Abstract
Let \(p_1, p_2\) be nonzero rational functions, \(\alpha _1, \alpha _2\) be nonzero constants, and \(P_d(z, f)\) be a difference polynomial in f of degree d. We prove that every finite order meromorphic solution of nonlinear difference equations \(f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}\) has few poles provided \(n\ge 2\) and \(d\le n-1\). This result shows there exists some difference between the existence of finite order meromorphic solutions of the above equations and its corresponding differential equations. We also give the forms of finite order meromorphic solutions of the above equations under some conditions on \(\alpha _1/\alpha _2\).
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References
Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)
Hayman, W.: Meromorphic Functions. Clarendon Press, Oxford (1964)
Halburd, R.G., Korhonen, R.: Finite order meromorphic solutions and the discrete Painlevé equation. Proc. Lond. Math. Soc. 94, 443–474 (2007)
Halburd, R.G., Korhonen, R.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006)
Halburd, R.G., Korhonen, R.: Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations. J. Phys. A. 40, 1–38 (2007)
Laine, I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin (1993)
Laine, I., Yang, C.C.: Entire solutions of some non-linear differential equations, Bull. De La Société Des Sciences Et Des Lettres De Lódí, LIX, pp. 19–23 (2009)
Li, P.: Entire solutions of certain type of differential equations. J. Math. Anal. Appl. 344, 253–259 (2008)
Li, P.: Entire solutions of certain type of differential equations II. J. Math. Anal. Appl. 375, 310–319 (2011)
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)
Liu, H.F., Mao, Z.Q.: Meromorphic solutions of certain types of non-linear differential equations. Comput. Methods Funct. Theory 1, 2 (2020). https://doi.org/10.1007/s40315-020-00313-0
Steinmetz, N.: Wertverteilung von Exponentialpolynomen. Manuscr. Math. 26, 155–167 (1978)
Yang, C.C.: A generalization of a theorem of P. Montel on entire functions. Proc. Amer. Math. Soc. 26, 332–334 (1970)
Yang, C.C., Li, P.: On the transcendental solutions of a certain type of nonlinear differential equations. Arch. Math. 82, 442–448 (2004)
Yang, C.C., Laine, I.: On analogies between nonlinear difference and differential equations. Proc. Jpn. Acad. Ser. A 86, 10–14 (2010)
Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers, New York (2003)
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This work is supported by the National Natural Science Foundation of China (No. 11661044).
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HL and ZM drafted the manuscript, read and approved the final manuscript.
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Liu, H., Mao, Z. Meromorphic Solutions of Certain Nonlinear Difference Equations. Results Math 76, 102 (2021). https://doi.org/10.1007/s00025-021-01414-5
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DOI: https://doi.org/10.1007/s00025-021-01414-5