Meromorphic solutions to certain differential-difference equations
Keywords:
Nevanlinna theory, meromorphic solutions, nonlinear, differential difference equation
Abstract
The aim of this paper is to investigate the growth and constructions of meromorphic solutions of the nonlinear differential-difference equation
$$f^n(z)+h(z)\Delta_cf^{(k)}(z)=A_0(z)+A_1(z)e^{\alpha_1z^q}+\cdots+A_m(z)e^{\alpha_mz^q},$$
where $n, m, q\in \mathbb{N}^+$, $\alpha_1,\cdots,\alpha_m$ are distinct nonzero complex numbers, $h(z)$ is a nonzero entire function and $A_j(z)~(0\leq j\leq m)$ are meromorphic functions. In particular, for $A_0(z)\equiv 0$, we give the exact form of meromorphic solutions of the above equation under certain conditions. In addition, our results are shown to be sharp.
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References
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[20] C. C. Yang, I. Laine. On analogies between nonlinear difference and differential equations. Proc. Jpn.
Acad., Ser. A, 86:10-14, 2010.
[21] F. Zhang, N. N. Liu, W. L¨u, C. C. Yang. Entire solutions of certain class of differential-difference
equations. Adv. Differ. Equ., 150, 2015.
[2] Y. M. Chiang, S. J. Feng. On the Nevanlinna characteristic of f(z + η) and difference equations in the
complex plane. Ramanujan J, 16(1): 105-129, 2008.
[3] Z. X. Chen. Complex differences and difference equations. Science Press, Beijing, 2014.
[4] G. G. Gundersen, W. K. Hayman. The strength of Cartan’s version of Nevanlinna theory. Bull. Lond.
Math. Soc., 36(4):433-454, 2004.
[5] W. K. Hayman. Meromorphic functions. Oxford Mathematical Monographs, Clarendon Press, Oxford,
1964.
[6] R. G. Halburd, R. J. Korhonen. Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn.
Math., 31(2):463-478, 2006.
[7] I. Laine. Nevanlinna theory and complex differential equations. Walter de Gruyter. Berlin, 1993.
[8] P. Li, C. C. Yang. On the nonexistence of entire solutions of certain type of nonlinear differential
equations. J. Math. Anal. Appl., 320:827-835, 2006.
[9] P. Li. Entire solutions of certain type of differential equations. J. Math. Anal. Appl., 344:253-259, 2008.
[10] P. Li. Entire solutions of certain type of differential equations II. J. Math. Anal. Appl., 375:310-319,
2011.
[11] L. W. Liao, C. C. Yang, J. J. Zhang. On meromorphic solutions of certain type of non-linear differential
equations. Ann. Acad. Sci. Fenn., Math., 38:581-593, 2013.
[12] N. N. Liu, W. L¨u, T. Shen, C. C. Yang. Entire solutions of certain type of difference equations. J.
Inequal. Appl., 63, 2014.
[13] Z. Latreuch. On the existence of entire solutions of certain class of nonlinear difference equations.
Mediterr. J. Math., 14(3):115, 2017.
[14] X. Q. Lu, L. W. Liao, J. Wang. On meromorphic solutions of a certain type of nonlinear differential
equations. Acta Math. Sin., 33:1597-1608, 2017.
[15] X. M. Li, C. S. Hao, H. X. Yi. On the growth of meromorphic solutions of certain nonlinear difference
equations. Mediterr. J. Math., 18:56, 2021.
[16] X. M. Li, C. S. Hao, H. X. Yi. On the existence of meromorphic solutions of certain nonlinear difference
equations. Rocky Mountain J. Math., 51(5):1723-1748, 2021.
[17] Z. Q. Mao, H. F. Liu. On meromorphic solutions of nonlinear delay-differential equations. J. Math.
Anal. Appl., 509(1):125886, 2022.
[18] C. C. Yang, H. X. Yi. Uniqueness theory of meromorphic functions. Kluwer Academic Publishers. New
York, 2003.
[19] C. C. Yang, P. Li. On the transcendental solutions of a certain type of nonlinear differential equations.
Arch. Math., 82:442-448, 2004.
[20] C. C. Yang, I. Laine. On analogies between nonlinear difference and differential equations. Proc. Jpn.
Acad., Ser. A, 86:10-14, 2010.
[21] F. Zhang, N. N. Liu, W. L¨u, C. C. Yang. Entire solutions of certain class of differential-difference
equations. Adv. Differ. Equ., 150, 2015.
Published
2023-12-24
How to Cite
Li, Y., & Niu, W. (2023). Meromorphic solutions to certain differential-difference equations. Journal of Progressive Research in Mathematics, 20(1), 74-89. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2215
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