (2013), 553–562.

    MathSciNet  MATH  Google Scholar 

  20. L. L. Wu, C. He, W. R. Lü, and F. Lü, Existence of meromorphic solutions of some generalized Fermat functional equations, Aequationes Math., 94 (2020), 59–69.

    Article  MathSciNet  Google Scholar 

  21. C. C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad Ser. A, 86 (2010), 10–14.

    Article  MathSciNet  Google Scholar 

  22. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers (Dordrecht, 2003).

    Book  Google Scholar 

Download references

Acknowledgement

The authors would like to thank the referee for helpful suggestions and comments.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Jinan, Jinan, Shandong, 250022, P. R. China

    X.-G. Qi

  2. School of Mathematics, Shandong University, Jinan, Shandong, 250100, P. R. China

    L.-Z. Yang

Authors
  1. X.-G. Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L.-Z. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X.-G. Qi.

Additional information

The work was supported by the NNSF of China (No. 11661052, 11801215, 12061042) and the NSF of Shandong Province (No. ZR2016AQ20, ZR2018MA021).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, XG., Yang, LZ. Meromorphic Solutions of Some Complex Non-Linear Difference Equations. Anal Math 47, 405–419 (2021). https://doi.org/10.1007/s10476-021-0085-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-021-0085-7

Key words and phrases

Mathematics Subject Classification

Search

Navigation