Bulletin of the Korean Mathematical Society (대한수학회보)

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ON TRANSCENDENTAL MEROMORPHIC SOLUTIONS OF CERTAIN TYPES OF DIFFERENTIAL EQUATIONS

  • Received : 2021.08.27
  • Accepted : 2021.11.12
  • Published : 2022.09.30
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Abstract

In this paper, for a transcendental meromorphic function f and α ∈ ℂ, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: $$f^n+{\alpha}f^{n-2}f^{\prime}+P_d(z,f)={\sum\limits_{i=1}^{k}}{p_i(z)e^{{\alpha}_i(z)},$$ where Pd(z, f) is a differential polynomial of f, pi's and αi's are non-vanishing rational functions and non-constant polynomials, respectively. When α = 0, we have pointed out a major lacuna in a recent result of Xue [17] and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian [2] by rectifying a gap in the proof of the theorem of the same paper. The case α ≠ 0 has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.

Keywords

Acknowledgement

The authors wish to thank the referee for thorough review. Abhijit Banerjee is thankful to DST-PURSE II programme for financial assistance. Tania Biswas is thankful to the University Grant Commission (UGC), Govt. of India for financial support under UGC-Ref. No.: 1174/(CSIR-UGC NET DEC. 2017) dated 21/01/2019. Sayantan Maity wish to thank Council of Scientific and Industrial Research (CSIR, India) for providing fellowship under File No: 09/106(0191)/2019-EMR-I.

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.-F. Chen and G. Lian, Expressions of meromorphic solutions of a certain type of nonlinear complex differential equations, Bull. Korean Math. Soc. 57 (2020), no. 4, 1061-1073. https://doi.org/10.4134/BKMS.b190744
  3. 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
  4. W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  5. P. Hu and M. Liu, Existence of transcendental meromorphic solutions on some types of nonlinear differential equations, Bull. Korean Math. Soc. 57 (2020), no. 4, 991-1002. https://doi.org/10.4134/BKMS.b190696
  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. Z. Latreuch, On the existence of entire solutions of certain class of nonlinear difference equations, Mediterr. J. Math. 14 (2017), no. 3, Paper No. 115, 16 pp. https://doi.org/10.1007/s00009-017-0914-x
  8. 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
  9. 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
  10. L.-W. Liao, Non-linear differential equations and Hayman's theorem on differential polynomials, Complex Var. Elliptic Equ. 60 (2015), no. 6, 748-756. https://doi.org/10.1080/17476933.2014.972390
  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. L.-W. Liao and Z. Ye, On solutions to nonhomogeneous algebraic differential equations and their application, J. Aust. Math. Soc. 97 (2014), no. 3, 391-403. https://doi.org/10.1017/S1446788714000305
  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. F. Lu, W. Lu, C. Li, and J. Xu, Growth and uniqueness related to complex differential and difference equations, Results Math. 74 (2019), no. 1, Paper No. 30, 18 pp. https://doi.org/10.1007/s00025-018-0945-z
  15. L. Mirsky, An Introduction to Linear Algebra, Oxford, at the Clarendon Press, 1955.
  16. J. X. Rong and J. F. Xu, Three results on the nonlinear differential equations and differential-difference equations, Mathematics 7 (2019), no. 6, 539. https://doi.org/10.3390/math7060539
  17. B. Xue, Entire solutions of certain type of non-linear differential equations, Math. Slovaca 70 (2020), no. 1, 87-94. https://doi.org/10.1515/ms-2017-0334
  18. C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003.
  19. J. 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.