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

Abstract

<abstract><p>In this paper, the representations of meromorphic solutions for three types of non-linear difference equations of form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f^{n}(z)+P_{d}(z, f) = u(z)e^{v(z)}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\lambda z}+p_{2}e^{-\lambda z} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z} $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>