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

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.

Keywords

• meromorphic solution
• difference equation
• difference polynomial