Meromorphic solutions of three certain types of non-linear difference equations

Min Feng Chen; Zhi Bo Huang; Zong Sheng Gao; Min Feng Chen; Zhi Bo Huang; Zong Sheng Gao

doi:10.3934/math.2021680

AIMS Mathematics

2021, Volume 6, Issue 11: 11708-11722. doi: 10.3934/math.2021680

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Research article

Meromorphic solutions of three certain types of non-linear difference equations

1.
School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou, 510006, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
3.
LMIB & School of Mathematical Sciences, Beihang University, Beijing, 100191, China

Received: 23 March 2021 Accepted: 05 August 2021 Published: 11 August 2021
MSC : 30D05, 34M10, 39A45, 39B32

In this paper, the representations of meromorphic solutions for three types of non-linear difference equations of form

$ f^{n}(z)+P_{d}(z, f) = u(z)e^{v(z)}, $

$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\lambda z}+p_{2}e^{-\lambda z} $

and

$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z} $

are investigated, where $ n\geq 2 $ is an integer, $ P_{d}(z, f) $ is a difference polynomial in $ f $ of degree $ d\leq n-1 $ with small coefficients, $ u(z) $ is a non-zero polynomial, $ v(z) $ is a non-constant polynomial, $ \lambda, p_{j}, \alpha_{j}\; (j = 1, 2) $ are non-zero constants. Some examples are also presented to show our results are best in certain sense.
- meromorphic solution,
- difference equation,
- difference polynomial
Citation: Min Feng Chen, Zhi Bo Huang, Zong Sheng Gao. Meromorphic solutions of three certain types of non-linear difference equations[J]. AIMS Mathematics, 2021, 6(11): 11708-11722. doi: 10.3934/math.2021680

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Abstract

In this paper, the representations of meromorphic solutions for three types of non-linear difference equations of form

$ f^{n}(z)+P_{d}(z, f) = u(z)e^{v(z)}, $

$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\lambda z}+p_{2}e^{-\lambda z} $

and

$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z} $

are investigated, where $ n\geq 2 $ is an integer, $ P_{d}(z, f) $ is a difference polynomial in $ f $ of degree $ d\leq n-1 $ with small coefficients, $ u(z) $ is a non-zero polynomial, $ v(z) $ is a non-constant polynomial, $ \lambda, p_{j}, \alpha_{j}\; (j = 1, 2) $ are non-zero constants. Some examples are also presented to show our results are best in certain sense.

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