DOI QR코드

DOI QR Code

EXISTENCE OF TRANSCENDENTAL MEROMORPHIC SOLUTIONS ON SOME TYPES OF NONLINEAR DIFFERENTIAL EQUATIONS

  • Hu, Peichu (School of Mathematics Shandong University) ;
  • Liu, Manli (School of Mathematics Shandong University)
  • Received : 2019.07.23
  • Accepted : 2019.11.06
  • Published : 2020.07.31

Abstract

We show that when n > m, the following delay differential equation fn(z)f'(z) + p(z)(f(z + c) - f(z))m = r(z)eq(z) of rational coefficients p, r doesn't admit any transcendental entire solutions f(z) of finite order. Furthermore, we study the conditions of α1, α2 that ensure existence of transcendental meromorphic solutions of the equation fn(z) + fn-2(z)f'(z) + Pd(z, f) = p1(z)eα1(z) + p2(z)eα2(z). These results have improved some known theorems obtained most recently by other authors.

Keywords

Acknowledgement

The authors would like to thank the referees for a careful reading of the manuscript and valuable comments and to China Scholarship Council (State Scholarship Fund No. 201906220075) for its financial support.

References

  1. Z. Chen, Complex Differences and Difference Equations, Science Press, Beijing, 2013.
  2. F. Gross, Factorization of meromorphic functions, U. S. Goverment Printing Office, Washington D. C., 1972.
  3. W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  4. Y. He and X. Xiao, Algebroid Function and Ordinary Differential Equations, Science Press, Beijing, 1988.
  5. R. Korhonen, A new Clunie type theorem for difference polynomials, J. Difference Equ. Appl. 17 (2011), no. 3, 387-400. https://doi.org/10.1080/10236190902962244
  6. I. Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, 15, Walter de Gruyter & Co., Berlin, 1993. https://doi.org/10.1515/9783110863147
  7. I. Laine and C.-C. Yang, Clunie theorems for difference and q-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556-566. https://doi.org/10.1112/jlms/jdm073
  8. B. Q. Li, On certain non-linear differential equations in complex domains, Arch. Math. (Basel) 91 (2008), no. 4, 344-353. https://doi.org/10.1007/s00013-008-2648-2
  9. P. Li, Entire solutions of certain type of differential equations II, J. Math. Anal. Appl. 375 (2011), no. 1, 310-319. https://doi.org/10.1016/j.jmaa.2010.09.026
  10. P. Li and C.-C. Yang, On the nonexistence of entire solutions of certain type of nonlinear differential equations, J. Math. Anal. Appl. 320 (2006), no. 2, 827-835. https://doi.org/10.1016/j.jmaa.2005.07.066
  11. L.-W. Liao, C.-C. Yang, and J.-J. Zhang, On meromorphic solutions of certain type of non-linear differential equations, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 2, 581-593. https://doi.org/10.5186/aasfm.2013.3840
  12. H. Liu and Z. Mao, On entire solutions of some type of nonlinear difference equations, Acta Math. Sci. Ser. B (Engl. Ed.) 38 (2018), no. 3, 819-828. https://doi.org/10.1016/S0252-9602(18)30786-0
  13. X. Q. Lu, L. W. Liao, and J. Wang, On meromorphic solutions of a certain type of nonlinear differential equations, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 12, 1597-1608. https://doi.org/10.1007/s10114-017-6484-9
  14. X. Qi, J. Dou, and L. Yang, The properties of solutions of certain type of difference equations, Adv. Difference Equ. 2014 (2014), 256, 10 pp. https://doi.org/10.1186/1687-1847-2014-256
  15. C. Yang, On the entire solutions of certain class of non-linear differential equations, J. Math. Anal. Appl. 33 (1971), 644-649. https://doi.org/10.1016/0022-247X(71)90084-9
  16. C. Yang, On entire solutions of a certain type of nonlinear differential equation, Bull. Austral. Math. Soc. 64 (2001), no. 3, 377-380. https://doi.org/10.1017/S0004972700019845
  17. H. Yi and C. Yang, Theory of the Uniqueness of Meromorphic Functions, Science Press, Beijing, 1995.