Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EXPRESSIONS OF MEROMORPHIC SOLUTIONS OF A CERTAIN TYPE OF NONLINEAR COMPLEX DIFFERENTIAL EQUATIONS

  • Chen, Jun-Fan (College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applications Fujian Normal University) ;
  • Lian, Gui (Ningbo Economic and Trade School)
  • Received : 2019.08.07
  • Accepted : 2019.12.20
  • Published : 2020.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^n+Qd(z,f)=\sum\limits_{i=1}^{3}pi(z)e^{{\alpha}_i(z)}$$ are studied by using Nevanlinna theory, where n ≥ 5 is an integer, Qd(z, f) is a differential polynomial in f of degree d ≤ n - 4 with rational functions as its coefficients, p1(z), p2(z), p3(z) are non-vanishing rational functions, and α1(z), α2(z), α3(z) are nonconstant polynomials such that α'1(z), α'2(z), α'3(z) are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.

Keywords

References

  1. M. F. Chen and Z. S. Gao, Entire solutions of certain type of nonlinear differential equations and differential-difference equations, J. Comput. Anal. Appl. 24 (2018), no. 1, 137-147.
  2. J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17-27. https://doi.org/10.1112/jlms/s1-37.1.17
  3. X. Dong and L. Liao, Meromorphic solutions of certain type of differential-difference equations, J. Math. Anal. Appl. 470 (2019), no. 1, 327-339. https://doi.org/10.1016/j.jmaa.2018.10.005
  4. R. Halburd and R. Korhonen, Growth of meromorphic solutions of delay differential equations, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2513-2526. https://doi.org/10.1090/proc/13559
  5. W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  6. J. Heittokangas, R. Korhonen, and I. Laine, On meromorphic solutions of certain nonlinear differential equations, Bull. Austral. Math. Soc. 66 (2002), no. 2, 331-343. https://doi.org/10.1017/S000497270004017X
  7. 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
  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, J. Math. Anal. Appl. 344 (2008), no. 1, 253-259. https://doi.org/10.1016/j.jmaa.2008.02.064
  10. 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
  11. 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
  12. 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
  13. L. Mirsky, An Introduction to Linear Algebra, Oxford, at the Clarendon Press, 1955.
  14. N. Steinmetz, First-order algebraic differential equations of genus zero, Bull. Lond. Math. Soc. 49 (2017), no. 3, 391-404. https://doi.org/10.1112/blms.12035
  15. C.-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
  16. C.-C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 1, 10-14. http://projecteuclid.org/euclid.pja/1262271517 https://doi.org/10.3792/pjaa.86.10
  17. C.-C. Yang and P. Li, On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math. (Basel) 82 (2004), no. 5, 442-448. https://doi.org/10.1007/s00013-003-4796-8
  18. C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003. https://doi.org/10.1007/978-94-017-3626-8
  19. J. Zhang, Further results on transcendental meromorphic solutions of certain types of nonlinear differential equations, Acta Math. Sinica (Chin. Ser.) 61 (2018), no. 4, 529-540.
  20. J. Zhang and L. Liao, Admissible meromorphic solutions of algebraic differential equations, J. Math. Anal. Appl. 397 (2013), no. 1, 225-232. https://doi.org/10.1016/j.jmaa.2012.07.030