Abstract
We consider transcendental meromorphic solutions with N(r, f) = S(r, f) of the following type of nonlinear differential equations:
where n ≥ 2 is an integer, Pn−2(f) is a differential polynomial in f of degree not greater than n−2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z) are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.
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References
Hayman, W. K.: Meromorphic Functions, Clarendon Press, Oxford, 1964
Laine, I.: Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin–New York, 1993
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)
Li, P., Yang, C. C.: On the nonexistence of entire solutions of certain type of nonlinear differential equations. J. Math. Anal. Appl., 320, 827–835 (2006)
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)
Yang, C. C., Li, P.: On the transcendental solutions of a certain type of nonlinear differential equations. Arch. Math. Basel, 82, 442–448 (2004)
Yang, C. C., Yi, H. X.: Uniqueness Theory of Meromorphic Functions, Science Press and Kluwer Academic Publishers, Bei**g, 2003
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We thank the referees for their time and comments.
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Sponsored by NNSF of China (Grant No. 11671191) and Natural Science Foundation of Shanghai (Grant No. 17ZR1402900)
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Lu, X.Q., Liao, L.W. & Wang, J. On meromorphic solutions of a certain type of nonlinear differential equations. Acta. Math. Sin.-English Ser. 33, 1597–1608 (2017). https://doi.org/10.1007/s10114-017-6484-9
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DOI: https://doi.org/10.1007/s10114-017-6484-9
Keywords
- Meromorphic solutions
- nonlinear differential equations
- small functions
- Nevanlinna’s value distribution theory