Skip to main content
SpringerLink
Log in

Entire Solutions of Two Certain Types of Non-linear Differential-Difference Equations

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

In this paper, we describe entire solutions for two certain types of non-linear differential-difference equations of the form

$$\begin{aligned} f^n(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=u(z)e^{v(z)}, \end{aligned}$$

and

$$\begin{aligned} f^n(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=p_1e^{\lambda z}+p_2e^{-\lambda z}, \end{aligned}$$

where q, Q, u, v are non-constant polynomials, \(c,\lambda ,p_1,p_2\) are non-zero constants, and \(\omega \) is a constant. Our results improve and generalize some previous 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. Alotaibi, A., Langley, J.K.: The separation of zeros of solutions of higher order linear differential equations with entire coefficients. Results Math. 63(3–4), 1365–1373 (2013)

    Article  MathSciNet  Google Scholar 

  2. Chen, M.F., Gao, Z.S., Zhang, J.L.: Entire solutions of certain type of non-linear difference equations. Comput. Methods Funct. Theory 19(1), 17–36 (2019)

    Article  MathSciNet  Google Scholar 

  3. Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan J. 16(1), 105–129 (2008)

    Article  MathSciNet  Google Scholar 

  4. Clunie, J.: On integral and meromorphic functions. J. Lond. Math. Soc. 37, 17–27 (1962)

    Article  MathSciNet  Google Scholar 

  5. Halburd, R.G., Korhonen, R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314(2), 477–487 (2006)

    Article  MathSciNet  Google Scholar 

  6. Halburd, R.G., Korhonen, R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31(2), 463–478 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Hayman, W.K.: Meromorphic Functions, vol. 78. Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  8. Laine, I.: Nevanlinna Theory and Complex Differential Equations, vol. 15. Walter de Gruyter, Berlin (2011)

    MATH  Google Scholar 

  9. Li, P.: Entire solutions of certain type of differential equations II. J. Math. Anal. Appl. 375(1), 310–319 (2011)

    Article  MathSciNet  Google Scholar 

  10. Li, P., Yang, C.C.: On the nonexistence of entire solutions of certain type of nonlinear differential equations. J. Math. Anal. Appl. 320(2), 827–835 (2006)

    Article  MathSciNet  Google Scholar 

  11. Liao, L.W., Ye, Z.: On solutions to nonhomogeneous algebraic differential equations and their application. J. Aust. Math. Soc. 97(3), 391–403 (2014)

    Article  MathSciNet  Google Scholar 

  12. Liao, L.W., Yang, C.C., Zhang, J.J.: On meromorphic solutions of certain type of non-linear differential equations. Ann. Acad. Sci. Fenn. Math. 38, 581–593 (2013)

    Article  MathSciNet  Google Scholar 

  13. Mues, E., Steinmetz, N.: The theorem of Tumura–Clunie for meromorphic functions. J. Lond. Math. Soc. 23(2), 113–122 (1981)

    Article  MathSciNet  Google Scholar 

  14. Reyzl, I.: On the theorem of Tumura–Clunie. Complex Var. Theory Appl. 28, 175–188 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Tumura, Y.: On the extensions of Borel’s theorem and Saxer–Csillag’s theorem. Proc. Phys. Math. Soc. Jpn. 19, 29–35 (1937)

    MATH  Google Scholar 

  16. Wen, Z.T., Heittokangas, J., Laine, I.: Exponential polynomials as solutions of certain nonlinear difference equations. Acta Math. Sci. Ser. B 28(7), 1295–1306 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Springer, Berlin (2004)

    Google Scholar 

  18. Yang, C.C., Li, P.: On the transcendental solutions of a certain type of nonlinear differential equations. Arch. Math. 82, 442–448 (2004)

    Article  MathSciNet  Google Scholar 

  19. Yang, C.C.: Applications of the Tumura–Clunie theorem. Trans. Am. Math. Soc. 151, 659–662 (1970)

    Article  MathSciNet  Google Scholar 

  20. Yi, H.X.: On a theorem of Tumura–Clunie for a differential polynomial. Bull. Lond. Math. Soc. 20, 593–596 (1988)

    Article  MathSciNet  Google Scholar 

  21. Zhang, J.J., Xu, X.P., Liao, L.W.: Meromorphic solutions of nonlinear complex differential equations (in Chinese). Sci. Sin. Math. 47, 919–932 (2017)

    Article  Google Scholar 

  22. Zhang, J., Liao, L.W.: On entire solutions of a certain type of nonlinear differential and difference equations. Taiwan. J. Math. 15(5), 2145–2157 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Sciences, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Wei Chen & Qiong Wang

  2. School of Mathematics, Shandong University, Jinan, 250100, Shandong, People’s Republic of China

    Peichu Hu

Authors
  1. Wei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peichu Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qiong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiong Wang.

Additional information

Communicated by Ilpo Laine.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work of both authors was partially supported by Basic and Advanced Research Project of CQ CSTC (Grant number: cstc2019jcyj-msxmX0107), and Fundamental Research Funds of Chongqing University of Posts and Telecommunications (CQUPT:A2018-125)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Hu, P. & Wang, Q. Entire Solutions of Two Certain Types of Non-linear Differential-Difference Equations. Comput. Methods Funct. Theory 21, 199–218 (2021). https://doi.org/10.1007/s40315-020-00343-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-020-00343-8

Keywords

Mathematics Subject Classification

Search

Navigation