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On analogies between nonlinear difference and differential equations. (English) Zbl 1207.34118
The authors study some similarities between results on the existence and uniqueness of finite order entire solutions of nonlinear differential equations and difference-differential equations of the form $$f^n + L(z, f) = h,$$ where $n \ge 2$ is an integer, $h$ is a given non-vanishing meromorphic function of finite order, and $L (z, f)$ is a linear difference-differential polynomials, with small meromorphic functions as coefficients. The authors prove for example the following theorem: Theorem 1. Let $p, q$ be polynomials. Then the nonlinear difference equation $$f^2 (z) + q (z) f (z +1)=p(z)$$ has no transcendental entire solutions of finite order.

34M05Entire and meromorphic solutions (ODE)
30D35Distribution of values (one complex variable); Nevanlinna theory
Full Text: DOI
[1] Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129. · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1
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[7] I. Laine, Nevanlinna theory and complex differential equations , de Gruyter, Berlin, 1993.
[8] I. Laine and C.-C. Yang, Clunie theorems for difference and $q$-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556-566. · Zbl 1132.30013 · doi:10.1112/jlms/jdm073
[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. · Zbl 1100.34066 · doi:10.1016/j.jmaa.2005.07.066
[10] C. Yang, On entire solutions of a certain type of nonlinear differential equation, Bull. Austral. Math. Soc. 64 (2001), no. 3, 377-380. · Zbl 0991.30019 · doi:10.1017/S0004972700019845
[11] 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. · Zbl 1052.34083 · doi:10.1007/s00013-003-4796-8
[12] C.-C. Yang and Z. Ye, Estimates of the proximate function of differential polynomials, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 4, 50-55. · Zbl 1122.30022 · doi:10.3792/pjaa.83.50 · euclid:pja/1177941417